math 107 by example - massey.limfinity.com
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Math 107 by ExampleVersion 8.8
Kenneth MasseyCarson-Newman
Mathematics Department
Jefferson City, TN
Geico
These notes and examples are designed to accompany Professor Massey’s Math 107 course at Carson-Newman. This material, answer keys, and other information are available on the course website:
http://massey.limfinity.com/107/
Topic Outline
1. the real numbers R, number line, sets, intervals, inequalities
2. absolute value interpreted as distance
3. arithmetic (fractions, order of operations, etc)
4. algebraic properties and manipulation (comm/assoc/dist, etc)
5. variables, constants, expressions
6. solving equations
7. linear and absolute value inequalities
8. exponents laws, scientific notation, arithmetic/geometric means
9. word problems, setting up equations, D = ST
10. the Cartesian plane, right triangles, Pythagorean theorem, distance, midpoint
11. points, graphs of equations, intercepts
12. linear relationships, slope, eqn of line, parallel, perpendicular, proportional
13. intersection of lines and systems of linear equations, word problems, substitution
14. circles: equation, center, radius, diameter, area, circumference, intercepts, unit circle
15. functions: notation, evaluating, intercepts, VLT, sketching graphs, constructing/applications (P, D, R, C, Y )
16. function library, shifts and scaling
17. function properties: symmetry, domain, range
18. graph properties (inc/dec, local/globla max/min, concavity, inflection)
19. secant lines, AROC
20. polynomials (degree, algebra, roots, factoring, solving eqns)
21. quadratics: properties, forms, vertex, completing the square, quadratic formula
22. solving polynomial/algebraic equations
23. constructing quadratics with given properties
24. quadratic word problems (dimensions, economics, projectiles, DST)
25. circles: completing sq to put in standard form, top/bot half as functions, dom/ran
26. polynomial inequalities
27. rational functions: simplifying, common denom, solving eqns, asymptotes, graphing, word problems
28. exponential functions: solving eqns, graphs, constructing, applications
1 - Class Examples
Numbers, Sets
God created the integers, all else is the work of man.– Leopold Kronecker
• A rational number can be written as the ratio of two integers; its decimal representation willterminate or repeat.
• The set of real numbers (R) is a continuum containing all rational and irrational numbers.
• Each point on the number line corresponds to a real number, increasing in size from left toright.
• Infinity is not a number, but a direction that may be followed forever.We write ∞ for right, and −∞ for left.
1. Is it real (R) ? is it rational (Q) ? is it an integer (Z) ?
(a)√
169Answer: R,Q,Z since√
169 = 13 = 131
(b) −2.25Answer: R,Q since−2.25 = −9
4
(c) 1.6Answer: R,Q since 1.6 =53
(d)√
2Answer: R
(e)√−9
Answer: not real
(f) 0Answer: R,Q,Z since0 = 0
1
(g) 10Answer: not defined
(h) 3.14Answer: R,Q since3.14 = 314
100 = 15750
(i) πAnswer: R
(j) π2Answer: R
(k)√
2√
8Answer: R,Q,Z since√
2√
8 =√
16 = 4
(l) 2√3−
√123
Answer: R,Q,Z since2√3−
√123 = 0
2 - Class Examples
A set of numbers can be described in several ways:
• a verbal description
• picture (shaded on the number line)
• list the members in braces (for finite sets only)
• intervals (closed if it includes the endpoint, otherwise open)
• inequalities (use a generic variable, commonly x or t)
If an object x is a member of a set S, we may write x ∈ S.Sets may be joined together using the union symbol ∪.
2. Write each set mathematically, and sketch it on the number line.
(a) The set of prime numbers less than 25.Answer: S = {2, 3, 5, 7, 11, 13, 17, 19, 23}
(b) The open interval of R between two and five.Answer: (2, 5)2 < x < 5
(c) The closed interval of R between two and five.Answer: [2, 5]2 ≤ x ≤ 5
(d) All non-negative real numbers less than 50.Answer: [0, 50)0 ≤ x < 50
(e) All R greater than 3.Answer: (3,∞)
3 - Class Examples
x > 3
(f) All R less than or equal to 7.Answer: (−∞, 7]x ≤ 7
(g) All R greater than 3, and less than or equal to 7.Answer: (3, 7]3 < x ≤ 7
(h) All R greater than 7, or less than or equal to 3.Answer: (−∞, 3] ∪ (7,∞)x ≤ 3 or x > 7
(i) All R except zero and one.Answer: x 6= 0, 1 (that x is a real number is understood from context)
3. A restaurant gives discounts to children under 12 and senior citizens over 60. Let S be an open setthat describes these ages.
(a) Sketch S on the number line
(b) Describe S with interval notation.Answer: (0, 12) ∪ (60,∞)
(c) Describe S with inequalities.Answer: 0 < x < 12 or x > 60
4 - Class Examples
(d) True or false? 60 ∈ SAnswer: false
Absolute Value
The distance between x and y is|x − y|
Absolute values are often used to set boundaries around a given center point.Look for keywords, like “within” or “at least.”
4. Find the distance between −5 and 2.Answer: | −5 − 2| = |2 −− 5| = 7
5. If |x − 4| = 3, then what could x be?Answer: The distance between x and 4 is three, so x could be 1 or 7.
6. Solve the equation |x + 3| = 5.Answer: Use the “double-negative trick” to write |x−− 3| = 5. This says the distance between x and−3 is 5. Therefore x = 2 or x = −8. Check your answers by plugging back in the original equation.
7. Solve |x| + 3 = 5.Answer: This says |x − 0| = 2, so x = ±2
5 - Class Examples
8. Consider the set of all solutions to |x − 7| ≤ 5.
(a) Say it in words.Answer: “The distance between x and 7 is less than or equal to (ltoet) 2”Think of this as a dog on a leash 5 feet long, tied down at 7.
(b) Draw it on the number line.
(c) Describe it with interval notation.Answer: [2, 12]
9. Consider the set of all solutions to |x − 3| > 1.
(a) Say it in words.Answer: “The distance between x and 3 is greater than 1”Think of this as a restraining order; x must stay one unit away from 3.
(b) Draw it on the number line.
(c) Describe it with interval notation.Answer: (−∞, 2) ∪ (4,∞)
10. Let S be the closed set of real numbers between −1 and 7.
(a) Picture S on the number line.
(b) Write S using interval notation.Answer: [−1, 7]
(c) Describe S with a regular inequality.Answer: −1 ≤ x ≤ 7
6 - Class Examples
(d) Describe S with an absolute value inequality.Answer: |x − 3| ≤ 4
11. Let S be the set (−∞, 3) ∪ (8,∞). Write S using an absolute value inequality.Answer: Draw the set first and see that it has two pieces, symmetric around 11
2 . Therefore S canbe described by |x − 11
2 | > 52 , or equivalently |2x − 11| > 5.
12. A potato chip bag claims to hold 14 ounces. The packaging plant manager wants the actual amount tobe within one-half ounce of the stated weight. Express this condition using an absolute value inequality.Answer: |W − 14| ≤ 0.5
13. You are programming a guided missile to hit a terrorist hideout positioned 5000 meters east of thelaunching point. There is a hospital located at 4800 meters, and there is a school at 5300 meters.Suppose you require a 50 meter margin of safety to avoid innocent civilians. Write an absolute valueinequality to describe the acceptable impact points for your missile, given that it must be centered atthe target.Answer: The interval is (4850, 5150), which can be described as |x − 5000| < 150
Arithmetic, Algebra, Expressions
To speak algebraically, Mr. M. is execrable, but Mr. G. is (x + 1) ecrable.– Edgar Allan Poe
7 - Class Examples
14. Compute .25((−2)2−5/12)
.3and write the answer as a reduced fraction (PEMDAS).
Answer:
.25((−2)2 − 5/12)
.3=
14 (4 − 5/12)
1/3=
14 (48/12− 5/12)
1/3=
14 (43/12)
1/3=
43/48
1/3=
43
48· 3
1=
43
16
15. What are the two operations on R, and what are their inverses?Answer: addition (inverse is subtraction) and multiplication (inverse is division).By definition 2 + −2 = 0 and 1
2 · 2 = 1. Also note that 32 = 1
2 · 3 = 2−1 · 3.
Algebra is the study of how members of a set interact with each other.
(a) It’s like a game where there are
• players/actors: e.g. the real numbers R
• actions: arithmetic operations, exponents
• rules of engagement:commutative 2 · 3 = 3 · 2associative (2 · 3) · 5 = 2 · (3 · 5)distributive 2(3 + 5) = 2 · 3 + 2 · 5
• specialists: additive and multiplicative identities, 0 and 1
• deductive facts: e.g. xy = 0 imples x = 0 or y = 0
(b) Constants are numbers or symbols that have a fixed value (e.g. π, or c = 186000 mps)
(c) Use variables to work with numbers generically, or to express relationships between un-known or changing quantities.
(d) An expression is an algebraic quantity, usually involving one or more variables.
16. Multiply out the expression, and combine like terms.
(a) (x + 2)(x − 2)Answer: the distributive property, commonly referred to as FOIL gives us x2+2x−2x−4 = x2−4
8 - Class Examples
(b) (x + 1)3
Answer:
(x + 1)3 = (x + 1)[(x + 1)(x + 1)]
= (x + 1)[x2 + x + x + 1]
= (x + 1)(x2 + 2x + 1)
= x3 + 2x2 + x + x2 + 2x + 1
= x3 + 3x3 + 3x + 1
(c) (2x − 3)2 − 7(1 − x)Answer:
(2x − 3)2 − 7(1 − x) = (2x − 3)(2x − 3) − 7 + 7x
= 4x2 − 6x − 6x + 9 − 7 + 7x
= 4x2 − 5x + 2
17. Which of these are generally true?
(a) 5 − x = −(x − 5)Answer: true, “factor out a negative trick” to reverse the subtraction
9 - Class Examples
(b) 2x2
8x = .25xAnswer: true, if x 6= 0
(c) (x + y)2 = x2 + y2
Answer: false
(d) 0x = 0Answer: true, except when x = 0
(e) x−y = −x
yAnswer: true
(f) 1x + 1
1 = 2x+1 ,
Answer: false, it should be 1+xx
(g) 2(x−1)+1x−1 = 3
Answer: false
(h) (2x − 3)2 = (3 − 2x)2
Answer: true
(i) 2 · x+1x = 2x+1
xAnswer: false
(j) 3x · x+1
6 = 12x (x + 1)
Answer: true
(k) 1−xx−1 = −1Answer: true
(l) If (x − 2)(5x + 6) = 1 then x − 2 = 1 or 5x + 6 = 1.Answer: false
(m) 1x−1 + 1
1−x = 0Answer: true
10 - Class Examples
Equations
Politics are only a matter of present concern. A mathematical equation stands forever.– Albert Einstein
• An equation expresses a relationship, analogous to a sentence where = is the verb.
• A solution to an equation makes the two sides equal. Check by plugging-in.
• An equation could have zero, one, or many solutions.
• Equality is maintained if you do the same invertible operation to each side.
18. Fill in the blank: 3 + = 7.Answer: Algebraically we write 3 + x = 7. Subtract 3 from both sides to get x = 4.
19. Solve the equation: 7x − 2 = 3(x + 2).Answer: First get rid of parentheses to get 7x − 2 = 3x + 6.Then move all the x’s to one side, and the constants to the other side to get 4x = 8.Therefore x = 2, which you can check in the original equation.
20. Solve the equation: x−143 = 7 + 5x.
Answer: Unravel this equation to isolate the variable x. You must do the same operation to bothsides of the equal sign to maintain equality.
x − 14 = 3(7 + 5x)
x − 14 = 21 + 15x
−14x = 35
x =35
−14=
−5
2
21. Which are solutions to x(x − 3) = x + 21?
11 - Class Examples
(a) −3 (b) 0 (c) 1 (d) 3 (e) 7
Answer: plug in to see that x = −3 and x = 7 are both solutions
22. Which are solutions to 2x3 + 5x2 = 4x + 3 ?
(a) 1 (b) 0 (c) −3 (d) −1/2 (e) 2
Answer: x = 1,−3,−1/2
23. Solve the equation, listing all real solutions.
(a) 25x = 1 − x
3Answer: To get rid of the fractions, use the old “multiply through by the denominator” trick.
15
(
2
5x = 1 − x
3
)
6x = 15 − 5x
11x = 15
x =15
11
12 - Class Examples
(b) 2x+1 = 1
3x−8
Answer: cross-multiply to get 2(3x − 8) = x + 1, so 6x − 16 = x + 1, so 5x = 17, and x = 175
(c) x2 = 12Answer: x = ±
√12 = ±2
√3
(d) x2 + 9 = 5Answer: x2 = −4 is impossible, so there is no real solution
(e) (2x − 5)2 = 9Answer: 2x − 5 = 3, so 2x = 8 and x = 4or 2x − 5 = −3, so 2x = 2, and x = 1
(f) |x − 13| = 7Answer: x − 13 = ±7, so x = 13 ± 7 = 6, 20
(g) |x − 5| + 1 = 0Answer: Equivalently |x − 5| = −1, which is impossible since distance can’t be negative, sothere is no solution.
(h) −2x(x2 − 9)(3x + 2) = 0Answer: When multiplied terms equals zero, set each term to zero separately.−2x = 0, so x = 0x2 − 9 = 0, so x = ±3
13 - Class Examples
3x + 2 = 0, so x = −2/3
(i) |2 + 5x| = 3Answer: 2 + 5x = 3, so 5x = 1 and x = 1
5or 2 + 5x = −3, so 5x = −5, and x = −1
(j) |x − 8| = |x − 2|Answer: x is equidistant between 8 and 2, so x = 5
(k) x2 = 5xAnswer: If you divide both sides by x, you get x = 5, which is a solution, but it’s not the onlyone. When you divide by an expression, you must assume it’s not zero. However, in this case zerois a solution, so the entire solution set is x = 0, 5.
24. Suppose 5y − x = 10.
(a) Solve for y.Answer: 5y = x + 10, so y = x+10
5
(b) Solve for x.Answer: x = 5y − 10
(c) What is y when x = 5?Answer: y = 3
14 - Class Examples
(d) What is x when y = −1?Answer: x = −15
25. Suppose y = xx+1 .
(a) Solve the equation for x.Answer: cross-multiply to get
y(x + 1) = x
yx + y = x
yx − x = −y
x(y − 1) = −y
x =−y
y − 1=
y
1 − y
(b) If y = 3, what is x?Answer: 3
1−3 = −3/2
Inequalities
Inequality, not mediocrity, individual superiority, not standardization,is the measure of the progress of the world.
– Felix Schelling
The solution set to an inequality may written using interval notation. Focus on these types:
• Linear inequalities may be solved by isolating x with basic algebra.Remember to reverse the inequality if you multiply/divide by a negative.
• Absolute value inequalities may be solved by finding the two boundaries, and picturingeither a leash (≤) or restraining order (≥).
26. Solve the inequality, writing your answer with interval notation.
15 - Class Examples
(a) −2(x + 1) < 6Answer: When you multiply/divide through by a negative, don’t forget to flip the inequality(for example 3 < 5 but −3 >− 5).x + 1 > −3, so x > −4, which we write as (−4,∞)
(b) −4 < 3x − 7 ≤ 5Answer: 3 < 3x ≤ 12, so 1 < x ≤ 4, which we write as (1, 4]
(c) |x − 5| ≤ 2Answer: The distance between x and 5 is LTOET 2, so the answer is [3, 7].
(d) |x − 5| ≥ 2Answer: The distance between x and 5 is GTOET 2, so the answer is (−∞, 3] ∪ [7,∞).
(e) |x + 5| < 2Answer: The distance between x and −5 is LT 2, so the answer is (−7,−3).
(f) |2x − 7| > 11Answer: This is a restraining order, so it must have two parts.If 2x − 7 is positive, then 2x − 7 > 11, so 2x > 18 and x > 9.But if 2x − 7 is negative, then 2x − 7 < −11, so 2x < −4 and x < −2.
16 - Class Examples
Therefore the solution is (−∞,−2) ∪ (9,∞)
(g) |2x − 7| < 11Answer: This is a leash, so the solution is (−2, 9).
(h) |3x + 5| ≤ 16Answer: Solve the equality: 3x + 5 = ±16, so 3x = −5 ± 16 = −21, 11, and x = −7, 11
3 . Thisis a leash, so the answer is [−7, 11
3 ].
(i) |x| + 1 = 0Answer: |x| = −1 has no solution
Exponents
Exponent notation is shorthand for repeated multiplication, e.g.
x · x · x · x = x4
Here the base is x, and the exponent is 4.Exponentiation is to multiplication what multiplication is to addition. These rules follow:
(a) xmxn = xm+n
(b) (xm)n = xmn
(c) x0 = 1
(d) x−1 = 1x
(e) xm
xn = xmx−n = xm−n
(f) x1/n = n√
x
(g) xm/n = (x1/n)m = ( n√
x)m
(h) (xy)m = xmym
17 - Class Examples
27. Evaluate:
(a) (−27)1/3
Answer: −3
(b) (12 )−3
Answer: 8
(c) 45−2
Answer: 100
(d) 4−1.5.Answer:
4−1.5 = 4−3/2 =(
41/2)−3
=(√
4)−3
= 2−3 =1
23=
1
8= 0.125
28. True or false?
(a) 10 + 01 =√
1Answer: true
(b) (−2x)4 = 16x4
Answer: true
(c) (x2 + 4)1/2 = x + 2Answer: false
(d) 25−2 = 251
2 = 5Answer: false
(e) (4x2)1/2 = 2xAnswer: true
(f) −x4 = (−x)4
Answer: true
(g) −x3 = (−x)3
Answer: false
(h)√
18x5 = 3x2√
2xAnswer: true
29. Simplify the expression, combine exponents, and rewrite without roots, parentheses, or fractions
18 - Class Examples
(a) y(7xy3)2
x−4x7
Answer: 49y7x−1
(b) x−1/3√
xAnswer: x−1/3x1/2 = x1/6
(c)(
x2y−1
y3
)1/2
Answer: (x2y−4)1/2 = xy−2
(d)(
2x2t−1
t−4x
)3
Answer: (2xt3)3 = 8x3t9
(e)
(
3√
x2y6
(2y)−3x1/6
)2
Answer:(
3
√
x2y6
(2y)−3x1/6
)2
=
(
x2/3y2(2y)3
x1/6
)2
=(
8x1/2y5)2
= 64xy10
(f) x3−2√
xx
19 - Class Examples
Answer: x2 − 2x−1/2
30. Rationalize the number√
512 .
Answer: There is no reason to do this except to match an answer, but anyway:
√
5
12=
√5√12
√12√12
=
√60
12=
2√
15
12=
√15
6
31. Write the number in scientific notation.
(a) Bill Gates is worth 61 billion dollars.Answer: 6.1 × 1010
(b) The radius of a hydrogen atom is 0.0000000000529.Answer: 5.29 × 10−11
32. Compute (6×108)(8×10−2)4×104 .
Answer: 6·84 × 108−2−4 = 12 × 102 = 1.2 × 103
33. Estimate how many calories Americans consume each year, and write the result in scientific notation.Answer: There are approximately 300 million Americans, say each consumes 2000 calories per dayfor 365 days.
300, 000, 000× 2, 000× 365 = 219, 000, 000, 000, 000 = 2.19 × 1014
20 - Class Examples
or 219 trillion calories.
34. Consider this list of numbers: {2, 4, 8}.
(a) Find the arithmetic mean.Answer: This is commonly called the “average.” We add up the numbers and divide by three:
ma =2 + 4 + 8
3=
14
3= 4.6
(b) Find the geometric mean.Answer: Instead of adding and dividing, we multiply and take the third root:
mg = (2 · 4 · 8)1/3 = 641/3 = 4
35. Consider this list of numbers: {1, 2, 3, 18, 72}.
(a) Find the arithmetic mean.Answer:
ma =1 + 2 + 3 + 18 + 72
5= 19.2
(b) Find the geometric mean.Answer:
mg = (1 · 2 · 3 · 18 · 72)1/5 = (6 · 6 · 3 · 6 · 6 · 2)1/5 = 6
21 - Class Examples
36. (calculator required) Consider two retirement funds, with annual gain/loss percentages listed:
year 1 2 3 4A 10 5 12 13B 20 50 0 -30
Compute the arithmetic and geometric means of the yearly multiplication factors that describe thegrowth of the account’s value.
Answer: Here are the factors:
year 1 2 3 4A 1.10 1.05 1.12 1.13B 1.20 1.50 1 0.70
For fund A, the arithemetic and geometric means are:
ma = (1.1 + 1.05 + 1.12 + 1.13)/4 = 1.1 mg = (1.1 · 1.05 · 1.12 · 1.13)1/4 = 1.0996
For fund B, the arithemetic and geometric means are:
ma = (1.2 + 1.5 + 1 + 0.7)/4 = 1.1 mg = (1.2 · 1.5 · 1 · 0.7)1/4 = 1.0595
22 - Class Examples
Word Problems
To solve word problems, keep these general procedures in mind:
• If appropriate, sketch a picture or make a table.
• Label quantities and variables.
• Decide which variable corresponds to the question being asked.
• Set up an equation.
• Many problems aren’t cookie-cutter, so use your reason and think.
37. The center square in Australian rules football has a perimeter of 200 meters. What is its area?Answer: Draw a picture. Let x be the length of a side. Then 4x = 200, so x = 50 m, therefore thearea is x2 = 2500m2
38. In 12 years, Jack will be twice as old as he was 10 years ago. How is he now?Answer: Let x be Jack’s current age. Then x + 12 = 2(x − 10). Solve to get x + 12 = 2x − 20, sox = 32.
39. You and I each have $30. How much must you give me so that I will have 50% more than you?Answer: Let x be the amount you give me. Then 30+x = 1.5(30−x). Solve to get 30+x = 45−1.5x,so 2.5x = 15, and x = 6.
23 - Class Examples
40. Poindexter got an 80 on his first test and a 97 on his second test. What grade must he get on the thirdof three equally weighted tests to maintain his 4.0 GPA?Answer: Let x be his third test grade. He needs to have a 90 average to get an A in the class, so
80 + 97 + x
3= 90
177 + x = 270
Which means x = 93.
41. After driving 210 miles of a 420 mile trip, you have averaged 60 MPH. How fast must you go the restof the way to average 70 MPH for the entire trip?Answer: We will use the formula D = ST repeatedly. Consider the trip components separately asin this table:
1st part 2nd part full tripD 210 210 420S 60 70T
Fill it in piece-by-piece to get
1st part 2nd part full tripD 210 210 420S 60 84 70T 3.5 2.5 6
So you must average 84 MPH the rest of the way.
42. A car gets 30 MPG highway and 20 MPG city. If you drove 320 miles on 12 gallons, how many ofthose were city miles?Answer: Let x be the number of city miles. Then 320 − x is the number of highway miles. In thecity, the car burned x
20 gallons, and on the highway it burned 320−x30 . This gives us the equation:
x
20+
320 − x
30= 12
24 - Class Examples
Solving shows that x = 80 miles.
43. A retired football player needs to lose weight, so he begins the John Basedow diet and exercise program.
If he exercises M minutes per day, his weight after t months will be W = 250 − M√
t4 .
(a) How much did he weigh at the start?Answer: Plug in t = 0 to get W = 250.
(b) How many minutes per day must he exercise to lose 10 pounds in the first month?Answer: We know t = 1, and that W = 240, so 240 = 250 − M
√
1/4. Solving for M , we get10 = M
2 , so M = 20 minutes.
(c) Continuing that, how long will it take to reach 200 pounds?Answer: We know W = 200 and M = 20, so 200 = 250 − 20
√
t/4. Solving for t, we get
50 = 20√
t2 , so 5 =
√t and t = 25 months.
44. Working alone, Bert takes 10 minutes to do the chores. Ernie takes 30 minutes. How long should ittake them working together?Answer: In one minute, Bert does 1/10 and Ernie does 1/30 of the work. Therefore solve 1/10 +
25 - Class Examples
1/30 = 1/t to get t = 7.5.
45. Suppose that I run one half mile, then get tired and walk another half mile at 3 miles per hour. If ittakes 13 minutes to finish the full mile, how fast did I run?Answer: Remember the formula that D = ST . Start by organizing the information you know.
run walk full mileD .5 .5 1S 3T 13
We can figure the time walking by T = D/S = .5/3 = 1/6 hour, or 10 minutes. That means that ittook 13−10 = 3 minutes to run. Converting this to hours, the running time is 1/20 hour, so the speedwas S = D/T = .5/(1/20) = 10 MPH.
Cartesian Plane
Cogito, ergo sum.– Rene Descartes
The Cartesian plane is a two dimensional coordinate system that helps us visualize the rela-tionship between two variables. Lines, circles, and other geometric objects may be anchored tothe coordinate plane to make problem solving easier.
46. Draw the Cartesian plane; label the axes, origin, and quadrants.Plot the point (5,−2) and explain the coordinates of the ordered pair.
26 - Class Examples
Answer: By convention, the first number in the ordered pair is position along the horizontal (x)axis, and the second number is the position along the vertical (y) axis.
47. Find the distance from the origin to P (4, 0), then from the origin to Q(4, 3).Answer: The first distance is obviously 4.For the second point, we must move diagonally (the shortest distance between two points is a straightline). Use the Pythagorean Theorem to get c2 = 42 + 32, so c = 5.
The Pythagorean Theorem (PT) for the sides of a right triangle, where c is the hypotenuse:
c2 = a2 + b2
Given two points (x1, y1) and (x2, y2) in the plane, the distance between them is√
(x2 − x1)2 + (y2 − y1)2
and the midpoint is(
x1 + x2
2,y1 + y2
2
)
48. Find the distance between P (−3, 1) and Q(9, 6).Answer: Complete a right triangle and use the PT. The two legs are | − 3− 9| = 12 and |1− 6| = 5,so the hypotenuse is:
dist(P, Q) =√
122 + 52 =√
169 = 13
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49. Find the midpoint between P (−3, 1) and R(5, 9).Answer: Just average the x and y coordinates to get the midpoint:
(−3 + 5
2,1 + 9
2
)
= (1, 5)
50. Plot the points P (3, 0), Q(−1, 4), and R(9, 2). Which is closer to P , Q or R?Answer: First calculate
dist(P, Q) =√
(3 −− 1)2 + (0 − 4)2 =√
32 = 4√
2
dist(P, R) =√
(3 − 9)2 + (0 − 2)2 =√
40 = 2√
10
Therefore Q is closer to P than R is.
51. Find the distance from the origin to the midpoint of (3, 6) and (−13, 14).Answer: First find the midpoint:
(
3 + −13
2,6 + 14
2
)
= (−5, 10)
Now the distance to the origin is
√
(−5 − 0)2 + (10 − 0)2 =√
125
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Note that this is slightly bigger than 11.
52. Prove that P (6,−7), Q(11,−3), and R(2,−2) form an isosceles right triangle. Find the area.Answer: The three sides have lengths:
dist(P, Q) =√
52 + 42 =√
41
dist(Q, R) =√
92 + 12 =√
82
dist(R, P ) =√
42 + 52 =√
41
Since two sides are the same, the triangle is isoscles. It is a right triangle if the sides satisfy thePythagorean theorem: √
412+√
412
=√
812
41 + 41 = 81
which it does! Finally, the area is
A =1
2bh =
1
2
√41√
41 =41
2= 20.5
53. If dist((x, 9), (2, 1)) = 10, what is x?Answer: This says the distance between (x, 9) and (2, 1) is 10. Drawing a picture indicates that weshould expect two solutions. Algebraically, we can solve the equation:
√
(x − 2)2 + (9 − 1)2 = 10
√
(x − 2)2 + 64 = 10
(x − 2)2 + 64 = 100
(x − 2)2 = 36
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x − 2 = ±6
x = 2 ± 6
x = 8,−4
54. Suppose you are at a height of x miles above the surface of a sphere with radius R miles.
(a) Show that the horizon is√
x2 + 2xR miles away.Answer: Draw a diagram and notice the right triangle with hypotenuse x + R and one leg R.The distance to the horizon is the other leg, which we can get from the Pythagorean theorem:
b2 = c2 − a2
b2 = (x + R)2 − R2
b2 = (x2 + 2xR + R2) − R2
b2 = x2 + 2xR
b =√
x2 + 2xR
(b) The Lighthouse of Alexandria (one of the 7 ancient wonders of the world) is said to have beenabout 440 feet high and visible from 30 miles away. Show that this must be slightly exaggerated,unless it was built up on a hill. (the radius of the Earth is about 4000 miles)Answer: Convert 440 feet into miles:
x = 440 feet · 1 mile
5280 feet=
1
12mile
Now plug into the formula from earlier:
b =√
(1/12)2 + 2(1/12)(4000) =
√
1
144+
4000
6≈ 25.82 miles
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55. Explain how absolute value and the distance formula are related.Answer: Absolute value is distance in one dimension. The distance formula gives distance in twodimensions. In fact, the formulas are basically the same, since
dist(x1, x2) = |x1 − x2| =√
(x1 − x2)2
Lines
I see you’re drinking one-percent. Is that ’cause you think you’re fat?’Cause you’re not. You could be drinking whole if you wanted to.
– Napoleon Dynamite
• The graph of an equation is the set of all points that make the equality true.
• To find a graph’s y-intercept(s) (where it crosses the y-axis), set x = 0.
• To find a graph’s x-intercept(s) (where it crosses the x-axis), set y = 0.
56. Find the intercepts, and sketch the graph of 2x + 3y = 12.Answer: To find the y-intercept, set x = 0 to get 3y = 12, so y = 4 and the y-int is (0, 4).To find the x-intercept, set y = 0 to get 2x = 12, so x = 6 and the x-int is (6, 0).We can find many other points by picking whatever we want for x, and then solving for y. For example,if x = 3 then solve 6+3y = 12 to get y = 2. Plot several such points to see that the graph is a straight
31 - Class Examples
line.
57. Find the intercepts of x2 = y + 4.Answer: Setting x = 0, we get the y-int to be (0,−4).Setting y = 0, we obtain x2 = 4, which has two solutions. There are two x-intercepts: (±2, 0).Clearly, this graph is not a straight line.
The slope between two points (x1, y1) and (x2, y2) is:
m =rise
run=
∆y
∆x=
y2 − y1
x2 − x1
58. Find the slope between (−1, 3) and (5, 7).
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Answer: m = 7−35−−1 = 4
6 = 23
A line is a graph for which the slope between any two points is the same.
• A line can be determined by any two distinct points in the plane.
• Successive points with fixed x increments will also have a constant change in y values.
• If the equation of a line is solved for y, then the slope is the coefficient of x.
• The study of lines is crucial, since if you zoom in enough, all smooth curves look linear.
59. Sketch the line between (−3, 6) and (7, 1), and find its slope.Answer: Plot the points and connect them. The slope is m = −5
10 = −12 .
60. Consider the line 6x − 2y = 18.
(a) Find the x-intercept.Answer: Set y = 0 to get 6x = 18, so x = 3 and the x-int is (3, 0).
(b) Find the y-intercept.Answer: Set x = 0 to get −2y = 18, so y = −9 and the y-int is (0,−9).
(c) Does the line go through (4, 3)?Answer: Plug in x = 4 and y = 3 to see if the equation holds:
6(4) − 2(3) = 24 − 6 = 18
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yes!
(d) Solve the equation for y.Answer: −2y = 18 − 6x, so y = −9 + 3x
(e) Find the slope.Answer: m = 3
61. Suppose 2x − 3(y + 1) = 10x + 3.
(a) Find the slope.Answer: Solve for y to get the slope:
2x − 3y − 3 = 10x + 3
−3y = 8x + 6
y =8x + 6
−3
Therefore m = −83 .
(b) Find the x-int.Answer: Set y = 0 to get 0 = 8x+6
−3 , so 8x + 6 = 0 and x = −3/4.
(c) Find the y-int.
Answer: Set x = 0 to get y = 8(0)+6−3 = −2.
62. Find the slope of each line. Does it go up or down? Which is the steepest?
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(a) y = 1 − 5x13
Answer: m = −513 , down
(b) y = 3 + 0.75(2x + 1)Answer: m = 1.5, up, steepest
(c) x = 3y − 8Answer: m = 1
3 , up
To find the equation of a line, first find
• any point (x1, y1)
• the slope m
The equation of the line isy = y1 + m(x − x1)
• A horizontal line has m = 0, so the formula reduces to y = y1.
• A vertical line has undefined (infinite) slope; the equation may be writen x = x1.
63. Find the equation of the line:
(a) that goes through (7, 3) with slope 12 .
Answer: y = 3 + 12 (x − 7)
(b) that goes though (1, 8) and (3, 2).Answer: m = 2−8
3−1 = −3, so y = 8 − 3(x − 1)
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Or you could use the other point and get y = 2−3(x−3). Either way, it simplifies to y = −3x+11.
(c) has slope 2 and y-intercept 5Answer: The known point is (0, 5), so we obtain y = 5 + 2(x − 0).
(d) has slope 2 and x-intercept 5Answer: The known point is (5, 0), so we obtain y = 0 + 2(x − 5).
(e) has x-intercept 3 and y-intercept 2.Answer: the points are (3, 0) and (0, 2), so m = 2
−3 , and we get
y = 5 − 2
3(x − 0)
64. Water freezes at 32 degrees fahrenheit, or 0 degrees celcius. It boils at 212◦ F, or 100◦ C.
(a) Find the linear conversion formula to convert celcius to fahrenheit.Answer: Use the points (0, 32) and (100, 212) to get m = 180
100 = 95 . Then
F = 32 +9
5C
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(b) What is the fahrenheit equivalent of 10◦ C?Answer: F = 32 + 9
5 (10) = 50
(c) What is the celcius equivalent of 77◦ F?Answer: plug in F = 77 to get 77 = 32 + 9
5C, so 45 = 95C and C = 25.
65. Find the equation of the line that goes through (2, 5) and (2,−1).Answer: The slope is undefined (or infinite)
m =−1 − 5
2 − 2=
−6
0
so we have encountered a vertical line. All points on this line have x-coordinate equal to 2, so the linemay be described by the simple equation:
x = 2
66. Find the equation of the horizontal line that goes through (2, 8).Answer: Since m = 0, we get y = 8 + 0(x − 2), or simply y = 8.
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67. Find the equation of the line, and the distance between P (−1, 5) and Q(3,−7).Answer: First find the slope m = −7−5
3−−1 = −3. Use either point, say we use P :
y = 5 +− 3(x −− 1)
If we used Q instead:
y = −7 +− 3(x − 3)
In either case, it boils down to y = −3x + 2. The distance between P and Q is√
42 + 122 =√
160.
68. A line goes through (−2, 5) and has a y-intercept of 1. Use those clues to find the x-intercept.Answer: We know two points (−2, 5) and (0, 1), so the slope is m = 5−1
−2−0 = −2. Therefore theequation of the line is
y = 5 − 2(x + 2)
To get the x-int, plug in y = 0 and solve for x.
0 = 5 − 2(x + 2)
2(x + 2) = 5
x + 2 =5
2
x =1
2
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Two lines are
• parallel (‖) if they have the same slope
• perpendicular (⊥) if they intersect at a right angle; their slopes are negative reciprocals
69. Sketch these parallel lines:
y = 3 − 2x y = −2(x + 4) + 7
Answer: They both have slope −2; with y-intercepts at (0, 3) and (0,−1) respectively.
70. Are these lines perpendicular?
6y − 4x = 12 y =1 − 3x
2+ 5
Answer: Solve the first equation for y to get y = 12+4x6 , so the slope is 4
6 = 23 . The slope of the
39 - Class Examples
other line is −32 . Since the slopes are negative reciprocals, they are ⊥.
71. Consider the line that goes through (−2, 5) and has slope m = 13 .
(a) Find the equation of this line.Answer: Using the point-slope form, we can just write down the answer!
y = 5 +1
3(x − −2)
(b) If the line also goes through (k, 7), what is k?Answer: We are given the y-coordinate to be 7, and we are replacing x with k
7 = 5 +1
3(k + 2)
2 =1
3(k + 2)
6 = k + 2
k = 4
(c) Find the equation of another line that goes through this same point, but perpendicular to ouroriginal line.Answer: The point is (4, 7) and the slope is m = −3, so the equation is
y = 7 − 3(x − 4)
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72. Find the equation of the line that goes through (5, 2) that is parallel to y = 12 (7x − 5).
Answer: The slope is m = 72 , so the eqn of the line is y = 2 + 7
2 (x − 5).
73. Find the equations of lines that pass through (3, 5) parallel and perpendicular to 2x + 9y = 1.Answer: Solving the given line for y, we find the slope to be −2
9 . The parallel line is
y = 5 − 2
9(x − 3)
and the perpendicular line is
y = 5 +9
2(x − 3)
74. Consider points P (−1, 8) and Q(7, 2). Find the line that forms the perpendicular bisector of ~PQ.Answer: The slope of the line thru P and Q is m = −3
4 . The bisector should pass through themidpoint, which is (3, 5). Therefore the desired line is
y = 5 +4
3(x − 3)
41 - Class Examples
75. At 2:00 you leave exit 417 headed west on I-40 at 84 MPH.
(a) Write a linear equation relating your mile-marker position and time.Answer: Let y be your position, and let t be the time. You know the point (2, 417) and yourvelocity is the slope, in this case m = −84. Therefore the equation is:
y = 417 − 84(t − 2)
Notice that t − 2 is your elapsed time, which is multiplied by your speed and subtracted fromyour starting position.
(b) Use the linear model to predict when you will reach the Mississippi River.Answer: Set y = 0 and solve for t:
0 = 417 − 84(t − 2)
84(t − 2) = 417
t − 2 = 417/84
t = 417/84 + 2 ≈ 7
Notice that the algebraic steps to solving this problem correspond to more informal mind calcu-lations that you may do in this situation.
(c) At 5:30 you get a speeding ticket. Where are you?Answer: Plug in t = 5.5 go get
y = 417 − 84(5.5 − 2) = 417− 84(3.5) = 123
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(d) Your speeding ticket was in a 70 MPH zone. It is for $ 80 plus $ 6 per MPH over the limit. Whatis your fine?Answer: 80 + 6(84 − 70) = 164
(e) Write a general mathematical formula for the fine amount in terms of the violator’s speed.Answer: Let s be the speed and F the fine. We can use the point (84, 164) and observe thatthe slope is m = 6.
F = 164 + 6(s − 84)
Notice that if we had used another point, say (70, 80), we would have an equivalent formula:
F = 80 + 6(s − 70)
76. Suppose that in 1900 the average global temperature was 70◦, and in 2000 it was 72◦. Find linearformula to describe the Al Gore memorial global warming trend.Answer: The two points are (1900, 70) and (2000, 72), so m = 2
100 = 150 degree per year.
T = 72 +1
50(y − 2000)
43 - Class Examples
Two quantities are proportional if one is a fixed constant times the other. This can be repre-sented by the equation
y = mx
which is a line through the origin. The slope m is called the constant of proportionality.Equivalently, the ratio of y to x is constant: y
x = m. Given two data points, you can write
y1
x1=
y2
x2
and cross-multiply.
77. Suppose the local property tax is proportional to the value of the property. If the tax on a $300,000farm is $1250, how much tax would you owe on a $180,000 house ?Answer: Solve 1250
300000 = x180000 to get x = 750.
The property tax formula is y = 1250300000x = 1
240x.
78. In fifth gear, my Mustang revs at 2000 RPM when I’m going 68 MPH. If it were possible to reach the5500 RPM redline, how fast would I be going?Answer: For manual transmissions, MPH (M) and RPM (R) are proportional. Therefore two pointsare (0, 0) and (2000, 68). From this we calculate the slope to be:
m =68 − 0
2000 − 0=
68
2000=
17
500
The linear equation is:
M =17
500R
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Plugging in R = 5500, we get
M =17
5005500 = 17 · 11 = 187
79. A biologist wants to estimate the local population of yellow-bellied sapsuckers (sphyrapicus varius).She will use the clever catch-release method. Suppose she tags 10 birds. The next day, she observes20 birds, 3 of which have tags. Approximately how many birds live in the area?Answer: Let x be the total bird population. Set up a proportion relating the ratios of tagged birds.
10
x=
3
20
3x = 200
x =200
3≈ 67
Another way to solve this problem is to use the line of proportionality
y =3
20x
where y is the number of tagged birds. Since y = 10, we can get our answer by solving
10 =3
20x
80. A Smoky Mountain park ranger tagged three bears. Later he observed that about 12% of bears spottedin the region were wearing tags. What is the approximate bear population?Answer: Let x be the total population. We have that
3
x=
12
100
45 - Class Examples
Solving we get 12x = 300 or x = 25.
Systems of Linear Equations
When you follow two separate chains of thought, Watson,you will find some point of intersection which should approximate the truth.
– Sherlock Holmes
Two non-parallel lines must intersect at a unique point. That point lies on both lines, andsatisfies both equations simultaneously.
Solving 2 linear equations involving 2 unknowns by substitution.
(a) Label the unknowns.
(b) Write equations involving the unknowns.
(c) Solve one of the equations for one of the variables (whichever is most convenient).
(d) Substitute into the other equation.
(e) Solve for the remaining variable.
(f) Plug-in to find the other variable.
(g) Check your solution.
81. Sketch the lines y = 2x − 9 and x + y = 3. Find the intersection by substitution.Answer: Since the first equation is already solved for y, substitute into the other equation to get
x + (2x − 9) = 3
3x − 9 = 3
3x = 12
x = 4
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Now plug x = 4 back in to get y = 2(4) − 9 = −1, so the intersection point is (4,−1).
82. Find the intersection of 7y + 2x = 1 and 3y − x = 6.Answer: Here it is probably easiest to sole the second equation for x to get x = 3y − 6, which weplug into the other eqn
7y + 2(3y − 6) = 1
7y + 6y − 12 = 1
13y − 12 = 1
13y = 13
y = 1
Now we have x = 3(1) − 6 = −3, so the solution is (−3, 1).
83. The Redskins defeated the Bills by 13 points in Super Bowl XXVI. Sixty-one total points were scoredin the game. What was the final score?Answer: Two equations are R − B = 13 and R + B = 61. Substitute R = B + 13 into the 2nd eqnto get (B + 13) + B = 61 or 2B + 13 = 61, 2B = 48, B = 24. Therefore R = 24 + 13 = 37, so the finalscore was 37-24.
47 - Class Examples
84. Jack is 6 years older than Jill. The sum of their ages is four times the age Jill was 8 years ago. Howold are they?Answer: Let B be Jack’s age, and G be Jill’s age. We translate the sentences into equations thatdescribe how the ages are related:
B = G + 6
B + G = 4(G − 8)
Since the first equation is already solved for B, we substitute as follows:
(G + 6) + G = 4(G − 8)
2G + 6 = 4G − 32
38 = 2G
G = 19
So the girl (Jill) is 19, and Jack must be 25. You should check that your answers do solve the originalproblem.
85. Brett bought 2 hot dogs and a drink for $8. Scott bought 1 hot dog and 5 drinks for $13. How muchdoes a hot dog cost? a drink?Answer: Let H and D represent the cost of a hot dog and drink respectively. Write down twoequations:
2H + D = 8
H + 5D = 13
Solving for D in the first equation givesD = 8 − 2H
which we substitute into the other equation:
H + 5(8 − 2H) = 13
and solve for H :H + 40 − 10H = 13
−9H = −27
H = 3
Plug back in to find D:D = 8 − 2(3) = 2
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So a hot dog is $3, and a drink is $2.
86. A movie theater sells tickets for $ 8. The student discount is $ 2. One night 525 tickets were sold fora total revenue of $ 3580. How many regular and student tickets were sold?Answer: Let R be the number of regular tickets, and S be the number of student tickets. Theequations are:
R + S = 525
8R + 6S = 3580
We can solve the first equation for R to get
R = 525 − S
Substitute into the 2nd equation:
8(525 − S) + 6S = 3580
4200 − 2S = 3580
620 = 2S
S = 310
So 310 student tickets were sold, leaving R = 525 − 310 = 215 regular tickets.
87. Heather is training for a biathlon. Yesterday, she ran 2 kilometers and swam 2 kilometers in 50 minutes.Today she ran 3 kilometers and swam 1 kilometers in 37 minutes.
49 - Class Examples
(a) Write two linear equations describing the situation. Explain what each variable represents.Answer: 2R + 2S = 50 and 3R + S = 37, where R is the number of minutes it takes her to run1k, and S is the number of minutes it takes to swim 1k.
(b) Solve the system.Answer: It is easiest to solve the second equation for S:
S = 37 − 3R
and plug back into the first equation:
2R + 2(37 − 3R) = 50
−4R + 74 = 50
−4R = −24
R = 6
Now plug back in to get S:
S = 37 − 3(6) = 19
So it takes 6 min to run 1k, and 19 min to swim 1k.
(c) If she enters a biathlon that requires a 10k run and 3k swim, how long will it take her?Answer: 10R + 3S = 10(6) + 3(19) = 117 minutes, or a little under two hours.
50 - Class Examples
88. Find the intersection of these lines:5y − 3x = 23
7x + 2y = 1
Answer: First we solve one equation for one variable; let’s solve the first equation for y to get
y =23 + 3x
5
Now plug that into the other equation:
7x + 2
(
23 + 3x
5
)
= 1
Multiply through by 5 to get rid of the fraction:
35x + 2(23 + 3x) = 5
This is an equation with only variable, so we can solve for x:
35x + 46 + 6x = 5
41x = −41
x = −1
Finally, plug this back into our expression for y:
y =23 + 3(−1)
5=
20
5= 4
Therefore the lines intersect at (−1, 4).
89. How close does the line y = 2x + 1 get to the origin?Answer: This is an open ended question, but we have all the tools to solve it. First sketch the lineand notice that the closest point connects to the origin via a perpendicular line segment. Let’s findthe equation of this line:
point: (0, 0) m = −1/2
51 - Class Examples
y = 0 +−1
2(x − 0) =
−x
2
The point of interest is the intersection of y = 2x + 1 and y = −x/2. Solving we get
2x + 1 = −x/2
4x + 2 = −x
5x = −2
x =−2
5
y = −x/2 =2
10
Finally we can calulate the distance from (0, 0) to (−.4, .2) to get
√
.42 + .22 =√
.16 + .04 =√
.2 ≈ 0.447
Circles
Noli turbare circulos meos.– Archimedes
A circle is a set of points equidistant from a center point.
• radius r is the distance from any point on the circle to the center
• diameter d = 2r
• circumference c = 2πr
• area a = πr2
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90. Find the equation of this circle:
Answer: The center is (2,−3) and it appears to have a radius of r = 5. Therefore we can mark anarbitrary point on the circle (x, y) and say
dist((x, y), (2,−3)) = 5
√
(x − 2)2 + (y + 3)2 = 5
(x − 2)2 + (y + 3)2 = 25
To find the equation of a circle, first find
• the center (x0, y0)
• the radius r
The equation of the circle is(x − x0)
2 + (y − y0)2 = r2
91. Find the equation of the circle centered at (−1, 4) with diameter 14.Answer: To find the eqn of a circle, we must find:
• centerThis is given in the problem to be (−1, 4)
• radiusSince d = 2r, we solve 14 = 2r to get r = 7 (the radius is half the diameter).
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The eqn of the circle is
(x −− 1)2 + (y − 4)2 = 49
92. Find the center, radius, diameter, circumference, and area of this circle:
2(y + 3)2 = 32 − 2(x − 8)2
Answer: First we must rearrange the equation to put it in standard form:
2(x − 8)2 + 2(y + 3)2 = 32
(x − 8)2 + (y + 3)2 = 16
Now we identify (8,−3) as the center, and r = 4. The other statistics are:
d = 2r = 8 C = 2πr = 8π A = πr2 = 16π
93. Find the x and y intercepts of (x − 6)2 + y2 = 49.Answer: To find the y-ints, set x = 0 to get 36+y2 = 49, so y = ±
√13, and the y ints are (0,±
√13).
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To find the x-ints, set y = 0 to get (x− 6)2 = 49, so x = 6± 7, and the x ints are (−1, 0) and (13, 0).
94. Find the equation of the circle centered at (2,−7) that has circumference 12π.Answer: 2πr = 12π, so r = 6 and the eqn of the circle is (x − 2)2 + (y + 7)2 = 36.
95. Find the equation of the circle with a diameter that goes from (−2, 1) to (6, 9).Answer: The center is at the midpoint (2, 5), and we can calculate the radius as the distance from(2, 5) to (6, 9), which is r =
√16 + 16 =
√32. Therefore
(x − 2)2 + (y − 5)2 = 32
You can check that both original points are on the graph of this circle.
96. A circle is centered at (3, 1) and has a y-int of −2.Find the equation of the circle, and then find its other y-intercept.Answer: You might want to sketch a picture first. To find the eqn of a circle, we must find:
• centerThis is given in the problem to be (3, 1)
• radiusThe y-int tells us the circle passes through (0,−2). The radius is the distance from the center to
55 - Class Examples
any point on the boundary, so we may compute
r =√
(3 − 0)2 + (1 −− 2)2 =√
18
The equation of the circle is
(x − 3)2 + (y − 1)2 = 18
Now we are supposed to find the other y-int, which means we set x = 0, yielding
(0 − 3)2 + (y − 1)2 = 18
(y − 1)2 = 9
y − 1 = ±3
y = 1 ± 3
Therefore, y = 1 + 3 = 4, or y = 1 − 3 = −2 (we already knew this one).
The unit circle is centered at the origin (0, 0) and has radius r = 1.It serves as the prototype for all circles.
97. If (12 , y) is on the unit circle in the 4th quadrant, what is y?
Answer: The equation of the unit circle is:
x2 + y2 = 1
Since x = 12 we have (1/2)2 + y2 = 1, so y2 = 3/4 and y = ±
√3
2 . In the 4th quadrant, y should be
negative, so we conclude that y = −√
32 .
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98. If the unit circle is inscribed in a square, what is the area of the border region?Answer: Draw a picture. The border is the area of the square minus the area of the circle, which isA = 4 − π.
Functions
Success is more a function of consistent common sense than it is of genius.– An Wang
A function is a machine that takes an input value and returns an output value.
x f y
• The input variable is independent, and is traditionally denoted x (horiz axis).
• The output variable is dependent, and is traditionally denoted y (vertical axis).
• By writing y = f(x), we mean “y is a function of x” or in other words “y depends on x.”
• A function may be represented by: formula, graph, table, list, or words.
• Given a function’s formula, plug-in to a make table of points, then sketch the graph.
99. Refer to this table of points:x 0 1 2 3y 1 3 5 7
(a) Plot these points in the plane and interpolate/extrapolate to make a graph.Answer: they lie in a straight line
(b) Infer a formula relating x and y.
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Answer: y = 2x + 1
(c) Describe this relationship in words.Answer: To get y, double x then add 1.
(d) Create a table, graph, and formula if the description were “add 1 to x then double the result.”Answer: The formula is y = 2(x + 1), and the table would look like
x 0 1 2 3y 2 4 6 8
Every y value is one unit more, so the original graph is moved up. Note that we could get anequivalent formula by using the point (1, 4) with m = 2 to get y = 4 + 2(x − 1).
100. Consider the points {(0,−1), (1, 0), (2, 3), (3, 8), (4, 15)}.
(a) Infer a formula for this function.Answer: Look for a pattern - it seems like the y values are all one less than a perfect square.You can check that y = x2 − 1 matches the table entries.
(b) If (7, y) is on the graph of this function, what is y?Answer: plug in x = 7 to get y = 48
(c) If (x, 99) is on the graph of this function, what is x?
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Answer: x = ±10
101. Are these two functions the same?
• y is the square root of x plus one
• y is the square root of x, plus one
Answer: no, since√
x + 1 6= √x + 1
102. This is a chart of Juan Pablo Montoya’s position during the 2008 Daytona 500 NASCAR race. Whatare the independent and dependent variables.
Answer: independent variable is the lap number, the dependent variable is his position
103. The weight of a puppy can be approximated by the formula
W = .04A
(
25t + 3
t + 4
)
where A is the dog’s adult weight, and t is age in months.
(a) Suppose A is a known constant, then what are the independent and dependent variables?
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Answer: t is independent, and W is dependent
(b) If an adult beagle weighs about 30 pounds, how much should a 3 month old puppy weight?Answer: W = 1.2(78/7) ≈ 13 pounds
(c) Solve the formula for A. If you now think of t as constant, what are the independent and dependentvariables?Answer: A = 25W
(
t+425t+3
)
; now W is independent and A is dependent
(d) A puppy of known parents weighs 12 pounds when he is 1 month old. Predict his adult weight.Answer: A = 300(5/28) ≈ 54 pounds
104. Here are the win totals for Carson-Newman football since 1997.
{ 11, 12, 13, 8, 6, 12, 11, 9, 8, 8, 10 }
Sketch the graph of wins versus the year. Is this a function? What are the independent/dependentvariables?Answer: Yes, it is a function, with independent variable year, and dependent variable wins.
105. Sketch a graph of these functions:
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(a) f(x) = |x + 1| − 2
(b) f(x) = x2 + 4
(c) f(x) =√
x2 + 4
(d) athletic ability as a function of age
(e) grade as a function of study time
(f) MPG as a function of MPH in your carAnswer: starts at (0, 0), increases to a maximum when at about 50 MPH, then graduallydecreases from there
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To be a function, a relation must have these properties:
• There is exactly one y for each x.
• The graph passes the vertical line test (VLT).
• The equation can be solved for y unambiguously in terms of x.
To find a point on the graph of a function, it is enough to know the x value. Just plug it in toobtain y. A generic point is:
(x, y) = (x, f(x))
The function’s argument substitutes for the independent variable in the stated formula,e.g. if f(x) =
√x, then
f(t) =√
t f(4x2) =√
4x2 f(,) =√
,
106. Which of these are functions?
(a) 2y + 3x = 4Answer: We can solve unambiguously for y, obtaining y = −3
2 x + 2. So it is a function.All non-vertical lines are functions.
(b) x2 + y2 = 1Answer: Solving for y we get y = ±
√1 − x2. Since there are two possibilities, this is not a
function. In fact, the graph of this equation is the unit circle, so it fails the VLT.We could pick say the top half only, and write f(x) =
√1 − x2.
(c) f(student) = social security numberAnswer: It is a function since each person has only one SS#.
(d) f(student) = email addressAnswer: Not a function since a student may have multiple email addresses.
(e) f(child) = motherAnswer: It is a function since each child has exactly one mother. But couldn’t two childrenhave the same mother?
(f) f(husband) = wife
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Answer: In most societies, yes this is a function since a husband has exactly one wife.
(g)t 7 2 1 5 0
y√
2 1 0 π 1Answer: yes, since each t value is paired with exactly one y value
(h)
Answer: the first graph is not a function (fails VLT), but the second graph is a function
(i) {(1, 3), (2, 3), (3, 1)}Answer: yes
(j) {(1, 3), (1, 2), (3, 1)}Answer: no, since x = 1 is paired with two different y values
(k) y2 = xAnswer: x is a function of y, but y is not a function of x, since y = ±√
x is ambiguous.
(l) the graph of a spiralAnswer: no, it fails the VLT
107. Find the x and y intercepts of y = (x − 2)2 − 9, then sketch the graph.Answer: A function can have at most one y-int, which you get by plugging in x = 0. It is (0,−5).To get the x-int(s), set y = 0 and solve 0 = (x − 2)2 − 9 to get (−1, 0) and (5, 0).
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Plug-in other x values to get a decent sketch.
108. Let f(x) = (2x − 5)(x + 3)
(a) Compute f(4)Answer: Plug in x = 4 to get f(4) = (2 · 4 − 5)(4 + 3) = 21
(b) Find the y-interceptAnswer: Set x = 0 to get (2 · 0 − 5)(0 + 3) = −15
(c) Find the x-intercept(s)Answer: Set y = 0, but y = f(x), so we solve 0 = (2x − 5)(x + 3).This means 2x − 5 = 0 or x + 3 = 0, so there are two x-int: x = 5/2,−3.
(d) Is (1,−8) on the graph of this function?Answer: Compute f(1) = (2 · 1− 5)(1 + 3) = −12, so (1,−12) is on the graph, but not (1,−8).
109. Let f(x) = 2x2 − 6x
(a) Make a table of points and sketch the graph.
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Answer: done in class
(b) Mark the points where x = −1 and x = 2, and find the slope between them.Answer: The two points are (−1, f(−1)) = (−1, 8) and (2, f(2)) = (2,−4).The slope between them is
m =−4 − 8
2 −− 1= −4
(c) What is f(−2x)?Answer: f(−2x) = 2(−2x)2 − 6(−2x) = 8x2 + 12x
(d) What is f(⋆)?Answer: f(⋆) = 2(⋆)2 − 6⋆
110. The time it takes to fall h feet under the force of gravity is T = 14
√h.
(a) The independent variable is and the dependent variable is .Answer: h is independent, and T is dependent
(b) If you jump off a 100 foot building, how long do you have to say your prayers before impact?Answer: T (100) = 1
4
√100 = 10
4 = 2.5 seconds
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(c) Suppose a daredevil needs 4 seconds to pick the handcuffs and open a parachute (which he mustdo with 120 feet to spare). How high must the cliff he jumps off of be?Answer: Set T = 4 and solve for h. 4 = 1
4
√h, so 16 =
√h and h = 256. But he needs 120 feet
to spare, so he must start out at 256+120=376 feet.
111. A waitress gets paid $3 per hour plus tips. If she waits on 4 tables per hour, the average table has 2.5people, the average meal costs $10, and the average tip is 18 %, write her take-home pay as a functionof hours worked.Answer: y = (3+4 ·2.5 ·10 · .18)t = 21t, so she is making $21 per hour. Since this is a linear functionthat passes through the origin, we say that her pay is proportional to hours worked.
112. It is about 500 miles from Bristol to Memphis.
(a) Suppose you drive non-stop at speed x. Write the time required as a function of x.Answer: f(x) = 500
x . We say that time is inversely proportional to speed.
(b) Suppose you make several stops that take a total of one hour and 15 minutes. Write the timerequired as a function of x.Answer: f(x) = 500
x + 1.25
113. A rectangular field has perimeter 400m.
(a) Write the area of the field as a function of its length.Answer: 2w + 2ℓ = 400 constrains the perimeter, and the area is A = ℓw. Eliminate the wvariable by solving for w = 200 − ℓ, and substitution:
A = ℓ(200 − ℓ)
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(b) Experiment to find the value of ℓ that maximizes the area.Answer: ℓ = 100 gives a maximum area of 10000m2, meaning that a square is optimal.
114. The perimeter of a track is 400m. It is composed of 2 semi-circles and a rectangle. Find the enclosedarea as a function of the radius.Answer: Let x be the length of a straight-away, and r be the radius. Then the perimeter and areaare
P = 2x + 2πr A = 2rx + πr2
Set P = 400 and solve for x to eliminate one variable: x = 200 − πr. Substitute this into the areaequation:
A = 2r(200 − πr) + πr2
115. Suppose a square warehouse is to be built. The walls/exterior cost $70 per foot, and the roof/floor/interiorcosts are $30 per square foot.
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(a) Write the total cost of the warehouse as a function of the square footage A.Answer: Draw a picture. Each of the four sides is
√A in length, so the total cost is
C = 280√
A + 30A
(b) What is the cost of a 16,900 square foot warehouse?Answer: C = 280(130) + 30(16900) = 543400
(c) Write the cost per square foot as a function of A.Answer: f(A) = C
A = 280A−1/2 + 30
116. A police car sits at (0, 1) as you drive along the road f(x) =√
x.
(a) Draw a picture. Write the distance between you and him as a function of x.Answer: You are at (x, y) = (x, f(x)) = (x,
√x), and he is at (0, 1). The distance between is
d(x) =
√
(x − 0)2 + (√
x − 1)2
=
√
x2 + (√
x − 1)2
=
√
x2 + x − 2√
x + 1
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(b) When x = 4, how far apart are you?
Answer: d(4) =√
42 + 4 − 2√
4 + 1 =√
17
(c) Sketch the graph of d(x) for x ≥ 0.Answer: a couple of easy points are (0, 1) and (1, 1), and we already found (4,
√17)
Economic concepts
• price: P , dollars per unit sold
• demand: D, quantity that can be sold
• supply: S, quantity produced
• cost: C = (fixed cost) + (unit cost)D
• revenue: R = PD, total (gross) cash received
• profit: Y = R − C, net gain/loss
In a free market, the price will adjust so that D = S.
117. A cell phone company charges $80 per month for unlimited usage. At that price they have 250 thousandcustomers.
(a) Market research shows that for every dollar price increase, they lose 4 thousand customers, andfor every dollar price decrease they gain 4 thousand customers. Write demand as a function ofprice.Answer: This is linear with slope m = −4 and a known point (80, 250), so we obtain
D = 250 − 4(P − 80)
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(b) Write revenue as a function of price.Answer: R = PD = P (250 − 4(P − 80))
(c) Find the revenue at P = 79, 80, 81.Answer: R(79) = 20066, R(80) = 20000, and R(81) = 19926, so they should probably lowertheir price slightly
118. The turkey leg vendor at Lane Stadium must pay $3000 for concession stand and equipment rental.Each leg costs $2 in raw materials. Let D be the number of legs sold at $5 each.
(a) Write the cost function.Answer: fixed cost plus unit cost is C = 3000 + 2D
(b) Write the revenue function.Answer: revenue is price times demand R = 5D
(c) Write the profit function.Answer: profit is revenue minus cost Y = R − C = 5D − (3000 + 2D) = 3D − 3000
(d) How many legs must he sell to break even?Answer: set Y = 0 to get D = 1000
119. Is it generally true that f(x1 + x2) = f(x1) + f(x2)?Answer: No, a counter-example is f(x) =
√x, e.g. note that
√x2 + 4 6= x+2. One class of functions
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this does work for is f(x) = mx.
Function Library / Shifts
120. Here are some basic building block functions. Plot points to sketch the graphs.
• f(x) = 1Answer: constant
• f(x) = xAnswer: line
• f(x) = x2
Answer: parabola, “U”
• f(x) = x3
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Answer: snake
• f(x) = |x|Answer: “V”
• f(x) = x1/2 =√
xAnswer: parabola on side, just one half since it must pass VLT
• f(x) = x−1 = 1x
Answer: hyperbola, 2 pieces
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Functions can be manipulated by the following transformations:
• vertical shift: f(x) ± c
• horizontal shift: f(x + c) left, f(x − c) right
• vertical scale: cf(x) (taller if |c| > 1, shorter if |c| < 1)
• vertical flip: −f(x)
• horizontal flip: f(−x)
Multiple transformations can be combined together. Generally, horizontal affects happen insidethe function argument x, and vertical affects happen outside to directly change y.
121. Illustrate all the above manipulations with f(x) =√
x and g(x) = |x|.
122. Let f(x) = x2. Write the formula for, and sketch the graph of:
(a) f(x) − 2Answer: x2 − 2, shift down 2
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(b) f(x − 2)Answer: (x − 2)2, shift right 2
(c) f(x + 2)Answer: (x + 2)2, shift left 2 (x = −2 makes the argument zero)
(d) 2f(x)Answer: 2x2, twice as steep
(e) − 12f(x)
Answer: − 12x2, opens down and half as steep
(f) f(−x)Answer: (−x)2 = x2, flip horizontally, which has no effect on this graph
(g) f(x − 1) + 3Answer: (x − 1)2 + 3, shift right 1 and up 3
123. Write the formula for a parabola that has y-intercept at (0, 1) and x-intercepts at (±2, 0).Answer: Start with f(x) = cx2 + 1. We can pick the constant c so that f(2) = 0. Solve 0 = 4c + 1
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to get c = −1/4, so
f(x) =−1
4x2 + 1
124. The formula for the function graphed below is: f(x) = .
1
4
0
3
2
-1
1
-2
x
32
Answer: We see that the basic shape is√
x. Shifting left one gives√
x + 1. Finally we see thatf(0) = 2, so it is twice as high. Therefore, f(x) = 2
√x + 1.
You should check that f(−1) = 0 and f(0) = 2.
125. Explain how the point-slope formula for a line is just a sequence of transformations on y = x.Use y = 2(x − 3) + 1 to illustrate.
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Answer: Starting with y = x, we shift right 3, up 1, and scale by a factor of 2.
126. Find the formula for a function with a graph that starts at the origin, goes straight to (3, 1), and thengoes straight to (6, 0). Then find the length of this section of the graph.Answer: Sketch the graph first to see that is an upside down short V shape with a vertex at (3, 1).Starting from y = |x|:
• shift right 3 to get y = |x − 3|
• shift up 1 to get y = |x − 3| + 1
• flip vertically to get y = −|x − 3| + 1
• scale vertically to get y = −13 |x − 3| + 1
The total length is twice the distance from (0, 0) to (3, 1), which is 2√
10.
127. Given the graph of f(x), the graph of 23f(x + 2) − 1 would be:
(a) shifted left or rightAnswer: left
(b) shifted up or downAnswer: down
(c) taller or shorterAnswer: shorter
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Function Properties
A function f(x) is
• even if f(−x) = f(x); symmetric across the y-axis
• odd if f(−x) = −f(x); symmetric about the origin
even odd
To test for even/odd symmetry:
• sketch the graph and look
• or evaluate f(−x) and compare to f(x)
128. Identify the symmetry (even/odd/neither) in these functions.
(a) f(x) = x2 − 1Answer: f(−x) = (−x)2 − 1 = x2 − 1 = f(x), so this is even (symmetric WRT the y-axis)
(b) f(x) = x3 + 5Answer: f(−x) = (−x)3 + 5 = −x3 + 5 is not the same as f(x) or −f(x), so it is neither evennor odd. However, it is symmetric WRT to the line y = 5.
(c) y = (x + 4)2
Answer: The parabola is shifted left 4, so it has axis of symmetry x = −4, but is neither even
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nor odd.
(d) f(x) = 1x + 2x
Answer: f(−x) = 1−x + 2(−x) = −(1/x + 2x) = −f(x), so it is odd (symmetric WRT the
origin)
(e) f(x) = 1−xx
Answer: f(−x) = 1−(−x)−x = 1+x
−x , so it is neither even nor odd. However, writing f(x) = 1x − 1
we see from the graph that it is symmetric WRT the point (0,−1).
(f) y = |x| + x2 − 3Answer: f(−x) = | − x| + (−x)2 − 3 = |x| + x2 − 3 = f(x), so it is even
The domain of a function f(x), denoted Dom(f), is the set of all possible input values (the x’s)To find the domain:
(a) If you have a graph, look for the left/right-most points (and any gaps in between).
(b) If you have a formula, find all x’s that avoid:
• dividing by zero
• taking an even root of a negative number
129. Find the domain of:
(a) The top half of the unit circle.
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Answer: draw it; the x values represented by the graph are [−1, 1]
(b) y = (x − 3)−1
Answer: Recognize that y = 1x−3 . Since the denominator can’t be zero, we get x − 3 6= 0 or
x 6= 3.
(c) f(x) =√
4 + 2xAnswer: You can’t take an even root of a negative number. Thus 4 + 2x ≥ 0, so 2x ≥ −4 andx ≥ −2, or [−2,∞)
(d) f(t) = 1 + x+2x(5−3x)
Answer: You can’t divide by zero, thus x(5 − 3x) 6= 0,so x 6= 0, 53
(e) y = 1x2−9
Answer: You can’t divide by zero, thus x2 − 9 6= 0, so x2 6= 9 and x 6= ±3
(f) y = 1x2+9
Answer: You can’t divide by zero, but x2 + 9 can’t be zero, so the domain is R.
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(g) y =√
|x| − 2Answer: |x| − 2 ≥ 0, so |x| ≥ 2, so the answer is (−∞,−2] ∪ [2,∞)
(h) f(x) = 2(x − 1)(x + 5)2 − |2 +√
x2 + 1|Answer: No divisions or possible roots of negatives, so Dom(f) = R.
The range of a function f(x), denoted Ran(f), is the set of all possible output values (the y’s).To find the range, sketch the graph and look for the lowest and highest points (and gaps inbetween).
130. Find the domain and range of
5−1−3
Answer: domain: [−3, 5), range: [−1,∞)
131. Find the range of:
(a) f(x) = 12 (x − 3)2 + 1
Answer: Sketch the graph. Since the parabola is CU and has vertex (3, 1), the range is [1,∞)
(b) y = 3 − |x|
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Answer: Sketch the graph and look for the low and high points. The range is (−∞, 3].
(c) y = −2x(x − 1)2
Answer: Plot some points to see that the graph rises to the left, and falls to the right. Therange is R.
132. Let f(x) = x4(9 − x2)−1
(a) Is this function even, odd, or neither?Answer: It is not easy to graph this function, so we will test for symmetry algebraically bycomputing f(−x).
f(−x) = (−x)4(9 − (−x)2)−1 = x4(9 − x2)−1
Since f(−x) = f(x), this function is even (y-axis symmetry).
(b) What is the domain?
Answer: First rewrite f(x) = x4
9−x2 . We cannot divide by zero, so we say
9 − x2 6= 0
x2 6= 9
x 6= ±3
So the domain is all R except for x = ±3.
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133. Let f(x) = 3 −√
x + 2.
(a) Find the domain.Answer: We cannot take the square root of a negative number, so we solve
x + 2 ≥ 0
x ≥ −2
So the domain is [−2,∞).
(b) Find the range.Answer: The range is the set of y-values, and these are easiest to find if we can sketch the graphfirst. Starting with y =
√x, we shift left with y =
√x + 2, flip with y = −
√x + 2, and then shift
up with y = −√
x + 2 + 3.
Observe that the graph heads down from a height of 3. Therefore the range is (−∞, 3].
Terminology that describes the shape or behavior of a graph at a given point:
• increasing/decreasing: the graph going up/down (as you move left to right)
• local max/min: the point is the highest/lowest in the vicinity
• global max/min: the point is the highest/lowest anywhere on the graph
• concave up/down: the graph is bending up/down
• inflection point: the graph switches concavity at this point
134. Classify the labeled points on this graph.
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a
b
c
d
e
f
g
h
i
(a) x-intAnswer: c,f,i
(b) y-intAnswer: d
(c) increasing
Answer: c,d,i
(d) decreasingAnswer: a,f,g
(e) local maxAnswer: e
(f) local minAnswer: b,h
(g) global maxAnswer: ∞
(h) global minAnswer: h
(i) concave upAnswer: b,h
(j) concave downAnswer: a,d,e,f
(k) inflection pointAnswer: c,g
135. Classify the labeled points on this graph.
a
b
c
d
f
h
i
g
e
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(a) x-intAnswer: e
(b) y-intAnswer: c
(c) increasing
Answer: a,e
(d) decreasingAnswer: c,g,h,i
(e) local maxAnswer: b,f
(f) local minAnswer: d,i
(g) global maxAnswer: f
(h) global minAnswer: d
(i) concave upAnswer: c,d,h
(j) concave downAnswer: f
(k) inflection pointAnswer: e,g
136. Suppose you graphed demand as a function of price. Would this graph be increasing or decreasing?Answer: Generally decreasing, people want less if they have to pay more, e.g. gasoline. One possibleexception would be status-symbol products, e.g. diamonds, artwork, luxury cars, tuition
A secant line connects two points on a graph.The slope of the secant line is called the average rate of change (AROC).
137. Sketch the graph of y = x2.
(a) Draw the secant line between the points (−1, 1) and (3, 9).
(b) What is the AROC?Answer: The slope of the secant line is m = 9−1
3−−1 = 2.
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(c) Draw the tangent line at (2, 4).
138. If f(x) = x − x1/2, find the equation of the secant line between x = 4 and x = 16.Answer: First we determine the two points by plugging in.
f(4) = 4 − 2 = 2 f(16) = 16 − 4 = 12
so the two points are (4, 2) and (16, 12). The slope between them is
m =12 − 2
16 − 4= 10/12 = 5/6
Using the pt (4, 2) and the slope, the eqn of the line is
y = 2 +5
6(x − 4)
139. Find the secant line of y = x(x − 5) between points where x = 3 and x = 7.Answer: Plug in to get the y-values. The points are (3,−6) and (7, 14). Therefore the AROC is
m = 14−−67−3 = 5 and the equation of the secant line is
y = 14 + 5(x − 7)
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140. Find the secant line that connects the x and y intercepts of f(x) = x−6x+2 .
Answer: Plug in x = 0 to get the y-int (0,−3). Solve f(x) = 0 to get 0 = x−6x+2 , so (6, 0) is the x-int.
The slope is m = −3−00−6 = 1
2 , and the eqn of the secant line is y = −3 + 12x.
141. A running back’s cumulative yardage totals are listed in this table:
games 0 1 2 3 4 5 6 7yards 0 80 120 190 240 370 570 700
(a) Compute the average rate of change (AROC) between 0 and 7.Answer: m = 700−0
7−0 = 100
(b) Compute the AROC between 1 and 5.Answer: m = 370−80
5−1 = 2904 = 72.5
(c) Compute the AROC between 5 and 7.Answer: m = 700−370
7−5 = 3302 = 165
(d) For each of your slope estimates, extrapolate to predict his total yardage after 11 games.Answer: He has 4 more games to play, so the estimates could be:
700 + 4 · 100 = 1100
700 + 4 · 72.5 = 990
700 + 4 · 165 = 1360
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142. On a trip down I-40 across Tennessee, you recorded your mile marker every hour:
t 0 1 2 3 4mm 417 367 293 249 157
(a) Compute your average speed in:
i. the 2nd hourAnswer:
∣
∣
∣
293−3672−1
∣
∣
∣= 77 MPH
ii. the middle two hoursAnswer:
∣
∣
∣
249−3673−1
∣
∣
∣= 59 MPH
iii. the first three hoursAnswer:
∣
∣
∣
249−4173−0
∣
∣
∣= 56 MPH
iv. the entire trip
Answer:
∣
∣
∣
157−4174−0
∣
∣
∣= 65 MPH
(b) What is the difference between speed and velocity?Answer: Velocity includes direction; speed can’t be negative.
(c) In what hour was your average speed the greatest?
Answer: For the 4th hour you averaged∣
∣
∣
157−2494−3
∣
∣
∣= 92 MPH.
(d) In what hour did you go the fastest?Answer: It’s impossible to know from the table.
(e) Contrast AROC with instantaneous speed.Answer: AROC gives a directional rate over a finite interval (secant line). Instantaneous speed
87 - Class Examples
refers to a non-negative rate of change measured over a infinitesimally small time period (tangentline).
143. Suppose the price of a certain stock during the first five years after its IPO is modeled by the function
S(t) = t3 − 8t2 + 19t + 2
(a) What was the initial price?Answer: plug in t = 0 to get S(0) = 2
(b) What was the price after 2 years, after 5 years?Answer: S(2) = 16 and S(5) = 22
(c) Compute the average rate of change (AROC) between t = 2 and t = 5.Answer: The average rate of change is the slope of the secant line. The two points are (2, 16)and (5, 22), so m = 22−16
5−2 = 2 dollars per year.
(d) Sketch graph and try to approximate local and global extrema on [0, 5].Answer: Make a table of points for t = 0, 1, 2, 3, 4, 5. It appears that the stock price has a localmax shortly before t = 2, and a local min around t = 3.5 years.
144. Let f(x) = (x − 2)2. Find the point on the graph such that the secant line between it and the y-intercept has slope 3.Answer: The y-int is (0, 4), and a generic point is (x, (x − 2)2). The slope is
m =(x − 2)2 − 4
x − 0=
x2 − 4x + 4 − 4
x=
x2 − 4x
x= x − 4
88 - Class Examples
Solving x − 4 = 3 gives us x = 7, and the y-coordinate is (7 − 2)2 = 25. Therefore the point is (7, 25).
Polynomials
Politics: (noun) from Greek: poly “many” and ticks “bloodsucking creatures.”– Larry Hardiman
• A polynomial is a linear combination of non-negative integer powers of x.
• The domain of any polynomial is R.
• Each term (monomial) consists of a coefficient times a power of x.
• The degree is the highest power of x in the polynomial’s expanded form.
• Polynomials may be added or multiplied to obtain other polynomials.
For example, ify = 2x4 − 3x2 + 5x + 1
the degree is 4, and the coefficient of the square term is −3.
145. Which of these are polynomials? If it is a polynomial, what is the degree?
(a) f(x) = 4(x − 3)2 + 2Answer: yesdeg(f) = 2
(b) y = 2x2 − 3|x|Answer: noabs val disqualifies
(c) f(x) = −3Answer: yesdeg(f) = 0
(d) y = x−14+x3
Answer: nofraction disqualifies
89 - Class Examples
(e) y =√
2(t2 − 3t) − πt5
Answer: yesdeg(y) = 5
(f) f(x) =√
x2 − 3x + 1Answer: nosqrt disqualifies
(g) q(x) = 2x(1 − x)2
Answer: yesdeg(q) = 3
(h) y = 3x−2 + 5x + 1Answer: noneg exponent disqualifies
(i) f(x) =√
x2 + 4Answer: nonote that f(x) 6= x + 2
(j) y = 3 + 23 (x − 1)
Answer: yesdeg(y) = 1
146. Let p(x) = 2(x + 3)(2 − x3)2 − 7(x + 1)5.
(a) What is deg(p)?Answer: The degree is the highest power of x in the expanded polynomial. The first set ofmultiplied terms indicate a x7 power, which has a higher power than the x5 term. Thereforedeg(p) = 7.
(b) What is the y-intercept?Answer: plug in x = 0 to get p(0) = 2(3)(2)2 − 7(1)5 = 17
(c) What is the domain?Answer: The domain of all polynomials is R.
90 - Class Examples
Define x to be a root of a function if f(x) = 0.
• This is just a synonym for x-intercept.
• To check if x is a root, plug in to see if y = 0.
• To find the root(s), set y = 0 and solve for x.
• It is easy to find roots of factored polynomials.
• For polynomials, the number of roots is ≤ the degree.
147. Let p(x) = 3(x2 − 1)5(x − 3)(2x − 5).
(a) What is deg(p)?Answer: 12
(b) Find the y-intercept.Answer: p(0) = −45
(c) Find the roots.Answer: x = ±1, 3, 5
2
148. If y = (x − 1)(3x + 2)2, what is the coefficient of the square term?Answer: We must multiply out:
y = (x − 1)(9x2 + 12x + 4)
= 9x3 + 12x2 + 4x − 9x2 − 12x − 4
= 9x3 + 3x2 − 8x − 4
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The coefficient of the square term is 3.
149. Is x = 1 a root of p(x) = x3 − 6x2 + 3x + 10? How about x = 2?Answer: plug in to see:p(1) = 1 − 6 + 3 + 10 = 8, so 1 is in not a rootp(2) = 8 − 24 + 6 + 10 = 0, so 2 is a root
150. Let p(x) = x + 4 and q(x) = x − 4. Find
(a) p + qAnswer: (x + 4) + (x − 4) = 2x
(b) 5p − 3qAnswer: 5(x + 4) − 3(x − 4) = 2x + 32
(c) p2
Answer: (x + 4)2 = x2 + 8x + 16
(d) q2
92 - Class Examples
Answer: (x − 4)2 = x2 − 8x + 16
(e) pqAnswer: (x + 4)(x − 4) = x2 − 16
(f) p2qAnswer: (x + 4)2(x − 4) = x(x2 + 8x + 16) − 4(x2 + 8x + 16) = x3 + 4x2 − 16x − 64
151. Check that (x + 1)2 = x2 + 2x + 1. Then is y =√
x2 + 2x + 1 a polynomial?Answer: no, because y =
√
(x + 1)2 = |x + 1|, which is not a polynomial because of the abs val
152. If possible, find the roots of these polynomials:
(a) f(x) = (x − 3)2 − 4Answer: 0 = (x − 3)2 − 4, so (x − 3)2 = 4, so x − 3 = ±2, thus x = 3 ± 2
(b) f(x) = (x − 3)2 + 4Answer: 0 = (x − 3)2 + 4, so (x − 3)2 = −4, which has no soln (sketch a graph to see why)
(c) y = π(x − 5)(7 + 2x)Answer: x − 5 = 0 or 7 + 2x = 0, so x = 5,−7/2
(d) y = 4 + 12 (x − 12)
Answer: 0 = 4 + 12 (x − 12) so −4 = 1
2 (x − 12), so −8 = x − 12, thus x = 4
93 - Class Examples
(e) p(x) = 3(x − 1)2(5x − 2)(x + 7)Answer: x = 1, 2
5 ,−7
153. Under what circumstances is it easy to find the roots of a polynomial?Answer: if it has degree ≤ 2, or if it is factored
Factoring is like reversing the distributive property. The goal is to write an expression as aproduct of simpler terms, so that roots can be easily found. Here’s a basic factoring strategy:
• common factor
• difference of squares (sum of squares doesn’t factor)
• perfect square
• quadratic trinomial
• grouping
154. Factor the following (if possible), then list the roots.
(a) x2 − 3xAnswer: x(x − 3), roots are x = 0, 3
(b) 12x3 + 3x2
Answer: 3x2(4x + 1), roots are x = 0, −14
(c) x2 − 9Answer: (x + 3)(x − 3), roots are x = ±3
(d) x2 + 4Answer: sum of squares does not factor (DNF)
94 - Class Examples
(e) 49x − x3
Answer: x(49 − x2) = x(7 − x)(7 + x), roots are x = 0,±7
(f) 4x2 − 81Answer: (2x + 9)(2x − 9), roots are x = ±9/2
(g) x4 − 1Answer: (x2 + 1)(x2 − 1) = (x2 + 1)(x + 1)(x − 1), roots x = ±1
(h) x2 + 6x + 9Answer: (x + 3)2, root is x = −3
(i) 4x2 − 20x + 25Answer: (2x − 5)2, root is x = 5/2
(j) 6x2 − 3x3 − 3xAnswer: −3x(x2 − 2x + 1) = −3x(x − 1)2, roots are x = 0, 1
(k) x2 + 2x − 15Answer: (x + 5)(x − 3), roots are x = −5, 3
(l) 4x − 2x2 + 6Answer: −2(x2 − 2x − 3) = −2(x − 3)(x + 1), roots are x = 3,−1
(m) 2x2 − 5x − 3Answer: (2x + 1)(x − 3), roots are x = − 1
2 , 3
(n) 12x2 − 29x − 8Answer: (3x − 8)(4x + 1), roots are x = 8/3,−1/4
95 - Class Examples
(o) x2 − 3x + 10Answer: DNF
(p) 9(x + 1) − (x + 1)x2
Answer: (9 − x2)(x + 1) = (3 + x)(3 − x)(x + 1), roots are x = ±3,−1
(q) 3x3 + 2x2 − 12x− 8Answer: x2(3x + 2)− 4(3x + 2) = (x2 − 4)(3x + 2) = (x + 2)(x− 2)(3x + 2), roots are ±2, −2
3
(r) x3 + 4x2 − 21x + (x − 3)(1 − 5x)Answer:
x(x2 + 4x − 21) + (x − 3)(1 − 5x) = x(x + 7)(x − 3) + (x − 3)(1 − 5x)
= (x − 3)(x(x + 7) + (1 − 5x))
= (x − 3)(x2 + 2x + 1)
= (x − 3)(x + 1)2
roots are x = 3,−1, (notice we multiply out only as a last resort)
Factoring is a tool to solve polynomial equations. Follow this procedure:
(a) move everything to one side
(b) factor
(c) find the roots
155. Solve the equation:
(a) x2 = 4Answer: x2 − 4 = 0, so (x + 2)(x − 2) = 0, so x = ±2
(b) x = 4xAnswer: x − 4x = 0, so −3x = 0, so x = 0
96 - Class Examples
(c) x2 = 4xAnswer: x2 − 4x = 0, so x(x − 4) = 0, so x = 0, 4
(d) 3x + 10x2 = 2(1 + x)Answer: 10x2 + 3x = 2 + 2x, so 10x2 + x − 2 = 0, so (2x + 1)(5x − 2), so x = −1
2 , 25
(e) x(x − 3) = 4Answer: x2 − 3x − 4 = 0, so (x − 4)(x + 1) = 0, so x = −1, 4
(f) (x − 2)(5 − x) = 2Answer: −x2 + 7x − 10 = 2, so x2 − 7x + 12 = 0, so (x − 3)(x − 4) = 0, so x = 3, 4
(g) 2x3 − 6x = 5x2 − 15Answer:
2x3 − 6x − 5x2 + 15 = 0
2x(x2 − 3) − 5(x2 − 3) = 0
(x2 − 3)(2x − 5) = 0
so x = ±√
3, 52
156. Find the domain of f(x) = x+218+3x−x2 .
Answer: the denom can’t be zero, so −(x2 − 3x − 18) 6= 0, or −(x − 6)(x + 3) 6= 0, so x 6= 6,−3
157. Demand for bootlegged DVD’s is given by D = 225 − (P + 1)2, where P is the price.
(a) How many DVD’s will you sell if you charge $5? What about $7?Answer: D(5) = 189 and D(7) = 161
97 - Class Examples
(b) If you raise your price too high, nobody will buy your DVD’s. What price is that?Answer: solve D = 0 by factoring to get (15 − (P + 1))(15 + (P + 1)) = 0, so P = 14
(c) Sketch D vs P .
(d) Write the revenue as a function of P .Answer: R = PD, so
R = P (225 − (P + 1)2)
(e) Assuming a fixed cost of $100, and a unit cost of $3, write the cost as a function of P .Answer:
C = 100 + 3D
= 100 + 3(225 − (P + 1)2)
(f) Write the profit as a function of P .Answer:
Y = R − C
= PD − (100 + 3D)
= (P − 3)D − 100
= (P − 3)(225 − (P + 1)2) − 100
98 - Class Examples
(g) Find the profit if you charge $10 per DVD.Answer: plug in P = 10 to get Y = 628 dollars
(h) Can you do better?Answer: yes, if P = 9, then Y = 650
158. The road from Preston to Bonida follows the path f(x) =√
2x + 1. Suppose a cell tower is located atthe origin, and has a range of 10 miles.
(a) Write the distance from a point on the road to the tower as a function of x, assuming x ≥ 0.Answer: A point is located at (x,
√2x + 1), so the distance to the origin is
d(x) =
√
x2 + (√
2x + 1)2 =√
x2 + 2x + 1 = x + 1
(b) If Banida is located along the road at x = 7, do they get a signal?Answer: yes, since d(7) = 8 ≤ 10
(c) At what point on the road would a call get dropped?Answer: solve d(x) = 10 to get x = 9, so the point is (9,
√19)
Quadratics
The analysis of quadratic functions soars to a pitch from whenceit may look proudly down on the feeble and vain attempts of geometry proper
to rise to its level or to emulate it in its flights.– James Joseph Sylvester
99 - Class Examples
A quadratic is a second degree polynomial. The graph of a quadratic is a parabola.Quadratics can be characterized by these important pieces of information:
• y-intercept
• roots
• vertex (the point of maximum curvature, highest or lowest point on the graph)
• concavity (CU or CD)
• tall/short/normal (compared with the standard parabola y = x2)
159. Find the given information about the quadratic y = −14 (x − 5)2 + 9.
(a) y-interceptAnswer: plug in x = 0 to get y = −1
4 (25) + 364 = 11
4
(b) vertexAnswer: (5, 9)
(c) rootsAnswer: solve 0 = −1
4 (x − 5)2 + 9, so (x − 5)2 = 36, so x − 5 = ±6, so x = 5 ± 6 = −1, 11
(d) is it concave up or down ?Answer: down since the coefficient of the x2 is negative
(e) is it tall or short ?Answer: short, since | − 1/4| < 1
160. Find the given information about the quadratic f(x) = −2(x + 1)(5 − x).
(a) y-interceptAnswer: f(0) = −10
100 - Class Examples
(b) rootsAnswer: x = −1, 5
(c) CU/CDAnswer: CU since the coefficient of x2 is 2. We could write f(x) = 2(x + 1)(x − 5).
(d) tall/short/normalAnswer: tall since |2| > 1
(e) vertexAnswer: The vertex is half-way between the roots, which is x = 2.Plug-in to get y = f(1) = −18, so the vertex is (2,−18)
Quadratics may be written in three different forms:
• general form (GF), e.g. x2 − 10x + 16; (EZ-button for y-int)
• factored form (FF), e.g. (x − 2)(x − 8); (EZ-button for roots)
• standard form (SF), e.g. (x − 5)2 − 9; (EZ-button for vertex)
161. Convert between to the designated form:
(a) 3(x − 3)(x + 7) to GFAnswer: just multiply out: 3(x2 + 4x − 21) = 3x2 + 12x − 63
(b) 12 (x − 4)2 + 3 to GFAnswer: just multiply out: 1
2 (x2 − 8x + 16) + 3 = 12x2 − 4x + 11
(c) 3x2 − 6x − 24 to FFAnswer: factor: (3x − 12)(x + 2)
101 - Class Examples
(d) (x − 3)2 − 4 to FFAnswer: we could convert to GF first, then factor:
x2 − 6x + 9 − 4 = x2 − 6x + 5 = (x − 5)(x − 1)
or we could use the difference of squares:
(x − 3)2 − 4 = (x − 3 + 2)(x − 3 − 2) = (x − 1)(x − 5)
(e) x2 − 6x + 2 to SFAnswer: complete the square:
(x2 − 6x ) + 2
(x2 − 6x + 9) + 2 − 9
(x − 3)2 − 7
Complete the square:
• Group the x2 and x terms.
• Factor out the coefficient of the x2 term.
• Add (12 coef x)2 inside the parentheses.
• Compensate outside the parentheses.
162. Complete the square to find the vertex of y = 5x2 + 10x − 2.Answer:
5(x2 + 2x ) − 2
5(x2 + 2x + 1) − 2 − 5
5(x + 1)2 − 7
102 - Class Examples
163. Find the roots and vertex of y = −2x2 − 16x + 8Answer: It doesn’t factor:
y = −2(x2 + 8x − 4)
So put it in SF:y = −2(x2 + 8x ) + 8
y = −2(x2 + 8x + 16) + 8 + 32
y = −2(x + 4)2 + 40
The vertex is (−4, 40). Solve to find the roots:
0 = −2(x + 4)2 + 40
2(x + 4)2 = 40
(x + 4)2 = 20
x + 4 = ±√
20
x = −4 ±√
20
164. Write the quadratic in all three forms, find all the information, and sketch the graph.
(a) y = 1 − (x − 2)2
Answer: multiply out to get y = 1 − (x2 − 4x + 4) = −x2 + 4x − 3 (GF); then factor to gety = −(x2 − 4x + 3) = −(x − 3)(x − 1) (FF)The roots are x = 1, 3; the y-int is (0,−3); the vertex is (2, 1); it is CD and normal
(b) y = 4x2 − 16x − 9Answer: factor to get (2x − 9)(2x + 1); complete the square to put it in SF:
y = 4(x2 − 4x + 4) − 9 − 16
y = 4(x − 2)2 − 25
103 - Class Examples
The roots are x = 9/2,−1/2; the y-int is (0,−9); the vertex is (2,−25); it is CU and tall.
(c) y = 12 (x + 5)(7 − x)
Answer: multiply out to get y = 12 (−x2 + 2x + 35) = − 1
2x2 + x + 352 (GF); then complete the
square:
y = −1
2(x2 − 2x + 1) +
35
2+
1
2
y = −1
2(x − 1)2 + 18
The roots are x = 7,−5; the y-int is (0, 35/2); the vertex is (1, 18); it is CD and short.
165. Find the vertex as efficiently as possible:
(a) y = 37 (x + 3)2 + 5
Answer: if in SF, the vertex is immediate (−3, 5).
(b) y = −2(x − 11)(3 + x)Answer: if in FF, the vertex is half-way between the roots. Here the roots are x = 11,−3, andthus the x-coord of the vertex is x = 4; plug in to get y = −2(4 − 11)(3 + 4) = 98, so the vertexis (4, 98).
(c) y = 12x + 7 − 2x2
Answer: if in GF, the x-coord of the vertex is −b2a = −12
2(−2) = 3; plug-in to get y = 12(3) + 7 −
104 - Class Examples
2(3)2 = 25, so the vertex is (3, 25).
Summary of methods for finding a quadratic’s vertex:
• SF: just look at the shifts
• FF: the x-coordinate of the vertex is half-way between roots, plug in to get y
• GF: use the vertex formula x = −b2a , plug in to get y
166. What is the range of f(x) = 3x2 − 15x + 1 ?Answer: This is a CU parabola, so find the vertex to get the lowest point. The x-coord is −b
2a =156 = 5
2 . Plug back in to get y = 3(25/4) − 15(5/2) + 1 = 75/4 − 75/2 + 1 = −75/4 + 1 = −71/4, sothe domain is [−71/4,∞).
167. If y = −x2 + bx + c has maximum at (4, 5), what are b and c?Answer: We want the vertex to be at (4, 5), so set −b
2a = 4 to get b = 8. Now we know thaty = −x2 + 8x + c, and when x = 4, y should be 5. Solve 5 = −(4)2 + 8(4) + c to get c = −11.
168. How many roots does the quadratic have? Explain why by sketching the graph.
(a) y = 2(x + 3)2
105 - Class Examples
Answer: just one root x = −3, since the vertex is on the x-axis
(b) y = 2(x + 3)(x − 5)Answer: two roots x = −3, 5; the parabola opens up and must have vertex below the x-axis.
(c) y = (x + 3)2 + 1Answer: no roots since the parabola opens up, and the vertex is already above the x-axis.
169. Find the roots as effeciently as possible:
(a) y = (x + 3)(3x − 7)Answer: if in FF, roots are immediate, x = −3, 7/3
(b) y = 8(x + 1)2 − 72Answer: if in SF, set y = 0 and solve for x
0 = 8(x + 1)2 − 72
8(x + 1)2 = 72
(x + 1)2 = 9
x + 1 = ±3
x = −1 ± 3 = 2,−4
(c) y = x2 + 7x − 18Answer: if in GF and easily factored, put in FF: y = (x+9)(x− 2), so the roots are x = −9, 2
106 - Class Examples
(d) y = x2 − 6x − 3Answer: if in GF and not easily factored, use the QF:
x =−(−6)±
√
(−6)2 − 4(1)(−3)
2(1)=
6 ±√
48
2= 3 ±
√12
The quadratic formula (QF) to find the roots of y = ax2 + bx + c
x =−b ±
√b2 − 4ac
2a
The term D = b2 − 4ac under the square root is called the discriminant.
• If D > 0 there are two real roots
• If D = 0 there is one real root
• If D < 0 there are no real roots
170. Use the QF to find the roots:
(a) y = 4x2 − 2x − 1Answer:
x =−(−2)±
√
(−2)2 − 4(4)(−1)
2(4)=
2 ±√
20
8=
2 ± 2√
5
8=
1 ±√
5
4
(b) y = 5x − 3x2 − 4Answer:
x =−5 ±
√
25 − 4(−3)(−4)
−6=
−5 ±√
25 − 48
−6
107 - Class Examples
Since the discriminent b2 − 4ac is negative, there are no roots.
171. Which of these is closest to a root of y = x2 − 6x − 3?
(a) 1 (b) 3 (c) 5 (d) 7 (e) 9
Answer: Use the QF to get x = 6±√
482 ≈ 6±7
2 = −12 , 13
2 . The best answer is x = 7.
Summary of methods for finding a quadratic’s roots:
• FF: set each term to zero
• SF: set y = 0, move the constant, and take the square root of both sides
• GF: factor if possible, otherwise use the QF
172. Find the roots of y = 3x2 + 7x + 2 by 1) the QF, 2) completing the square, and 3) factoring. Verifythat you obtain the same answer by each method.Answer: The QF gives
x =−7 ±
√
49 − 4(3)(2)
6=
−7 ±√
25
6=
−7 ± 5
6= −1/3,−2
Completing the square gives us
y = 3(x2 +7
3x +
49
36) + 2 − 49
12
y = 3(x + 7/6)2 − 25
12
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Setting y = 0 we get (x + 7/6)2 = 25/36 or x = −7/6± 5/6 = −1/3,−2.If we factor, y = (3x + 1)(x + 2), and we obtain the same roots.
173. Find the constant c, such that y = 3x2 + 2x + c has exactly one root.Answer: It has one root if b2 − 4ac = 0, so we solve 22 − 4(3)(c) = 0 to get c = 1
3
To solve polynomial equations:
• move everything to one side, setting it equal to zero
• find the roots (e.g. by factoring or the QF)
174. Solve the equation:
(a) x(x + 1) = 2Answer: x2 + x = 2, so x2 + x − 2 = 0, so (x + 2)(x − 1) = 0 so x = −2, 1
(b) x+1x+3 = 3
x−1
Answer: cross-multiply to get (x+1)(x−1) = 3(x+3), so x2 −1 = 3x+9, so x2 −3x−10 = 0,so (x − 5)(x + 2) = 0, so x = 5,−2
(c) x2 − 1 = 3x
109 - Class Examples
Answer: x2 − 3x − 1 = 0 doesn’t factor, so use QF to get x = 3±√
132
(d) x =√
x + 2Answer: square both sides (whenever you do this, remember to check your answer in the originalequation) to get x2 = x + 2 or x2 − x− 2 = 0, so (x− 2)(x + 1) = 0 and x = 2,−1. Of these, onlyx = 2 works in the original eqn.
175. Find the intersection(s) of y = x + 10 and y = (x − 2)2.Answer: use substitution to get
x + 10 = (x − 2)2
x + 10 = x2 − 4x + 4
x2 − 5x − 6 = 0
(x − 6)(x + 1) = 0
So x = 6,−1 and the intersections are (6, 16) and (−1, 9).
176. Sketch and find the intersection(s) of xy = 3 and y + 1 = 2x.Answer: The first is a hyperbola y = 3
x , and the second is a line. Solve the second eqn for y andsubstitute to get y = 2x − 1, and thus
x(2x − 1) = 3
2x2 − x − 3 = 0
(2x − 3)(x + 1) = 0
x =3
2,−1
110 - Class Examples
Plug in to get the y-vals, so the intersections are (3/2, 2) and (−1,−3).
To find a/the formula for a quadratic with given properties:
(a) Based on the information available, choose which form to use
• if you know the vertex, use SF
• if you know the roots, use FF
(b) Write the chosen form, with a mystery coefficient a out front.
(c) Find a that is consistent with the given conditons:
• If no other point is known, pick a to suit.
• If another point is known, then substitute for x and y, and solve for a.
177. Write the equation of a/the quadratic with the given properties:
(a) vertex at (−1, 7), concave down, tallAnswer: use the SF, and pick a constant consistent with the conditions:
y = −2(x + 1)2 + 7
(b) roots 1,−2; y-int (0,−6)Answer: we choose the FF to get y = a(x − 1)(x + 2); plug-in the other point and solve for a:
−6 = a(0 − 1)(0 + 2)
−6 = −2a
a = 3
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So the desired formula is y = 3(x − 1)(x + 2).
(c) vertex (2,−4); passes thru (5, 11)Answer: we choose the SF to get y = a(x − 2)2 − 4; plug-in the other point and solve for a:
11 = a(5 − 2)2 − 4
15 = 9a
a =5
3
So the desired formula is y = 53 (x − 2)2 − 4.
(d) root at x = 3; vertex at x = 5; y-int at y = −7Answer: We aren’t given complete root or vertex information, but we can deduce that the otherroot is at x = 7. Use the FF to get y = a(x − 3)(x − 7), and solve for a:
−7 = a(0 − 3)(0 − 7)
−7 = 21a
a =−1
3
So the desired formula is y = −13 (x − 3)(x − 7).
178. If a parabola goes through (4, 0), (8, 0), and (1, 7), then it also goes through:
(a) (−1, 5) (b) (0, 10) (c) (5,−1) (d) (6, 0) (e) (7, 2)
Answer: You are given the roots and one other point, so you may derive the formula to be
y =1
3(x − 4)(x − 8)
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Plug in the x values to see that the only option that is on the graph is (5,−1).
Quadratic Word Problems
179. One rectangular room in a museum has a perimeter of 130 feet, and an area of 1000 square feet. Whatare the dimensions of the room?Answer: Draw a picture and label the room’s length (ℓ) and width (w). We set up two eqns:
2ℓ + 2w = 130
ℓw = 1000
Using substitution, we get ℓ = 65−w and then (65−w)w = 1000. Solve this quadratic eqn by movingeverything to one side, factoring, and finding the roots. We get (w − 25)(w − 40) = 0, so w = 25, 40.
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Whichever value we take for w, the other value becomes ℓ.
180. The ASPCA needs to build 5 equally sized dog pens partitioned as in the diagram. They have 120 feetof fence.
L
W dog
dog
dog
dog
dog
(a) Write an equation relating L, W , and the total fencing available.Answer: 2L + 6W = 120 or L + 3W = 60
(b) What values of L and W will maximize the total area?Answer: A = LW = (60 − 3W )W = 60W − 3W 2; the vertex is at W = 10; so L = 30
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(c) How many square feet of space does each dog get?Answer: Each dog gets a pen of size 10 × 6, so 60 square feet.
181. I once punted a football over a large tree so that the path was y = 16− 115 (x− 15)2 yards. Sketch the
path. How high did the punt go? How far did it go?Answer: The parabolic path begins at the y-intercept: (0, 1). The maximum height will occur atthe vertex, which is (15, 16), so the ball goes 16 yards, or 48 feet high.
To find out how far it goes, determine the landing point, which is a root. Solve
0 = 16 − 1
15(x − 15)2
to get x = 15 ±√
240. The larger of these is x = 15 +√
240 ≈ 30.5 yards.
182. A circus act requires a human cannonball be launched from a 72 foot platform. If she would reach amaximum height after 5 seconds, and land after 12 seconds, will she crash into the 150 ft roof of thedomed stadium?Answer: Draw a picture and sketch her height as a function of time. One root is at t = 12, and thevertex occurs at t = 5. Therefore (provide reasoning) the other root is at t = −2, and we use the FF
y = a(t − 12)(t + 2)
Using the other know point (0, 72) we can solve for a. Since 72 = a(0 − 12)(0 + 2) we get a = −3, andtherefore
y = −3(t− 12)(t + 2)
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Her maximum height occurs at the vertex; when t = 5 we plug in to get y = 147, so she comes within3 feet of the roof.
183. A boat takes one hour longer on a 24 mile trip upstream than it does going downstream. The waterspeed of the boat is 10 MPH. How fast is the current?Answer: Remember the formula D = ST and its variants. Let x be the speed of the current. Thenthe trip up takes 24
10−x hours, and the trip back takes 2410+x hours. Therefore:
24
10 + x+ 1 =
24
10 − x
A trick to eliminate the fraction is to multiply through by the common denominator (10 + x)(10− x).
[
24
10 + x+ 1 =
24
10 − x
]
(10 + x)(10 − x)
24(10− x) + (10 + x)(10 − x) = 24(10 + x)
240 − 24x + 100 − x2 = 240 + 24x
0 = x2 + 48x − 100
(x + 50)(x − 2) = 0
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Of the two solutions x = −50, 2, only x = 2 makes sense, so the current is 2 MPH.
184. Suppose a Columbian drug-lord orders you to take a shipment down river. The boat takes one hourlonger on a 24 mile trip upstream than it does going downstream. The water current is 3 MPH. Howlong does the entire trip take?Answer: This is similar to the previous problem, but this time we know the current but not the boatspeed; we’ll call it x. The trip up takes 24
x−3 hours, and the trip back takes 24x+3 hours. Therefore:
24
x + 3+ 1 =
24
x − 3
Solve this equation as follows:
24
x + 3+
x + 3
x + 3=
24
x − 3
27 + x
x + 3=
24
x − 3
(27 + x)(x − 3) = 24(x + 3)
x2 − 153 = 0
The positive solution is x =√
153 ≈ 12.37 MPH. The entire trip takes
24√153 + 3
+24√
153 − 3≈ 4.12 hours
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185. On Planet X, a projectile launched from a 15 foot cliff reaches a maximum height of 27 feet after 2seconds. When does it land?Answer: We can tell that the independent variable is time, t. The y-int is (0, 15), and the vertex is(2, 27). Therefore we use the SF to write
y = a(t − 2)2 + 27
Plug in t = 0 and y = 15 to solve for a. We obtain:
y = −3(t − 2)2 + 27
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It will land when y = 0, so we solve to get t = 2 ± 3 = −1, 5. Obviously t = 5 is the landing time.
186. A cable company currently has 60 thousand subscribers at $40 per month. Market research indicatesthat for each dollar they raise the price, they will lose 5 thousand customers. Similarly, for every dollarthey lower the price, they will gain 5 thousand customers. What price should they charge to maximizerevenue?Answer: Demand is D = 60 − 5(P − 40), and revenue is R = PD = 260P − 5P 2. The maximumrevenue occurs at the vertex, P = −260
−10 = 26. So they should lower their price to 26 dollars.
187. A caveman was taking the moving sidewalk between concourses in an airport. After passing a Geicosign, he became offended, and needless to say, a little T’dO. Suppose the sidewalk took him 6 feet pastthe sign before he turned around and walked back for another look. If he walked 6 feet per second,
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and it took him 4.5 seconds for the round trip (riding past the sign, and walking back to it), how fastwas the moving sidewalk going?
Geico
Answer: Let x be the speed of the sidewalk. As he rides past the sign, he goes D = 6 with S = x.Coming back, D = 6 still, but now S = 6−x (we assume he can walk faster than the sidewalk moves).The sum of the times gives us the equation:
6
x+
6
6 − x= 4.5
[
6
x+
6
6 − x= 4.5
]
x(6 − x)
6(6 − x) + 6x = 4.5x(6 − x)
36 − 6x + 6x = 27x − 4.5x2
4.5x2 − 27x + 36 = 0
x2 − 6x + 8 = 0
(x − 2)(x − 4) = 0
There are two possible solutions: x = 2, 4
188. Demand for NuPont fiber-woven 24 piece sets (with the mini sailboat included) at price P is given byD = 217
P+3 . Supply is S = P − 21. Find the equilibrium price. How many sets can Uncle Rico sell atthat price?Answer: We know that in a free market, the equilibrium will occur when supply equals demand.
217
P + 3= P − 21
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Cross multiply, factor, and find the roots to get P = −10, 28. Of course, only P = 28 makes sense. Itfollows that S = 28 − 21 = 7 sets will be sold.
189. Suppose a volleyball is served from a height of 2 18 meters. It reaches a maximum height of 3 meters
after traveling 7 meters horizontally. If the 2.25 meter high net is half-way along the 20 meter court,does the ball go over the net, and would it land in bounds?Answer: Draw a picture! We use the SF to get y = a(x − 7)2 + 3 and solve for a by plugging in thepoint (0, 17/8)
17/8 = a(0 − 7)2 + 3
−7/8 = 49a
a =−1
56
The trajectory of the ball is modeled by the equation y = −156 (x−7)2 +3. The net is located at x = 10,
so plugging in we get y = −956 + 3 = 2 47
56 > 2.25, so it goes over. The baseline is at x = 20, so we get
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y = −156 (13)2 = −169
56 + 3 < 0, so it would land in.
Circles Revisited
• Put the equation of a circle into standard form to reveal the center and radius.
(a) Group the x and y terms on the left, and move the constants to the right.
(b) Complete the square for the x’s and y’s separately, remembering to compensate onthe right.
• Although a circle fails the VLT and is not a function, the top and bottom halves are.Solving for y, choose the positive/negative square root for the top/bottom halves respec-tively.
• You can find a semi-circle’s domain and range by considering its graph.
190. If it is a circle, find its center and radius.
(a) x2 + 6(x − 1) + y2 = 9 + 2yAnswer:
(x2 + 6x) + (y2 − 2y) = 15
(x2 + 6x + 9) + (y2 − 2y + 1) = 15 + 9 + 1
(x + 3)2 + (y − 1)2 = 25
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So the center is (−3, 1) and r = 5.
(b) x2 + y2 + 4y + 5 = 0Answer: x2 + (y2 + 4y + 4) = −5 + 4, so x2 + (y + 2)2 = −1, which is impossible
(c) 2x2 + y2 = 9Answer: This is not a circle (it’s an ellipse) since the coefficients of x2 and y2 do not match.
191. Write a function for the top half of the unit circle.Answer: x2 + y2 = 1, solving for y we get y2 = 1− x2. Since we want the top half, take the positivesquare root:
y =√
1 − x2
192. Write a function for this semi-circle:
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3
7
Answer: The center is (0, 3) and r = 4. Therefore x2 + (y − 3)2 = 16. Solve for y to get
(y − 3)2 = 16 − x2
y − 3 =√
16 − x2
y = 3 +√
16 − x2
193. Write a function for the bottom half of (x + 2)2 + (y − 1)2 = 49, and find the domain and range.Answer:
(y − 1)2 = 49 − (x + 2)2
y − 1 = −√
49 − (x + 2)2
y = 1 −√
49 − (x + 2)2
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The center is (−2, 1) and r = 7. Therefore the domain is [−9, 5] and the range is [−6, 1].
194. Given this equation of a circle:3x2 + 3y2 + 12y = 30(x − 2)
(a) Find the center and radius:Answer: First note that we can divide thru by 3
x2 + y2 + 4y = 10(x − 2)
Now group the x’s and y’s together, with the constants on the other side:
(x2 − 10x) + (y2 + 4y) = −20
Complete both squares
(x2 − 10x + 25) + (y2 + 4y + 4) = −20 + 25 + 4
(x − 5)2 + (y + 2)2 = 9
We now see that r = 3 and the center is (5,−2).
(b) Write a function that describes the top half of this circle.Answer: Solve for y, taking the positive square root:
(y + 2)2 = 9 − (x − 5)2
y + 2 = +√
9 − (x − 5)2
125 - Class Examples
y = −2 +√
9 − (x − 5)2
(c) Find the domain and range.Answer: Moving left and right 3 units from 5, the domain is [2, 8]. Moving up 3 units from -2,the range is [−2, 1].
195. Sketch the function f(x) = 5 −√
169 − (x + 7)2.
(a) Find the domain.Answer: Since this is the bottom half of a circle centered at (−7, 5) with r = 13, we can go leftand right 13 units from −7 to get [−20, 6].
(b) Find the range.Answer: Go down 13 units from 5 to get [−8, 5]
(c) Find the y-intercept.Answer: Set x = 0 to get f(0) = 5 −
√120 ≈ 5 − 11 = −6
(d) Find the x-intercept(s).Answer:
0 = 5 −√
169 − (x + 7)2
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√
169 − (x + 7)2 = 5
169 − (x + 7)2 = 25
(x + 7)2 = 144
x + 7 = ±12
x = −7 ± 12 = −19, 5
Inequalities Revisited
196. Solve 2x − 5 ≥ 1 − 4x.Answer: Graph both lines to see where left one is higher.Rearrange to get 6x ≥ 6, so x ≥ 1, which you could write [1,∞).
197. Solve the linear inequality
2 < 5 − 3x ≤ 11
Answer: This says that the quantity 5 − 3x is between 2 and 11. First subtract 5 all the way thru
−3 < −3x ≤ 6
Now divide by -3 (remember to flip the inequalities when you multiply/divide by a negative number)
1 > x ≥ −2
Other ways to write the solution are:
−2 ≤ x < 1
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or in interval notation [−2, 1).
198. Solve |x − 2| ≤ 3.Answer: Visualize by sketching the graphs.The distance between x and 2 is LTOET 3, so we get [−1, 5].
199. Solve the absolute value inequality|2x + 5| < 13
Answer: We have a leash, so there is only one piece to the solution. We have the sandwich inequality:
−13 < 2x + 5 < 13
−18 < 2x < 8
−9 < x < 4
Which we could write as (−9, 4).
200. Solve the absolute value inequality|2x + 5| > 13
Answer: We have a restraining order, so there are two pieces to the solution. Two separate inequal-ities:
2x + 5 > 13 2x + 5 < −13
2x > 8 2x < −18
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x > 4 x < −9
Which we could write as (−∞,−9) ∪ (4,∞)
To solve polynomial inequalities:
(a) Move everything to one side.
(b) Find the roots, e.g. by factoring.
(c) Plot the roots on the number line.
(d) Label each interval with a + or −. Use one of these methods:
• Plug in a test point from each interval to see if the expression is positive or negative.
• Label the right-most interval with the sign of the leading coefficient.Work your way left, switching signs for each term with an odd multiplicity.
(e) Write the solution interval: positives if >, negatives if < zero.
201. Solve x2 ≥ 2x.Answer: Sketch the graph. If we divide by x, then we get x ≥ 2, which is only part of the solution.The problem is that we must assume x > 0 to divide by x. Proceed the safe way instead.
x2 − 2x ≥ 0
x(x − 2) ≥ 0
+ 0 - 2 +
We want the “plus” intervals, so the answer is (−∞, 0] ∪ [2,∞).
129 - Class Examples
202. Solve (x + 1)2 ≥ 4.Answer: Sketch the graphs and visualize the two pieces to the solution.
x2 + 2x + 1 ≥ 4
x2 + 2x − 3 ≥ 0
(x + 3)(x − 1) ≥ 0
+ −3 - 1 +
We want the “plus” intervals, so the answer is (−∞,−3] ∪ [1,∞).
203. Solve x2 ≤ 3x + 10.Answer: Sketch both sides to get a graphical picture of what this inequality means. Move everythingto one side, then factor.
x2 − 3x − 10 ≤ 0
(x − 5)(x + 2) ≤ 0
Mark the roots -2 and 5 on a number line. Since the leading coefficient is positive, mark the right-mostinterval with a plus sign. Both roots have odd multiplicity, so switch signs at each root.
+ −2 - 5 +
The ≤ inequality says we want the “minus” intervals, so the answer is [−2, 5].
204. Solve the inequality −45 (x − 3)2(x − 7) ≥ 0.
Answer: It is already factored, so look at the number line:
+ 3 + 7 -
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Notice that the sign starts negative to the right, and doesn’t change at x = 3. The answer is (−∞, 7].
205. Suppose your business profit is a function of price: Y = 2P (10 − P ).
(a) What price will maximize your profit? What will your profit be at that price?Answer: This is a concave down quadratic, so the max occurs at the vertex, halfway betweenthe roots 0 and 10. So P = 5 is the optimal price, and your profit would be Y = 2 ·5(10−5) = 50.
(b) What range of prices will guarantee you a profit of at least Y = 42?Answer: The keywords “at least” indicate that we need to solve Y ≥ 42.
2P (10 − P ) ≥ 42
P (10 − P ) ≥ 21
10P − P 2 − 21 ≥ 0
−(P 2 − 10P + 21) ≥ 0
−(P − 3)(P − 7) ≥ 0
Use the number line method to obtain
- 3 + 7 -
So the answer is [3, 7], which means you can charge anywhere between $3 and $7. This range may
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correspond to sale and regular prices.
Rational Functions
It has been said that man is a rational animal.All my life I have been searching for evidence which could support this.
– Bertrand Russell
A rational function is the ratio of two polynomials.We refer to the numerator and denominator by N and D respectively.
206. Let f(x) = (x−9)(x+2)3−5x .
(a) Find deg(N) and deg(D).Answer: deg(N) = 2 and deg(D) = 1
(b) Find the y-intercept.Answer: f(0) = −18
−3 = 6
(c) Find the roots.Answer: Solve f(x) = 0. To set a fraction equal to zero, you cross-multiply to get rid of thedenominator. Therefore we have 0 = (x − 9)(x + 2), so the roots are x = 9,−2.
(d) Find the domain.
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Answer: set D 6= 0 to get 3 − 5x 6= 0, so x 6= 35
207. If f(x) = 4x+9x2−12 then:
(a) Find f(1).Answer: plug in x = 1 to get f(1) = −13
11
(b) Solve f(x) = 1.Answer: solve 1 = 4x+9
x2−12 to get x2 − 12 = 4x + 9, so x2 − 4x − 21 = (x − 7)(x + 3) = 0 andx = 7,−3
208. Demonstrate the concept of a common denominator by computing 16 + 8
15 .Answer:
1
2 · 3 +8
3 · 5 =1 5 + 8 2
2 · 3 · 5 =21
30=
7
10
209. Write the expression 1n − 1
n+1 with a common denominator.Answer:
1 n+1 − 1 n
n(n + 1)=
1
n(n + 1)
For example, 16 − 1
7 = 142 .
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210. Get a common denominator to simplify the expression: xx+1 − 3
x2−x−2 .Answer:
x
x + 1− 3
x2 − x − 2=
x
x + 1− 3
(x + 1)(x − 2)
=x x-2 − 3 1
(x + 1)(x − 2)
=x2 − 2x − 3
(x + 1)(x − 2)
=(x − 3)(x + 1)
(x + 1)(x − 2)
=x − 3
x − 2
211. Write with a common denominator.
1 + 2x−1 − 3
x2 − 5x
Answer:
1 +2
x− 3
x(x − 5)
1 x(x-5) + 2 x-5 − 3
x(x − 5)
x2 − 5x + 2x − 10 − 3
x(x − 5)
x2 − 3x − 13
x(x − 5)
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212. Write with a common denominator, then find the roots.
y =x + 1
x − 2− 10(x2 − 4x + 4)−1
Answer:
y =x + 1
x − 2− 10
(x − 2)2
=(x + 1) x-2 − 10 1
(x − 2)2
=x2 − x − 12
(x − 2)2
=(x − 4)(x + 3)
(x − 2)2
To set a fraction equal to zero, just set the numerator to zero. The roots are x = −3, 4.
213. Solve x+3x(x−1) + 4
1−x = 6
Answer: First notice that 1 − x = −(x − 1) and rewrite:
x + 3
x(x − 1)− 4
x − 1= 6
Get a common denominator:(x + 3) 1 − 4 x
x(x − 1)= 6
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−3x + 3
x(x − 1)= 6
−3(x − 1)
x(x − 1)= 6
−3
x= 6
x = −1/2
Another approach is to multiply through the equation by the common denominator at the start to getrid of the fractions.
214. Find the roots and domain of f(x) = xx+1 + 6
x−1 .Answer:
f(x) =x x-1 + 6 x+1
(x + 1)(x − 1)
=x2 + 5x + 6
(x + 1)(x − 1)
=(x + 2)(x + 3)
(x + 1)(x − 1)
Therefore the roots are x = −2,−3 and the domain is x 6= ±1.
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215. Let y = (x − 1)−1 + 2.
(a) Sketch the graph.Answer: Write as y = 1
x−1 +2. This has the same shape as 1/x, but shifted right 1 and up 2.
(b) Get a common denominator to write as one fraction.
Answer: y =1 1 +2 x-1
x−1 = 2x−1x−1
(c) Find deg(N) and deg(D).Answer: deg(N) = 1 and deg(D) = 1
(d) Find the y-intercept.Answer: plug in x = 0 to get y = 1
(e) Find the x-intercept.Answer: Solve 0 = 2x−1
x−1 to get 2x − 1 = 0, so x = 12 .
(f) What is the horizontal asymptote (HA)?Answer: We see from the graph that it is the line y = 2.
(g) What is the vertical asymptote (VA)?Answer: We see from the graph that it is the line x = 1.
(h) Find the equation of the secant line between points where x = −1 and x = 2.Answer: The points are (−1, 3/2) and (2, 3). The slope is
m =3 − 3/2
2 − −1=
3/2
3=
1
2
Using the point (2, 3), we get the secant line:
y = 3 +1
2(x − 2)
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The horizontal asymptote (HA) is a horizontal line that the graph gets (and stays) arbitrarilyclose to as x → ±∞. Remember the L’s that the HA characterizes:
• the long term behavior
• the limiting value of y as x gets large (also called the limfinity)
• where the graph levels out
The HA of a rational function is determined by its leading terms. There are three cases:
• deg(N) = deg(D): the HA is the ratio of the leading coefficients.
For example y = 3x2
2x2+1 has HA determined by 3x2
2x2 , so the HA is y = 1.5.
• deg(N) < deg(D): the HA is automatically y = 0, since the denominator gets much biggerthan the numerator, driving the fraction to zero.For example y = 3x
2x2+1 has HA of y = 0.
• deg(N) > deg(D): there is no HA since the numerator grows much bigger than thedenominator.For example y = 3x2
2x+1 has no HA.
216. Find the HA:
(a) y = x2+15x+3
Answer: no HA(b) y = (4x+1)2
2(x+1)(x−1)
Answer: y = 162 = 8
(c) f(t) = x−2(x − 3)Answer: y = 0
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A vertical asymptote (VA) is a vertical line that the graph approaches as x gets near a fixedvalue. The VA acts like a barrier or force field that the graph cannot cross. Instead the functionshoots up to ±∞. There may be more than one VA for a function. Division by zero causesVA’s. Therefore, to find the VA of a rational function, follow these steps: .
• Write with a common denominator and factor completely.
• Cancel terms if possible.
• Any x value that makes D = 0 is a VA.
217. Find the VA(s):
(a) y = x+52(x−3)
Answer: x = 3
(b) y = x2−49x−14−x2
Answer: y = (x+2)(x−2)−(x−2)(x−7) , so the VA is x = 7
(c) y = 1x2−1
Answer: x = ±1
(d) y = x−3x−4 − 3
x2−5x+4Answer:
y =x − 3
x − 4− 3
(x − 1)(x − 4)=
(x − 3)(x − 1) − 3
(x − 1)(x − 4)=
x2 − 4x
(x − 1)(x − 4)=
x(x − 4)
(x − 1)(x − 4)=
x
x − 1
so the VA is x = 1.
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218. Let f(x) = (x+1)2(2x−7)(x−2)(x2+5)2
(a) Find the y-intercept.Answer: f(0) = −7
−50 = 0.14
(b) Find the roots.Answer: When setting a fraction to zero, only the numerator must be zero, so we get x =−1, 7/2.
(c) What is the HA?Answer: Since deg(N) < deg(D), the HA is y = 0.
219. Let f(x) = 3x(6−3x)2x2−8 .
(a) Factor completely.
Answer: f(x) = 9x(2−x)2(x+2)(x−2)
(b) Find the HA.Answer: The ratio of leading terms is y = −9
2 .
(c) Find the VA(s).Answer: When x 6= 2, we can simplify to f(x) = −9x
2(x+2) . Therefore x = −2 is a VA.
220. Given the rational function:
f(x) =x + 14
(x − 1)(3x + 2)+
2x + 1
x − x2
(a) Get a common denominator and simplify completely.Answer: First factor:
x + 14
(x − 1)(3x + 2)+
2x + 1
−x(x − 1)
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(x + 14) -x + (2x + 1) 3x+2
−x(x − 1)(3x + 2)
−x2 − 14x + 6x2 + 7x + 2
−x(x − 1)(3x + 2)
5x2 − 7x + 2
−x(x − 1)(3x + 2)
(5x − 2)(x − 1)
−x(x − 1)(3x + 2)
5x − 2
−x(3x + 2)
(b) What are the root(s)?Answer: x = 2/5; note that f(1) is not defined, so x = 1 is not a root.
(c) What is the y-intercept?Answer: plugging in x = 0 is undefined, so there is no y-int
(d) What is the HA?Answer: since deg(N) < deg(D), the HA is y = 0
(e) Find the VA(s)?Answer: x = 0 and x = −2/3
141 - Class Examples
(f) Solve the equation f(x) = 1.Answer: 5x−2
−x(3x+2) = 1, so cross-multiply to get 5x − 2 = −3x2 − 2x. Rearranging we obtain
3x2 + 7x − 2 = 0, and by the QF, the solutions are
−7 ±√
49 − 4(3)(−2)
6=
−7 ±√
73
6
To sketch a rational function’s graph:
• Find and plot the asymptotes.
• Plot points as necessary, e.g. the intercepts.
• Connect the points, using the asymptotes as guides.
221. Sketch the graph of:
(a) y = 1(x−2)2
Answer: The VA is at x = 2, and the HA is y = 0.
(b) f(x) = x1+x2
Answer: Note that the HA is y = 0, and there are no VA’s. Plot a few points; it will help tonotice the function is odd, so the graph is symmetric with respect to the origin. This graph is
142 - Class Examples
called a “serpentine.”
Interestingly, this graph does cross its HA, contradicting a popular urban legend.
(c) y = 11+x2
Answer: Again, the HA is y = 0 and there are no VA’s. The graph stays above the axis in abell shape. This graph is named the “Witch of Agnesi”
(d) y = x2
x2+1Answer: The HA is y = 1 and there is no VA. We get an upside-down bell shape that goesthrough the origin. Show that this is equal to y = 1 − 1
1+x2 .
222. Blaze Johnson’s 40 yard dash time is y = 17t+104t+2 , where t is the number of years of training.
(a) How many seconds did Blaze take to run the 40 yard dash when he began training?
143 - Class Examples
Answer: plug in t = 0 to get y = 10/2 = 5 seconds
(b) After one year, how long did it take?Answer: plug in t = 1 to get y = 27/6 = 4.5 seconds
(c) To become a 1st round NFL draft pick, Blaze needs to run a 4.4 second 40. How many years oftraining will that take?Answer: Solve 17t+10
4t+2 = 4.4 = 22/5.
5(17t + 10) = 22(4t + 2)
85t + 50 = 88t + 44
6 = 3t
t = 2
(d) Theoretically, what is the limit to how fast he can run the 40 yard dash?Answer: “limit” indicates we want the HA, which is the ratio of leading coefficients: y = 17/4 =4.25
223. In a certain town there are 80 democrats and 20 republicans. Each year 5 people move in, 3 republicansand 2 democrats.
(a) Write a function for the percentage of republicans after t years.Answer: After t years there will be 20 + 3t republicans, and a total of 100 + 5t people, so thepercentage is
P =20 + 3t
100 + 5t
144 - Class Examples
(b) What percentage of population is republican after 10 years?Answer: plug in t = 10 to get P = 50/150 which is about 33%
(c) How many years later are the parties equal in number?Answer: Solve P = .5
20 + 3t
100 + 5t=
1
2
2(20 + 3t) = 100 + 5t
40 + 6t = 100 + 5t
t = 60
(d) What is the long-term republican percentage?Answer: Find the HA using the leading coefficients: y = 3
5
224. A congressman is proposing a “progressive” income tax rate that will follow the formula:
R(x) =36x − 108
x + 17
where x is a person’s income (in thousands of dollars), and R(x) is the percentage he must pay intaxes. If R(x) is negative, then the person owes no tax.
145 - Class Examples
(a) How much can your kid make mowing lawns in the summer without having to pay taxes?Answer: Solve 36x− 108 ≤ 0 to get x ≤ 3, so he can make $3,000 tax-free.
(b) If you make $43,000, how much tax will you pay?
Answer: Your rate is R(43) = 36(43−3)43+17 = 24 percent. He owes 24% of $43,000, which is $10,320.
(c) If your neighbor pays a 30% tax rate, what was his yearly income?Answer: Solve R(x) = 30:
36(x − 3)
x + 17= 30
36(x − 3) = 30(x + 17)
36x − 108 = 30x + 510
4x = 618
x = 103
So he made $103,000.
(d) What is the maximum tax rate that anyone (e.g. Howard Stern) would ever have to pay?Answer: The HA is R = 36
225. Suppose the earth’s population (in billions) is modeled by the rational function:
P =2(29t + 9)
4t + 3
where t is in hundreds of years since 2000 A.D.
146 - Class Examples
(a) What was the population in the year 2000?Answer: plug in t = 0 to get P = 6 billion
(b) According to the model, what is the best estimate for the year when the population will reach 10billion people?
i. 2055 ii. 2066 iii. 2077 iv. 2100 v. never
Answer: Solve 10 = 2(29t+9)4t+3 to get
10(4t + 3) = 2(29t + 9)
40t + 30 = 58t + 18
12 = 18t
so t = 2/3, and the year 2066 is closest
(c) At what value will the Earth’s population level out?Answer: HA is y = 58
4 = 14.5 billion
226. Suppose your blood alcohol level t hours after drinking n shots of liquor is
f(t) =nt
2(5 + 8t2)
Find all asymptotes, then plot the graph, sketching as many points as necessary. As time passes, whathappens to the alcohol level?
147 - Class Examples
Answer: It starts at zero, rises to a maximum, and then tapers off to zero as you sober up.
Exponential Functions
The greatest shortcoming of the human race is our inability to understand the exponential function.– Albert Bartlett
An exponential equation has a variable in the exponent. To solve,
• Make the bases equal, so that you can compare the exponents apples-to-apples.
• Set the exponents equal and solve.
227. Solve the exponential equations:
(a) 49x+3 = 7x
Answer: Get a common base, then set the exponents equal.
72(x+3) = 7x
2(x + 3) = x
2x + 6 = x
x = −6
(b) 91+x = 3x2−1
Answer:
32(1+x) = 3x2−1
148 - Class Examples
2 + 2x = x2 − 1
x2 − 2x − 3 = 0
(x − 3)(x + 1) = 0
x = 3,−1
(c) (.25)x = 85x−2
Answer:
(1/4)x = 85x−2
2−2x = 23(5x−2)
−2x = 15x − 6
17x = 6
x =6
17
228. Make a table of points and sketch the graph of:
(a) y = 9x
Answer: This is an example of exponential growth.
149 - Class Examples
(b) y =(
23
)t
Answer: This is an example of exponential decay.
We will study exponential functions of the general form:
y = abt + c
• c is the horizontal asymptote (vertical shift)
• a scales the graph, so that a + c is the y-intercept
• b is the multiplicative factor by which y − c changes as t increases by one
• If b > 1 then it grows away from the HA; if 0 < b < 1 then it decays toward it.
To sketch the graph of an exponential function, plot the HA and two points (esp. the y-int).
229. Sketch the graphs of:
(a) y = 2t
(b) y = 2t − 8
(c) y = 3 · 2t
(d) y = 5 − 2t
(e) y = (.8)t
(f) y = 7 − 5(.8)t
230. Sketch y = 2−t and explain the effect of the negative exponent.
Answer: Since 2−t =(
12
)t, the graph decays (horizontal flip). People often use negative exponents
150 - Class Examples
to avoid fractions.
To find the equation of an exponential function y = abt + c, follow these steps:
• Let c be the HA. If y grows/decays by a fixed factor or percentage, then the HA is zero.
• Plug in the y-int to find a.
• Plug in any other point to find b.
Remember the “CAB” order.
231. Find the exponential function that fits the data:
(a) thru (0, 3) and (1, 7) with HA y = 1Answer:
y = abt + 1
3 = ab0 + 1
a = 2
y = 2bt + 1
7 = 2b1 + 1
b = 3
y = 2(3)t + 1
(b) thru (0,−1) and (1, 0) with HA y = 3Answer:
y = abt + 3
151 - Class Examples
−1 = ab0 + 3
a = −4
y = −4bt + 3
0 = −4b1 + 3
b = .75
y = −4(.75)t + 3
(c) thru (0, 0) and (2, 2) with HA y = −1Answer:
y = abt − 1
0 = ab0 − 1
a = 1
y = bt − 1
2 = b2 − 1
b = (3)1/2
y = 3t/2 − 1
152 - Class Examples
232. A bank offers 5% APY (annual percentage yield) on a savings account. If you deposit $1000, write aformula for the value of your investment after t years.Answer: Every year, the bank pays you 5% of your previous balance, which is added to your accountto earn interest in subsequent years. This is called compounded interest. Let y be the value of youraccount. Each year it is multiplied by
100% + 5% = 105% = 1.05
t y
0 10001 10502 1102.503 1157.6320 2653.30
In general, we can jump to any future time t by utilizing the exponent notation:
y = 1000(1.05)t
Note that true exponential growth can’t continue indefinitely.
233. Suppose a legendary investor (e.g. Warren Buffet) earned 24% compounded annually on his invest-ments. If he started with 50 thousand dollars, find a formula for his account’s value t years later. Howmuch was it worth after 35 years?Answer: The HA is y = 0, so we have an exponential function of the form
y = abt
Since the y-intercept should be 50, we get
y = 50bt
and since he earns 24% per year (on top of the original 100%), the growth rate factor is b = 1.24.
y = 50(1.24)t
When t = 35 we get
y = 50(1.24)35 ≈ 93053
thousand, which is about 93 million dollars.
153 - Class Examples
In this example, the exponential function is a model of reality. Despite the fluctuations, the exponentialfunction accurately describes the general trend.
234. Suppose a Honda Accord, purchased new for $23 thousand, depreciates by 15 percent per year. Writean exponential function for the car’s value after t years, then predict the resale value if you trade it inafter five years.Answer: The multiplication factor is 100% - 15% = 85%.
y = 23(.85)t
When t = 5, the car is worth 23(.85)5 ≈ 10.2 thousand.
235. In 1990, Las Vegas had a population of 258 thousand. By 2005, it had swelled to 535 thousand. Ifexponential growth continues at this rate, predict the population in 2020.Answer: Imagining backward in time, the HA is y = 0, so
y = abt + 0
The first time in the problem should be identified with t = 0, so the y-int is (0, 258) and
258 = ab0
So a = 258 andy = 258bt
Now use the other point (15, 535)535 = 258b15
535
258= b15
b =
(
535
258
)1/15
154 - Class Examples
Therefore the exponential eqn is
y = 258
(
535
258
)t/15
The year 2020 corresponds to t = 30, so we get
y = 258
(
535
258
)30/15
≈ 1109
thousand people.
236. A can of Mountain Dew contains 55 mg of caffeine. Suppose that two hours after chugging it, you have11 mg left in your bloodstream. How much was left after 30 minutes?Answer: Sketch the graph with points (0, 55) and (2, 11). Eventually all the caffeine would beeliminated, so the HA is y = 0. Therefore we have c = 0 and
y = abt + 0
Now plug in the y-int to get
55 = ab0
So a = 55 and
y = 55bt
Finally plug in the other point:
11 = 55b2
1
5= b2
b =
(
1
5
)1/2
Therefore the formula is
y = 55
(
1
5
)t/2
To answer the question, plug in t = 12 to get
y = 55
(
1
5
)1/4
≈ 36.8
155 - Class Examples
237. Suppose you put a room temperature 70◦ can of Pepsi into a 30◦ freezer, and that after 30 minutesit had cooled to 50◦. Write a formula for the temperature of the Pepsi after being in the freezer for tminutes.Answer: This time the HA is not zero, because the drink can’t get any colder than the freezer itself.The HA is y = 30, so
y = abt + 30
Use the y-int to get a.
70 = ab0 + 30
a = 40
Now we have
y = 40bt + 30
Using the other point
50 = 40b30 + 30
20 = 40b30
1
2= b30
b =
(
1
2
)1/30
Therefore the formula is
y = 40
(
1
2
)t/30
+ 30
Note that since this is decay, we could use a negative exponent:
y = 40 · 2−t/30 + 30
156 - Class Examples
238. Your car can go 0-60 MPH in 7 seconds, and has a top speed of 140 MPH. Use an exponential modelto predict the speed after 10 seconds.Answer: The HA is 140 and we have points (0, 0) and (7, 60). Find ”c-a-b” in succession:
y = abt + 140
0 = ab0 + 140
y = −140bt + 140
60 = −140b7 + 140
−80 = −140b7
4/7 = b7
b = (4/7)1/7
y = −140(4/7)t/7 + 140
Now plug in t = 10:y = −140(4/7)10/7 + 140 ≈ 77
239. A radioactive isotope has a half-life of 9 hours. How much will be left after 2 days?Answer: Half of the sample decays during each half-life. Assume we start with 100%, then after 9hours there is 50% left, after 18 hours there is 25%, after 27 hours there is 12.5%, and so on. Evidently,the HA is y = 0. Use the points (0, 100) and (9, 50) to get the equation:
y = abt + 0
157 - Class Examples
y = 100bt
50 = 100b9
0.5 = b9
b = (1/2)1/9
y = 100(1/2)t/9
Plug in t = 48 hours to get
y = 100(1/2)48/9 ≈ 2.5
percent.
240. Suppose that if you don’t study for a test, you will get a 35. If you study two hours you’ll make a 60.Use an exponential model to predict your grade if you study 5 hours.Answer: The benefits of studying illustrate diminishing returns. The HA is y = 100, and we can usepoints (0, 35) and (2, 60).
y = abt + 100
35 = ab0 + 100
−65 = a
y = −65bt + 100
60 = −65b2 + 100
−40 = −65b2
8/13 = b2
b = (8/13)1/2
y = −65(8/13)t/2 + 100
Plug in t = 5 to get
y = −65(8/13)5/2 + 100 ≈ 81
158 - Class Examples
241. Suppose we start a rumor in class; say 25 people hear it. After one hour, each of us tell three otherpeople. Suppose there are 2000 people on campus. Create two models for the spread of the rumor,and sketch the graphs.Answer: Known points are (0, 25) and (1, 100). First we assume pure exponential growth and get.
y = 25(4)t
However, if we wish to use the HA of y = 2000 we get:
y = abt + 2000
25 = ab0 + 2000
a = −1975
y = −1975bt + 2000
100 = −1975b1 + 2000
1900 = −1975b
b =76
79
y = −1975
(
76
79
)t
+ 2000
These two models are drastically different. Reality is probably somewhere between (a logistic model).
1 - Homework
Numbers, Sets
1. Given the set S = {√
7, 0, 23 ,√−4, π
2 ,−7, 3√−8, 1.357, 2.81 }
(a) Which are integers?
(b) Which are irrational?
(c) Which one is not real?
(d) Plot the real numbers in S on the number line.
2. Write out the set of the first five prime numbers.
3. Write the set of all real numbers except for zero and pi.
4. Given this set:
3 11
(a) Write it with interval notation. (b) Write it as an inequality.
5. Write the set described by −1 ≤ x < 7 in interval notation.
6. Consider the interval set S = (−∞, 5].
(a) Write S as an inequality.
(b) Describe S with an English sentence.
7. Define the set S = (−∞, 1] ∪ (5, 7].
(a) Sketch S on the real number line.
(b) Which of these are true?
i. 2 ∈ S ii. 2π ∈ S iii. 5 ∈ S iv. −1763.912 ∈ S
8. Using interval notation, describe the set of real numbers that are either between 1 and 3 inclusive, orgreater than 10.
9. Write the set of real numbers in interval notation, then as an inequality.
(a) open set of numbers between 2 and 7
(b) closed set of numbers between 2 and 7
10. Consider the set of all positive real numbers.
(a) Write this set as an interval.
(b) Write this set as an inequality.
(c) Is the set open or closed?
(d) What is the smallest positive real number?
2 - Homework
Absolute Value
11. What is the distance between −3 and 12?
12. If |x − 5| = 3, then what are the possible values of x?
13. If |t + 2| = 7, then what are the possible values of t?
14. The statement |x + 3| ≥ 1 could be read as:
The between and is .
15. |x − 8| < 3 describes the (open/closed) set of numbers within units of .
16. Describe the closed set of numbers at least one unit from 3 using an absolute value inequality.
17. Describe the inequality |x + 2| ≤ 3 as an interval.
18. A recent Gallup poll found that Pedro has 61% of the vote, and is accurate to within ±3%. If P is theactual percentage that will vote for Pedro, write the set of possible P as an absolute value inequality.
19. Consider the closed set of real numbers within 3 units of 1.
(a) Describe the set with a regular inequality.
(b) Describe the set with an absolute value inequality.
(c) Describe the set as an interval.
20. Use a single absolute value inequality to describe the set (−1, 7).
21. Use a single absolute value inequality to describe the set (−∞, 4] ∪ [10,∞).
22. Does | − x| = x for all possible values of x? If not, find a counter-example.
Arithmetic, Algebra, Expressions
23. Compute 56 − 0.7 and write the answer as a reduced fraction.
24. Simplify the expression completely:2(6 − x) − 3(5 − 2x)
25. Multiply out and simplify completely by collecting like terms.
(x + 5)2 − 3(2x − 3)(1 − x)
26. Is the statement generally True or False?
(a)√
x2 + 4 = x + 2
(b) xy = 0 if and only if x = 0 or y = 0
(c) 10 = 0
(d) aa = 1 if a 6= 0
(e) −(1 − x) = x − 1
(f) (x + 2)(x − 4) = x(x + 2) − 4(x + 2)
(g) x3 = 1
3x
(h) 2(
xy
)
= 2x2y
(i) (x + 3)2 = x2 + 9
(j) |1 − x| = |x − 1|
3 - Homework
Equations
27. Which of these are solutions to the equation√
2x + 1 = x(x+1)10 + 1
(a) -1 (b) 0 (c) 1 (d) 4
28. If x2 = 49, then list all possible values of x.
29. Solve the equation for x. List all solutions.
(a) −3(x − 5) = 7 − x
(b) 1x = 2
3x+5
(c) (x − 2)2 = 9
(d) |5 + 2x| = 11
(e) |3x − 1| = 7
(f) x(x − 3)(2x + 5) = 0
30. Supose 2y − 3x = 12
(a) Solve for y.
(b) Solve for x.
(c) What is y when x = 6?
(d) What is x when y = 3?
Inequalities
31. Solve the linear inequalities:
(a) 2x − 1 < 5x + 8 (b) −5x ≤ 20 (c) 12x + 1 ≥ 7x
32. Solve the sandwich inequalities:
(a) 3 ≤ 5x + 8 ≤ 28 (b) −1 < 3 − 2x ≤ 7.
33. Solve the absolute value inequalities:
(a) |x + 4| ≤ 2
(b) |x − 7| ≥ 3
(c) |3x + 1| < 5
(d) |2x − 3| > 7
34. Suppose eggs need to be kept between 2 and 7 degrees Celcius. If
C =5
9(F − 32)
what is the safe temperature range in Farenheit?
Exponents
35. Compute the following:
4 - Homework
(a) 3√−8 (b) 27
23 (c) 20
9−2 (d) 16.25
36. Rationalize the denominator of√
3√8.
37. Simplify the expression, removing all radicals, parentheses, and negative exponents.
(a) 1(√
x)−4 (b)(
27x6y
y−2 3√
x2
)1/3(c)
(
x−3tt−2
√x
)−1
38. Is it true for all real numbers x for which both expressions are defined?
(a) x−2 = x1/2
(b)√
x4 = 1
2
√x
(c)√
x2 = |x|
(d) 2x−2 = 14x2
(e) 1x+x−1 = 1+x
x
(f) 1+x−1
1−x−1 = x+1x−1
39. Remove the fraction, and write as a sum of terms involving fractional powers of x.
x2 + 1√x
40. Suppose a car’s braking distance in feet, from a speed of s MPH, is approximated by
D =(s
5
)7/3
How much further does it take to stop from 70 MPH than it would from 55 MPH? (use a calculator)
41. Write 0.0003172 in scientific notation.
42. Estimate how many seconds you have been alive, and write it in scientific notation with 3 significantdigits.
43. Suppose C-N football is ranked {3, 6, 8, 9} by various polls. Compute the arithmetic and geometricmean rankings.
Word Problems
44. In 4 years, Kip will be twice as old as he was 13 years ago. How old is Kip now?
45. A rental car costs $50, plus 20 cents per mile driven over 100 miles.
(a) If x is the number of miles driven, write an expression for the total cost.
(b) If you drive 200 miles, how much does it cost?
(c) If your bill comes to $82, how far did you drive?
46. Bertha scored 15, 7, and 23 points in her first three basketball games. How many points must shescore in the fourth game to bring her average up to 16 points per game?
47. The time it takes to fall h feet under the force of gravity g is given by the formula
t =
√
2h
g
5 - Homework
(a) On earth, g = 32. How long does it take to fall 50 feet?
(b) On the moon, g = 5.2. How long would it take to fall 50 feet there?
(c) You observe a stick go over a waterfall. If it takes 3 seconds to reach the bottom, how high arethe falls?
48. If the area of a pizza is 64π, what is its circumference?
49. An exerciser walks up a hill and then runs back down. She runs twice as fast as she walks, and for theentire trip she averages 5 MPH. How fast does she run?
50. Suppose you drive m miles on g gallons of gas.
(a) Write an expression for your gas mileage.
(b) If you drive 390 miles on 15 gallons of gas, what is your mileage?
(c) How many gallons of gas did you use if you got 30 MPG on a 240 mile trip?
Cartesian Plane
51. Find the distance between (3,−2) and (8, 10).
52. Which is closer to P (1,−7); Q(2, 0) or R(4,−1)?
53. Which of the points, (3, 2) or (1, 0), is closer to the point (5,−1)?
54. Consider the points P (−1, 5) and Q(4,−7).
(a) Plot both points, and tell what quadrant they’re in.
(b) What is the distance between P and Q?
(c) Find the midpoint of ~PQ.
(d) How far is Q from the origin?
(e) How far is Q from the y-axis?
55. A wheelchair ramp is to be built to rise 4 feet, with a horizontal base of 20 feet. Which is the bestapproximation of the total length of the ramp?
(a) 20.4 (b) 21.4 (c) 22 (d) 24
56. If a baseball runner is half-way between first and second base, how far is he from home plate?(assume the baseball diamond is square of 90 feet per side)
57. If the point (x, 3) is 5 units from (6,−1), what are the possible values of x?
58. Sketch the triangle with corners at P (2, 2), Q(3,−1), and R(−3,−3).
(a) Find the length of each side.
(b) Is this a right triangle?
(c) Find the area of the triangle.
59. Use the Pythagorean theorem to estimate how far away can you see a storm cloud that is 2 miles abovethe ground. (assume the Earth’s radius is 4000 miles, and use a calculator)
6 - Homework
Lines
60. Consider the line y = 3−2x5 .
(a) What is the slope?
(b) If x = 4, what is y?
(c) If y = 3, what is x?
61. If 4x − 2(y − 1) = 8,
(a) Solve for y.
(b) What is the slope of this line?
62. Given the two points P (−1,−5) and Q(2, 1).
(a) Find the slope between P and Q.
(b) Find the equation of the line between P and Q.
(c) What is the y-intercept of this line?
(d) What is the x-intercept of this line?
63. Which of these lines go through the origin?
(a) x − 3y = 0 (b) y = 2x + 1 (c) x = y (d) 2(x − 3) = y − 6
64. The line 7x = 2y − 3 goes through which of these points?
(a) (0, 0) (b) (3, 12) (c) (−1,−2) (d) (2, 5)
65. Write the equation of the line that:
(a) goes through (1, 3) and (7, 5).
(b) passes through (2,−1) with slope 3.
(c) has slope 15 and y-intercept −2?
(d) has x-intercept 12 and y-intercept 4
66. For each line, find the x-intercept, y-intercept, and slope.
(a) y = 12x − 3
(b) y = 2 + 3(x − 1)
(c) 6x − 2y = 12
(d) y = 2
67. A line has slope -2 and x-intercept (1, 0).
(a) Sketch the line.
(b) What is the y-intercept?
(c) Find the equation of the line.
68. Suppose a line passes through (1,−3) and (3, 5).
7 - Homework
(a) What is the slope?
(b) Find the equation of the line.
(c) Find the intercepts.
(d) Does this line also pass through (4, 9)?
69. Use the intercepts to sketch the line x2 − y
5 = 1.
70. The line that goes through (−1, 8) and (k, 0) has a slope of m = −2. Find the value of k.
71. If y is proportional to x, and y = 3 when x = 5, then find a formula for y in terms of x.
72. Suppose you get pulled over for speeding, and the ticket amount is $35 plus $5 for each mile an hourover the speed limit.
(a) Write a formula for the ticket amount.
(b) Is your ticket amount proportional to speed?
(c) If you were going 85 in a 70 MPH zone, what is your ticket amount?
(d) If your ticket came to $100, how much over were you going?
73. Are the lines 2x − 3y = 7 and 3x = 5 − 2y parallel, perpendicular, or neither?
74. Find the equation of the line that:
(a) Passes through (4,−2) and has no y-intercept.
(b) Passes through the origin, parallel to y = 4x − 7.
(c) Passes through (1, 5), perpendicular to x + 2y = 4.
(d) Has y-intercept (0, 3) and is parallel to y = 1 − x4
(e) Is vertical and goes through (3, 7).
(f) Is horizontal and goes through (−2, 5).
75. During a shift, a waitress makes $20 in base salary plus tips. Suppose the total bill of all the tablesshe waits on comes to x dollars for the night, and she gets 15% tips.
(a) Write a formula for y, the total amount of money the waitress earns in a shift, in terms of x.
(b) If x = 500 how much does she make?
(c) If she made $65 last night, what was x?
76. Three years ago you planted a tree. In the last year, it has grown two feet, and is now 11 feet tall. Leth be the height of the tree, and let t be the number of years since you planted it.
(a) Write a linear formula for h in terms of t.
(b) How tall was the tree when you planted it?
(c) When will the tree be 35 feet high?
77. The height of an object is proportional to the length of its shadow. Suppose a 4 foot tall girl casts a 7foot shadow. How tall is a giraffe that casts a 30 foot shadow?
78. A game warden needs to estimate the coyote population in a certain region. He catches, tags, and thenreleases 6 coyotes. Over the next month, 8% of coyotes sighted are wearing tags. Use a proportion toestimate the coyote population.
79. In your car, revolutions per minute (RPM) is proportional to your speed (MPH). Suppose that in 5thgear, you can go 55 MPH at 2200 RPM.
(a) Write an equation relating RPM and MPH.
(b) If your car redlines at 6000 RPM, how fast can you go?
8 - Homework
Systems of Linear Equations
80. Where do these lines intersect?
y = 2x − 5
y = −3x + 5
81. Where do these lines intersect?
3x + y = 5
4x + 5y = −8
82. In 1995, Virginia Tech played Texas in the Sugar Bowl. A total of 38 points were scored in that game,and Virginia Tech won by 18 points.
(a) Write two equations involving V and T , the Virginia Tech and Texas respective scores.
(b) What was the final score of the game?
83. During the offseason, a soccer player trains by biking and running. On Monday she did 1/2 hour ofeach and covered 10 miles. On Tuesday, she did twenty minutes of running and one hour of biking,and went 16 miles. If she runs and bikes at constant speeds,
(a) Write two linear equations for the variables R and B, her running and biking speeds respectively.
(b) How fast does she run? How fast does she bike?
84. Megan and T.J. went to Taco Bell for lunch. Megan bought two tacos and one burrito for $4.00. T.J.bought 5 tacos and 3 burritos for $10.80.
(a) Write two equations describing this scenario. (ignore tax)
(b) How much does a taco cost? How much does a burrito cost?
85. At a certain party, there were 3 more guys than girls. When Ben showed up, Ashley and Sarah left.At that point, there were twice as many guys as girls.
(a) Write an equation for the original situation.
(b) Write an equation for the new situation.
(c) Solve the simultaneous equations to figure out how many guys and girls were originally at theparty.
86. Suppose your house is located at the origin, and a road runs along y = 3 − 3x.
(a) Sketch a diagram. Label your house and the road.
(b) You want to build a driveway that goes directly from your house to the road. Draw it in yourdiagram.
(c) How would you describe the intersection of the road with your driveway?
(d) Find the equation of the line that describes your driveway.
(e) What are the coordinates of the intersection?
(f) How long is the driveway?
9 - Homework
Circles
87. Find the equation of the circle with the given properties:
(a) center @ (−2, 3), radius 6
(b) center at the origin, radius 7
(c) center @ (1, 0), diameter 6
(d) center @ (2,−5), passes through the origin
(e) center @ (−1, 6), passes through (4, 3)
(f) a diameter goes from (−3, 1) to (3,−7)
(g) has circumference 8π, centered @ (2, 1)
88. Write the equation that describes all points (x, y) that are four units from (1, 2).
89. Write the equation of the circle that has area 4π, is symmetric about the y-axis, and passes throughthe origin at its lowest point.
90. Which of these define circles?
(a) y2 = 7 − x2 (b) x2 + 7 = y2 (c) 7x2 + y2 − 1 = 0 (d) 1 − x2 − y2 = 0
91. Find the center and radius of these circles.
(a) (x − 2)2 + (y + 5)2 = 9
(b) x2 + (y − 2)2 = 36
(c) 2(x + 1)2 = 32 − 2y2
92. Find the area and circumference of this circle:
(x + 2)2 = 25 − y2
93. If a circle’s area equals its circumference, what is its radius?
94. Draw the circle (x − 3)2 + y2 = 25 and find its x and y intercepts.
95. Suppose the point (.5, y) is on the unit circle. What are the possible values of y?
96. If the point (x, 4/5) is on the unit circle, and in the second quadrant, what is x?
Functions
97. Does it represent a function or not?
(a) The set of points {(1, 3), (2, 5), (−1, 1), (3, 5), (0, π)}(b) The set of points {(1, 2), (2, 4), (1, 5), (3, 7)}
(c) The table:x 1 2 3 4 5y 2 1 0 1 2
(d) y2 = x
(e) y3 = x
10 - Homework
(f) 2y − x2 = 3
(g) The relationship between the independent variable vehicle and the dependent variable VIN #.
(h) The relationship between the independent variable parent and the dependent variable child.
(i) This graph:
(j) This symbol: ∧(k) This symbol: ♥(l) The top half of a circle.
(m) The left half of a circle.
(n) y = mx + 3
(o) A vertical line
(p) A horizontal line
98. Write a formula for the function that adds two, and then squares the result.
99. Sketch your height (in inches) as a function of your age (in years).
100. Sketch a function that represents the average daily high temperature as a function of the month.
101. This table is an estimate of the world population (in millions) as a function of the year. Sketch it.t -2000 -1000 0 1000 1500 1800 1900 2000P 27 50 170 254 425 813 1550 6100
102. Given the table of values, find a formula to write y as a function of x. State whether your function islinear or not.
(a)x 0 1 2 3 4y -1 1 3 5 7
(b)x 1 2 3 4 5y 2 1 0 1 2
(c)x 1 2 3 4y 2 5 10 17
(d)x 4 6 8 10y 5 4 3 2
103. Draw a circle centered at the origin, with radius 3.
(a) What are the x-intercepts? (b) What are the y-intercepts? (c) Is this a function?
104. Let f(t) = 3t − 5.
(a) What is the independent variable?
(b) Compute f(−1), f(1), f(4/3), and f(3).
(c) If (7, y) is on the graph, what is y?
11 - Homework
(d) If (t, 7) is on the graph, what is t?
(e) Is the point (5, 10) on the graph?
(f) Find the y-intercept.
(g) Find the t-intercept.
105. Let f(x) = x(1 + x2)−1
(a) Compute f(1), f(12 ), and f(3).
(b) Does this function pass through the origin?
(c) If (−2, y) is on the graph, what is y?
(d) Is (7, 1) on the graph?
106. Is it possible for a function to have two x-intercepts? If so, sketch an example.
107. Is it possible for a function to have two y-intercepts? If so, sketch an example.
108. Make a table, plot the points, and sketch the graph of these functions:
(a) f(x) = |x − 2|(b) f(x) = x3 − 4x
(c) f(t) = 9 − t2
(d) f(x) =√
x2 + 1
109. Suppose y = f(x) =√
2x+171+|2−x|+3(x−1)2−x3/2
(a) is the independent variable and is the dependent variable.
(b) if (4, y) is on the graph of this function, what is y?
110. Let y = f(x) = x2 − 4x.
(a) f(6) =
(b) f(−1) =
(c) f(−x) =
(d) −f(x) =
(e) 2f(x) =
(f) f(x) + 2 =
(g) f(2x) =
(h) f(x2) =
(i) f(x + 2) =
(j) f(t) =
(k) f(♥) =
(l) f(x + h) =
111. Write the area of a circle as a function of its diameter. Identify the independent and dependentvariables.
112. Suppose that after work, you have to run some errands that will take 30 minutes, and then drive 15miles home. If your average driving speed is x MPH, write a function for the total time (in minutes)it will take you to get home after work. If you average 45 MPH, how long will it take?
113. Supose Poindexter got 63, 77, and 80 on his three tests. His final grade in the class is the average ofhis three tests and the final exam.
(a) Write his final average, as a function of his exam grade x.
(b) What is the highest grade he could make in the class?
(c) What is the lowest grade he could make in the class?
(d) What does he need to get on the exam to guarantee a “C” for the class (70 average)?
12 - Homework
114. Let f(x) = 3x2 − x + 1.
(a) Evaluate this function at x = −1 and x = 1 to obtain two points on the graph.
(b) Find the slope between these two points.
(c) Find the equation of the line between these points.
115. An astronaut weighs 200 lbs on earth, but in orbit h miles above the earth he will weigh
w(h) = 200
(
4000
4000 + h
)2
Sketch the graph of this function by filling in this table first (use a calculator)h 0 200 500 1000 4000 16000w
116. Suppose y = 5 − 2x.
(a) Sketch this line.
(b) Write the distance between the point (x, y) and the origin as a function of x.
(c) If (2, y) is on the line, how far is it from the origin?
117. Suppose you leave Dandridge (exit 417) heading west on I-40. If you travel 70 MPH, write your milemarker position as a function of time.
118. Old MacDonald needs to build a fenced in chicken pen next to his barn. He has 60 feet of fence. Oneside of the rectangular pen is the barn, so the fence only needs to enclose the other three sides.
(a) Draw a picture and label the pen’s width and length.
(b) Write the area of the pen as a function of its width.
(c) Experiment to find dimensions that maximize the enclosed area.
119. From a height of h miles above the earth, you can see D(h) =√
8000h + h2 miles to the horizon. Usea calculator and plot several points (e.g. h = 0, 1, 2, 3, 5, 10, 50, 100). Connect the dots to obtain asketch of this function’s graph.
120. Sebastian wants to make some sweet moolah with a math tutoring business. Suppose it costs $1900 tohire a plane to advertise at the football game. Also, estimate that unit costs are 50 cents per tutoringhour (gas money, paper, pencils). Let D be thie demand for tutoring, i.e. the the number of hours hetutors.
(a) Write cost as a function of D.
(b) If he charges $10 per hour, write revenue as a function of D.
(c) Write his profit as a function of D.
(d) How many hours must he tutor to break even?
Function Library / Shifts
121. Sketch from memory the graphs of:
13 - Homework
(a) f(x) = 1
(b) f(x) = x
(c) f(x) = x2
(d) f(x) = x3
(e) f(x) = 1x
(f) f(x) = |x|(g) f(x) =
√x
122. For each of the functions above, sketch and describe the transformed version:
(a) f(x) + 2
(b) f(x) − 2
(c) f(x + 2)
(d) f(x − 2)
(e) 2f(x)
(f) 12f(x)
(g) f(−x)
(h) −f(x)
123. Sketch the graph of:
(a) y = x2 − 5
(b) f(x) = (x + 2)3(c) y = −
√t
(d) f(w) = 2 − |w|(e) f(x) = (1 + x)−1
(f) y = 2(x − 3) + 1
124. Sketch the graph of f(x) = |x − 2| − 1. Find the x and y intercepts.
125. Sketch the graph of f(x) = (x + 2)2 − 9. Find the x and y intercepts.
126. The graph of f(x + 1) + 5 is the same as the graph of f(x), except shifted:
(a) up 5, right 1 (b) up 1, left 5 (c) up 5, left 1 (d) up 1, right 5
127. Write a formula for this graph:
0-1
2
1.5
-2
1
0.5
-3 21
f(x) =
128. Write a formula for this graph:
1
0.8
0.6
0.4
0.2
0210-1-2
f(x) =
14 - Homework
129. Write a formula for this graph:
0
x
10.50-0.5-1
3
2.5
2
1.5
0.5
1
f(x) =
130. Identify the shift and sketch the graph of f(x) = |x| − 1.
131. Sketch the function f(x) = −x−1 and compare it to the graph of f(x) = 1x .
132. Use shifts to draw the function f(x) = |x + 2| − 3.
133. Sketch the graph of f(x) = 4 − x2.
134. What is the formula for the parabola y = x2 shifted right 2 and down 4?
135. Sketch and label y = x2, y = 12x2, and y = 2x2.
136. Is the graph of cf(x) shorter or taller than the graph of f(x) if |c| > 1?
Function Properties
137. Describe the symmetry of these graphs. Is it a function? If so, is it even, odd, or neither.
138. For each function, compute f(−x) and −f(x), and sketch both of these graphs. Then decide if f(x) iseven, odd, or neither.
(a) f(x) = x3
(b) f(x) = x4 − 3|x|(c) f(x) =
√x
139. Is the function odd, even, or neither?
15 - Homework
(a) y = x3 − 1 (b) f(x) =√
|x| (c) y = 2x−1 (d) y = (x + 1)2
140. Can a function be symmetric about the x-axis? If so, give an example.
141. With one exception, a function’s graph cannot be both odd and even. Come up with a graph that isboth odd and even, but not a function.
142. Let z = t2 − 4t.
(a) The independent variable is , and the dependent variable is .
(b) Make a table of function values, plot those points, and sketch the function’s graph.
(c) Does this graph pass the VLT?
(d) What is the domain?
(e) What is the range?
(f) What is the axis of symmetry?
143. Find the domain of the following functions.
(a) y = (x−2)(3+x)(x−7)(2x−3)
(b) f(t) =√
6 − 2t
(c) g(x) = x+1x−1
(d) f(x) =3√
xx2+9
(e) y = (x2 − 4)−1
(f) f(x) =√
|x − 1| − 3
144. Find the domain and range of the following functions.
(a) y = x−2
(b) y = 3 − x2
(c) f(x) = |x − 1|(d) f(x) = 1+x
x
(e) y =√
x2 + 1
(f) f(x) = 2(x − 1)2(g) y = t3
(h) y =√
1 − x2
145. Let f(x) =√
3x + 1. Find the equation of the secant line between x = 1 and x = 5.
146. If g(t) = 3t+21t2+1 , find the equation of the secant line between t = −1 and t = 3.
147. Consider this graph.
-2
-1
0
1
2
3
4
5
-6 -4 -2 0 2 4 6
A
B
C
D
E
F
G
H
I
J
K
line 1
(a) Does it pass the VLT?
(b) What is the range?
(c) Sketch the secant line between points C and J , then find the equation of this secant line.
(d) Sketch the tangent line at J . Is the graph CU or CD there? What can you say about how thelocation of the tangent line depends on concavity?
(e) At which labeled points is it
16 - Homework
i. x-intercept
ii. y-intercept
iii. increasing
iv. decreasing
v. concave up
vi. concave down
vii. inflection point
viii. local max
ix. local min
x. global max
xi. global min
148. This table lists the average SAT score of entering freshmen at Rick’s college:
year 1985 1990 1995 2000 2005SAT 1080 1120 1130 1150 1300
(a) What was the average rate of change between 1985 and 2005?
(b) What was the average rate of change between 2000 and 2005?
(c) What was the average rate of change between 1990 and 2000?
(d) Find the secant line between the years 1985 and 2005.
(e) If this trend continues, predict the average SAT score in 2010.
149. This table shows the value of one share of Limfinity Matrix Group (LMG) stock over the last few years.year price
2000 152001 122002 172003 202004 282005 302006 352007 412008 39
(a) What was the average rate of change between 2000 and 2008?
(b) What was the average rate of change between 2003 and 2006?
150. Let f(x) = x2 − 6x + 5.
(a) Find the slope of the secant line from x = 2 to x = 6.
(b) Find the equation of the secant line.
(c) Sketch the function, along with the secant line. Label the two points of intersection.
(d) Where does the secant line cross the x-axis?
151. Phillip recently ran a drag race up to 100 MPH. Speed can be written as a function of time:
s(t) =140t
t + 6
17 - Homework
0
20
40
60
80
100
0 2 4 6 8 10 12 14 16
spee
d
time
(a) What is the domain?
(b) What is the range?
(c) What was his 0-60 time?
(d) What was his speed at t = 8?
(e) What is the slope of the secant line during the first two seconds?
(f) What is the slope of the secant line during the next two seconds?
(g) How is the slope related to acceleration?
(h) Is the graph concave up or down? What does this mean about acceleration?
Polynomials
152. Is it a polynomial? If so, what is its degree?
(a) y = (x + 2)3
(b) f(x) = x7 − 2x2 + 1
(c) y = 1x2+1
(d) y = x−1 + 2x
(e) y = πt5
(f) y = t2 − t(2t − 7)3
(g) y =√
x2 + 2x − 1
(h) y = x3(1 − x2)2(1 + x)
(i) f(x) = 7
153. Find the degree and y-intercept of the polynomial.
(a) f(x) = 9 − 7x
(b) y = 2x4 − 3x + 1
(c) p(x) = (x2 + 1)3(2x − 1)
(d) f(t) = t(t − 2)2 + t4 − 3t
154. What is the coefficient of the cubic term of p(x) = 2 + 5x2 − 7x3 + x4?
155. Compute the following, multiply out, and collect like terms.
(a) (x − 3)(x + 3)
(b) (x + 3)(3 − x)
(c) (x + 3)2
(d) (x − 3)2(e) (3 − x)2
(f) (3x − 1)2
156. Multiply out using the distributive property, and collect like terms.
(x − 2)(x2 + 2x + 4)
157. Multiply out the expression completely, and collect like terms:
x(x − 2)2(5 − 2x)
158. Let p(x) = x − 2 and q(x) = x2 + 3x − 1. Find the following, multiply out, and collect like terms.
18 - Homework
(a) p + q
(b) q − 2p
(c) pq
(d) p2q
159. In general, does (x + y)2 = x2 + y2 ? What specific condition would be required to make that true?
160. Suppose p(x) and q(x) are polynomials of degree 2 and 3 respectively. (you can make up your ownexamples)
(a) What is the degree of the sum p + q? Will that always be the case, regardless of what you choosefor p and q?
(b) What is the degree of the product pq? Is that always true?
Roots and Factoring
161. Find the roots of these polynomials:
(a) y = 3 − 5t
(b) y = −2x(x + 1)
(c) f(x) = x4 + 1
(d) f(x) = x2(1 − x)
(e) y = x2 − 25
(f) y = x2 + 3
(g) y = (2 + x)(3x − 5)2(x − 7)
(h) y = t(t2 + 4)(2t − 1)
(i) p(x) = (x − 1)(8 − 2x2)
162. Define the polynomial p(x) = x(x2 + 1)3(3x − 9).
(a) What is the degree?
(b) List the roots.
163. Is 2 a root of f(x) = 2x3 − 9x + 2?
164. Is −1 a root of p(t) = t4 − t3 + t2 + 4t − 3?
165. Factor out the common factor.
(a) y = x3 − 4x
(b) y = 10 − 25x + 5x2
(c) y = 12t4 + 6t3 − 24t2
(d) f(x) = 2(x − 1) + 6(x − 1)2
166. Factor out the common factor from this expression
3x2y3 + 6x3y − 12x3y2
167. Find the x and y intercepts of these polynomials.
(a) y = x(x2 − 4)(2x − 3)
(b) y = (2 − x)2(c) f(x) = 2(x − 3)(x + 1)
(d) y = 18x − 2x3
168. From this graph, what are the roots and y-intercept?
19 - Homework
-6
-4
-2
0
2
4
6
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
169. Factor the expressions if possible, and list the roots.
(a) 7t − 35
(b) 12 − 3x2
(c) x2 + 8x + 16
(d) x4 − 9
(e) x2 − 2x − 8
(f) x2 − 2x + 7
(g) 1 − 9x2
(h) 20 + 8x − x2
(i) 40x − 12x2 − 4x3
(j) x6 − 2x4 − 3x2
(k) x5 − x7
(l) 3x2 − 4x − 7
(m) x2 − 2xy − 3y2
(n) 3x2 + 15x + 12
(o) 2x2 + 13x + 15
(p) 2x2 − 13x + 15
(q) 2x2 − 13x − 15
170. Factor by grouping.
(a) x2 − 4x − 2x + 8 (b) x4 − x3 + x − 1 (c) t3 + 2t2 − t − 2
171. Factor and find the roots of y = 8x − 2(x + 1)2x.
172. Find the roots:
(a) p(x) = x3 + 2x2 − 15x (b) y = 2x2 − 32 (c) f(x) = 8x − 7 − x2
173. Find the domain of y = (x2 − 10x + 16)−1
174. Factor the expressions and list the roots.
(a) 3x2 − 4x − 7
(b) 2x2 + 13x + 15
(c) 2x2 − 13x + 15
(d) 2x2 − 13x − 15
(e) 8x2 − 23x − 3
(f) 8x2 − 2x − 3
175. Factor by grouping.
(a) x2 − 4x − 2x + 8 (b) x3 − x2 − x + 1 (c) t3 + 2t2 − t − 2
176. Solve the equation - find all solutions.
(a) x2 = x + 6
(b) 2x2 + 7x = 4
(c) 6x = 9 + x2
(d)√
x2 − 6x + 9 = 1
177. Suppose demand for your product is the following function of price
D = 72 − 2P 2
20 - Homework
(a) How many can you sell if you charge $4?
(b) At what price does nobody buy your product?
(c) If your fixed costs are $30, and the unit cost is $2, write the cost function in terms of P .
(d) Write revenue as a function of P .
(e) Write profit as a function of P .
(f) What is your profit if you charge $4?
(g) Find a better price to charge.
178. The demand for premium lemonade depends on the price being charged. Suppose
D = 10(P + 7)(3 − P )
(a) If you charge $1 per cup, how many will you sell? How about if you charge $2 per cup?
(b) If we we give it away, how many cups will be requested?
(c) At what price will no cups be sold?
(d) What is the revenue function?
(e) If each cup costs fifty cents to produce, what is the cost function?
(f) What is the net profit function? Write it in factored form.
(g) Find the revenue and profit if you charge $1.
(h) Find the revenue and profit if you charge $2.
(i) Approximate the optimal price to maximize profits. How much profit will be made?
179. A circle with radius r is inscribed in a square.
r
(a) What is the area of the circle?
(b) What is the area of the square, in terms of r?
(c) What is the area of the shaded region.
(d) What is the perimeter of the shaded region.
Quadratics
180. Find the vertex of each parabola.
(a) (x − 2)2 + 4 (b) −5(x + 1)2 − 2 (c) 3(x − 7)2 − 11
181. For each quadratic, find the y-intercept, tell whether it is concave up/down, and whether it is normal,tall, or short. If it is in standard form, find the vertex. If it is in factored form, find the roots.
21 - Homework
(a) y = 4 − (x + 3)2
(b) y = 13x2 − 2x + 5
(c) y = −2(x + 1)(2x − 3)
182. Find the roots:
(a) y = 4(x − 1)(7 − x)
(b) y = x2 − 7x + 12
(c) y = (2 − x)2 − 25
(d) y = 2(x − 5)2 − 8
(e) f(x) = 9 − (2x + 1)2
(f) y = (t − 3)2 + 4
183. Find the vertex, y-intercept, and roots. Sketch the parabola.
(a) y = (x − 5)(x + 3) (b) y = 2x2 − 20x + 42 (c) y = (x − 5)2 − 9
184. Find the roots, vertex, and y-intercept by the most efficient means possible.
(a) y = 2(x − 9)(x + 3)
(b) p(x) = −3(x − 1)2 + 12
(c) p(x) = 2x2 + 2x − 4
185. Find the vertex of each parabola. Is it concave up or down?
(a) y = (x − 3)2 + 1
(b) y = (3 − x)2 + 1
(c) y = (x − 3)2 − 1
(d) y = (x + 3)2 + 1
(e) y = 1 − (x − 3)2
(f) y = (2x − 6)2 + 1
(g) y = 2(x − 3)2 + 1
(h) y = −2(x + 5)2 + 7
(i) y = x2 − 8x + 4
(j) y = 3x2 + 24x − 2
(k) y = 5 + 7x − x2
(l) f(t) = (t + 2)(t − 8)
(m) y = 12x − 3x2
(n) y = (x − 3)(8 − 2x)
186. Complete the square to write the quadratic in standard form.
(a) x2 − 4x + 9 (b) x2 + 2x − 2 (c) 3x2 − 30x + 76
187. Write the quadratic in all three forms: standard, factored, and general. Then find the vertex, roots,and y-intercept. Sketch the graph.
(a) x2 − 4x − 5 (b) (x − 1)(x + 7) (c) 25 − (x + 3)2
188. Let f(x) = −2x2 + 16x − 14.
(a) What is the y-intercept?
(b) Is the graph CU or CD?
(c) Write in FF and find the roots.
(d) Write in SF and find the vertex.
189. Use the quadratic formula to find the roots:
22 - Homework
(a) y = x2 − 6x + 4 (b) y = 6x2 − 13x + 7
190. Find the roots:
(a) f(x) = (x−2)(4x−9)3
(b) y = 2(x − 3)2 − 32
(c) y = x2 + 2x − 6
(d) y = x3 − 6x2 + 7x
(e) y = 2x2 − 10x + 11
(f) y = 36x2 − 60x − 24
191. Which of these is closest to a root of y = x2 − 6x − 41?
(a) 2 (b) 4 (c) 6 (d) 8 (e) 10
192. How many real roots does the quadratic have (0, 1, or 2)?
(a) y = (x − 1)2
(b) y = 2(x + 3)2 − 7
(c) y = (x − 4)2 + 16
(d) y = x2 − 4x + 9
(e) y = 3x2 + x − 5
(f) y = 1 − 6x + 9x2
193. Find the constant b so that y = 2x2 + bx + 18 has exactly one root.
194. For what values of c does y = x2 − 6x + c have at least one real root?
195. Find the intersections of y = 5 − 2x and y = x2 − 3.
196. Find the intersections of y = x + 1 and y = (x − 3)2.
197. Solve the equations:
(a) x2 + 10 = 7x
(b) 4x2 = 8x − 3
(c) x2 − 4x = 1
(d) 2x−1 = x
x+1
(e) (x − 4)2 = 8
(f) x = 3√
4 − x
(g)√
5w + 6 − w = 2
(h) x + 4√
x = 21
(i) xx−10 = 2
3−x
198. Write the quadratic with the given properties.
(a) Roots at x = −3, 1; y-int (0, 12).
(b) Roots at x = 2, 5; y-int (0,−2).
(c) Roots at x = 0, 8; passes through (6, 4).
(d) Vertex at (3,−2); root at x = 1.
(e) Vertex at (−2, 7); y-int (0,−5).
(f) Vertex at (1, 3); passes through (2,−2).
(g) One root at x = 3, vertex at x = 7, y-int (0,−6).
23 - Homework
Quadratic Word Problems
199. What is the area of the largest rectangular field you can enclose with 100 feet of fence?
200. If the rectangular field must be divided into three equal pens, and you still have 100 feet of fence, whatdimensions will maximize the total enclosed area?
201. A rectangular room has a perimeter of 62 feet, and an area of 228 square feet.
(a) Draw a picture and label variables.
(b) Write an equation for the perimeter.
(c) Write an equation for the area.
(d) Use substitution to solve for the dimensions of the room.
202. If you drop a ball from a height of 144 feet, it’s position after t seconds is y(t) = 144 − 16t2. Whendoes it hit the ground?
203. Uncle Rico punted a football from 3 feet above the ground, which reached a maximum height of 67feet two seconds later.
(a) Find a quadratic function for the height of the ball at time t.
(b) How high is the ball after one second?
(c) When is the ball 35 feet above ground?
(d) If the ball is allowed to hit the ground, what was the punt’s hangtime?
204. A ball thrown off the roof of a 48 foot building reaches it’s maximum height after one second, andlands after three seconds. Write the ball’s height as a function of time. What is it’s maximum height?
205. He-man throws a boulder from the top of a cliff (at x = 0). The path of its flight makes the parabolicarc: y = 1
5 (x + 1)(5 − x) (in hundreds of feet).
(a) How high is the cliff?
(b) How far away does it land?
(c) What was its maximum height?
206. The height of a ball thrown (at t = 0) off the roof of a building is given by
y = −16(t2 − 2t − 8)
(a) Factor y.
(b) What are the roots?
(c) What is the t-coordinate of the vertex?
(d) What is the y-intercept?
(e) Sketch the ball’s height as a function of time.
(f) How high is the building?
(g) How high does the ball go?
(h) When does it land?
207. A missile is fired from a 500 foot cliff. Suppose the height is given by
h(x) = −32(x/400)2 + x + 500
where x is the horizontal distance from the edge of the cliff.
24 - Homework
(a) What is the maximum height of the projectile?
(b) How far from the cliff will the missile strike the water?
208. A cannon ball fired from an old navy ship (at sea level) rises to a maximum height of 144 feet afterthree seconds. Let h(t) be a quadratic function that describes the ball’s height after t seconds.
(a) List two points that you know.
(b) Write h(t).
(c) By how much will the shot clear a 75 foot wall that it passes at t = 5?
(d) The shot hits on a hill 30 feet above sea level. How long was it in the air?
209. A motorboat heads upstream for 24 miles against a 3 MPH current. If the round-trip takes 6 hours,what was the boat’s speed relative to the water?
210. Marge and Walter drove from Knoxville to Albuquerque (1500 miles) and back. Marge drove out, andWalter drove back. If Walter drove 10 MPH faster than Marge, and the the return trip was 3 hoursshorter, how fast did Marge drive?
211. If revenue is related to price by R(p) = − 12p2 + 1900p, what price should you charge to maximize
revenue?
212. On the scalpers’ market, 300 people would accept a free ticket to see a presentation of The Barber of
Seville. Nobody would pay more than $20 for a ticket.
(a) Write demand as a linear function of ticket price.
(b) Write revenue as a function of ticket price.
(c) Suppose you know the stage manager and can bribe him with $50 to let you have as many ticketsas you want for $3 each. Write your cost as a function of demand.
(d) Write cost as a function of price.
(e) Write profit as a function of price.
(f) How much should you charge to maximize your profit?
213. Suppose demand for lemonade is given by the function D = 500− 200P , and costs total C = 80 + 12D
(a) Write the revenue function R = DP in terms of P .
(b) Write the cost function in terms of P .
(c) Write the profit function Y in terms of P .
(d) What price should you charge to maximize revenue?
(e) What price should you charge to maximize profit?
214. The cost to produce x widgets is 50 + 4x. The revenue from selling x widgets is 100x − x2. Tobreak even, how many widgets would you produce? As factory manager, how many widgets would youproduce to maximize profit?
215. At a price of x, the supply and demand for a certain product are
S =100x− 50
x + 1D =
100
x
At market equilibrium, supply equals demand. What is the equilibrium price?
25 - Homework
216. Under normal driving conditions, the fuel economy of a Honda Civic going s MPH is approximately
f(s) =−2
125(s − 50)2 + 40
(a) How fast should you drive to get the best fuel economy?
(b) How many MPG do you get at that speed?
(c) What MPG do you get driving 70 MPH?
(d) What if you drive 0 MPH?
Circles
217. Find the center and radius of these circles. If it is a semi-circle, tell whether the function gives the topor bottom half. Find the domain and range.
(a) f(x) = 3 +√
100− (x + 7)2
(b) y = 2 −√
4 − x2
(c) y = −1 +√
1 − (x − 3)2
218. Write the bottom half of this circle as a function:
(x − 1)2 + y2 = 9
219. Write the top half of this circle as a function:
(x + 1)2 + (y − 3)2 = 4
Sketch the graph, then find the domain and range.
220. The simplest, and most useful circle is the unit circle, which is centered at the origin and has r = 1.
(a) Draw the unit circle.
(b) Write the equation for the unit circle.
(c) Write the top half as a function of x.
(d) Write the bottom half as a function of x.
(e) Find at least 5 points on the unit circle.
221. Is it a circle? If so, find the center and radius. You may need to complete the square.
(a) 2x2 + 2(y − 1)2 = 8
(b) x2 − y2 = 9
(c) 3(x + 2)2 + (y + 2)2 = 1
(d) x2 − 4x + y2 + 3 = 0
(e) x2 + 6x + y2 − 2y = −6
(f) x2 = 1 − y2
(g) x2 + y2 − 4y + 5 = 0
26 - Homework
Inequalities
222. Solve the inequalities:
(a) |3x − 5| ≥ 25
(b) |1 − 2x| < 7
(c) (x − 2)(x + 5) < 18
(d) x2 ≥ 5x + 24
(e) (x + 1)(5 − 2x) ≥ 0
(f) x2 + 9 < 6x
223. Your business model indicates that profit (in millions) is the following function of price:
Y = −P 2 + 17P − 66
(a) At what prices will you guarantee a profit?
(b) At what prices will you guarantee at least 4 million dollar profit?
Rational Functions
224. Is it a rational function? If so, find the deg(N) and deg(D).
(a) y = (1−x)2
2x−5(b) y = −2
x3−2x+1 (c) y = x√x2+1
225. Consider the rational function:
f(x) =2x
x − 4+ 2(x2 − 6x + 8)−1
(a) Write with a common denominator.
(b) Find the roots.
(c) Solve the equation f(x) = 2.
226. Which of these is closest to a solution to
1
x + 3+ 2 =
x
1 − x
(a) −3/7 (b) 2/3 (c) 1 (d) 2 (e) 5/2
227. Simplify (get a common denominator, factor, cancel, etc) the expression as much as possible:
y =x − 1
x + 1+
4x − 4
x2 − 2x − 3
228. If y = x2+4x−122(7x+3)(1−3x) ,
(a) Find the y-intercept.
(b) Find the roots.
229. Find the vertical asymptote(s).
27 - Homework
(a) y = 2x4x−x3 (b) y = x−1
x2+2x−3 (c) y = x−42x2+13x−7
230. Find the horizontal asymptote if there is one.
(a) y = 1+x2
2x3
(b) y = 10x+14−5x
(c) y = 3x+14−x2
(d) y = (6x−2)(4x+1)3x2+1
(e) y = x2+1x
(f) y = 3x2(1−5x3)(2x−1)5
231. Find asymptotes, intercepts, and plot a few other points to sketch the graph of:
(a) y = 13−x
(b) y = x2+1x
(c) y = 1(x−2)2
(d) y = x1+x2
(e) y = 1−2xx−3
(f) y = x2
x−1
232. If f(x) = 12x−1x2+2 . Solve the equation f(x) = 4.
233. Consider this rational function:
f(x) =x(x − 3)
2x2 − 8+
x3 − 2x
(x − 2)(x + 2)2
(a) Get a common denominator and simplify completely.
(b) What are the x-intercepts?
(c) What is the HA?
(d) What is the VA?
(e) Solve the equation f(x) = 1.
234. If f(x) = x+3x2+1 , find the equation of the secant line between x = 0 and x = 2.
235. The blood concentration (in mg per liter) of a drug administered at t = 0 is given by
f(t) =12t
t2 + 5
(a) Find the concentration after one hour, and after 12 hours.
(b) At what time(s) does the concentration equal 2 mg per liter?
(c) Sketch a graph of the concentration as a function of time.
(d) Explain the significance of the horizontal asymptote.
236. The population of an endangered species is expected to recover according to the function
p(t) =70t + 800
2 + 0.05t
(a) What is the population now (t = 0)?
(b) Predict the population in 5 years.
(c) What is the long-term population of this species?
237. Hot water (170◦) is being poured into a tub that initially contained 2 gallons of cold water (50◦).
(a) Write the average water temperature as a function of the total number of gallons poured into thetub.
(b) How many gallons of hot water should be poured in to bring the average temperature to 120◦?
28 - Homework
238. Julio started pumping iron to impress the ladies. The number of pounds he could benchpress, after tdays of working out, is given by this function:
B(t) =25t + 500
.1t + 4
(a) How many pounds could he lift at the beginning (t = 0)?
(b) How much could he lift after one month (t = 30)?
(c) If he keeps working out, what is the maximum that he will ever be able to lift?
Exponential Functions
239. Let f(x) = 4x and evaluate:
(a) f(0)
(b) f(1)
(c) f(2)
(d) f(−1)
(e) f(1/2)
(f) f(2.5)
240. Does 3x2
= (3x)2 ?
241. Solve the equation for x.
(a) 2x = 18
(b) 53x = 25x+3
(c) 42x+2 = 8x+2
(d) 4x−2 = 22−x
(e) 3x2
= 9x+4
(f) (2x)x = 1162x+3
242. For each exponential function: find the HA, find the y-int and one other point, determine whether itis growth/decay, and sketch the graph.
(a) y = 5(1.2)t
(b) y = 2x − 3
(c) f(x) = −3x + 1
(d) y = 3(1/2)x
(e) y = 12 (4)x − 2
(f) y = 20(.8)t
(g) y = 1 − (2/3)x
(h) f(t) = 1 + 2−t
(i) y = 9.5x
243. Find the exponential function in the form y = abt + c that satisfies the given conditions.
(a) HA: y = 5, passes thru (0, 0) and (1, 3)
(b) HA: y = −1, passes thru (0, 1) and (1, 6)
(c) HA: y = 30, passes thru (0, 57) and (3, 38)
244. A municipal bond pays the holder 7% APY for 20 years. If you initially invest $3000, how much willit be worth at maturity?
245. A bacteria population starts at 100 and doubles in size every 3 days.
(a) Write the population as a function of t.
(b) How big is the population after 2 weeks?
246. A new car was purchased for $28 thousand. After three years, the car’s resale value is expected to be$16 thousand.
(a) Find a formula for the car’s value after t years.
(b) Estimate how much the car is worth when it is 10 years old.
29 - Homework
247. The population of the US increased from 230 million in 1980 to 280 million in 2000.
(a) Write population as a function of time in years since 1980.
(b) Predict the population in 2020.
248. A nurse injects a patient with 200 mg of a drug. The effective half-life is 4 hours. Find a formula forthe amount of the drug that remains in the bloodstream t hours after injection.
249. A turkey is removed from a 30 degree freezer and allowed to thaw in a 72 degree room. After twohours it had warmed to 37 degrees. Find a formula for the temperature of the turkey t hours afterbeing removed from the freezer.
250. A dragster has a top speed of 240 MPH. From a standstill, it can reach 60 MPH in two seconds.
(a) Sketch the speed graph.
(b) Write speed as an exponential function of time.
(c) How fast is it going after 5 seconds?
1 - Practice
These sample problems are a fair, but not comprehensive or verbatim, representation of what to expect onthe test. Also study your class notes and homework. You should be able to work these without the aid ofcalculators or notes.
Practice
1. Circle the rational numbers:
13
73√−8 −
√4 1.41421356
√2 − 3.406
2. Evaluate |6 − 2(9 − 2(−1 + 3))|
3. Suppose |x + 3| < 7.
(a) The distance between x and is less than .
(b) Sketch this set on the number line.
(c) Write this set in interval notation.
(d) Write this set as a regular inequality.
(e) Is the set open or closed?
4. Use a single absolute value inequality to describe the set (−∞, 1] ∪ [9,∞).
5. Solve the equation:
(a) 2(x − 7) = 6 − 5x
(b) x2 = 14
(c) (3x − 1)2 = 121
(d) |x + 1| = 7
(e) |2x + 1| = 7
(f) 1x+1 = 2
3−x
(g) −2x(5x − 4)(x2 + 3) = 0
6. Circle the solution(s) to the equation:√
x + 8 = x − 4.
(a) -7 (b) -4 (c) 1 (d) 4 (e) 8
7. Multiply out and simplify completely by collecting like terms:
−2x(x + 7) + 3(x − 4)2
8. Circle the letters corresponding to true statements.
(a) |x| ≥ 2 describes the interval(−∞,−2] ∪ [2,∞)
(b) x−yy−x = −1
(c) (x − 2)(1 + x) = x2 − x − 2
(d)√
a2 + b2 = a + b
(e) | − 3| =√
(−3)2
(f) (x + h)2 = x2 + h2
(g) 1
( 2
x )= 1
2x
(h) 1x + 1 = 2
x+1
9. Solve the inequality |3x − 7| ≥ 5, and write your solution in interval notation.
10. Evaluate the following.
(a) 01 + 10 − 01 (b) 9
3
2 (c) 45−2 (d)
(
23
)−1 − 12 − 20
2 - Practice
11. Simplify the following by removing radicals and combining exponents.
(
y−4√
x5
8x−1/2y2
)−1/3
12. Find the arithmetic and geometric means of these numbers: {1, 1, 2, 8}
13. The human life expectancy is 2.4 billion seconds. Write this number in scientific notation.
14. In twelve years, Bubba will be 80 percent older than he was four years ago.
(a) Write an algebraic equation for this problem.
(b) How old is Bubba now?
15. In the SAC basketball tournament, Andy scored 25 points in the first game and 17 points in the secondgame.
(a) If he scores x points in the third game, write an expression for his scoring average.
(b) How many points must he score in the third game in order to average 30 points per game in thetournament?
16. Suppose you leave Knoxville at 12:00, and need to be in Roanoke, VA by 4:00. This is a 260 mile trip.If it takes you 1:30 to go the first 100 miles, how fast must you drive the rest of the way to get thereon time?
17. Find the distance between:
(a) 2 and -7
(b) P (2,−5) and Q(−4, 3)
(c) P and R, where R is the midpoint of PQ.
18. Find the point that is one-quarter of the way from (−3, 5) to (9, 1).
19. Consider the line 3y − 2x = 12
(a) Find the x-intercept.
(b) Find the y-intercept.
(c) Find the slope.
20. Find the equation of the line that goes through (4,−3) and (9,−1).
21. Find the equation of the line that has x-intercept 2, and also goes through (−1, 6).
22. Suppose that a 5 year old boy weighs 43 pounds, and that every year after that he gains 12 pounds.
(a) Write the equation of a line that describes this child’s weight. Let x be his age, and y his weight.
(b) When will he weigh 175 pounds?
23. Circle the letters corresponding to true statements.
(a) Any triangle that satisfies the PythagoreanTheorem must be a right triangle.
(b) P (−2, 1) is in the third quadrant.
(c) The slope of 3x + 9 = 1−y2 is m = −6
(d) 9−2 = 91
2
(e) If a line’s x and y intercepts are both nega-tive, then the slope is positive.
(f) (5, 5) is farther from the origin than (7, 1)
24. Find the equation of the line that goes through (2, 5) parallel to 3x − 8y = 10.
25. Dwight Schrute receives a base salary of $30 thousand dollars. In addition, for each client he signsbeyond the first four, he receives a $2 thousand bonus.
(a) If he signs 7 clients, what is his total income?
3 - Practice
(b) Write a general formula if he signs x clients.
(c) If he made $44 thousand, how many clients did he sign?
26. A European tourist arrived in New York and exchanged 700 Euros for 1000 dollars. Returning home,she had 50 dollars left over, which she converted back into Euros.
27. Given two points P (4, 3) and Q(−2, 11), find the equation of the line that intersects the midpoint ofPQ at a right angle.
28. Suppose the number of women that Casanova dates in a month is proportional to the number of pickuplines that he uses. Last month he used 63 pickup lines and went on 14 dates. This month he used 45pickup lines. How many dates did he get?
29. Brutus and Popeye went out to dinner. Brutus ordered 4 steaks and one side of spinach for $32. Popeyeordered 2 steaks and 7 sides of spinach for $29.
(a) Write two linear equations describing this situation.
(b) A steak costs and spinach costs .
30. Where do these lines intersect?
3x = 5y + 11 2x − 3y = 8
31. Santa’s elves can make 5 dolls and 3 trains in 81 minutes. They can make 2 dolls and 7 trains in 73minutes.
(a) Write a system of linear equations describing this situation.
(b) Use substitution to solve the equations.It takes minutes to make a doll, and minutes to make a train.
32. Circle the letters corresponding to true statements.
(a) x = 4 is a vertical line
(b) The lines y = 3x − 1 and 3y + x = 4are parallel.
(c) The perimeter of a square is proportional to
the length of a side.
(d) The viewing area of a TV screen is propor-tional to the length of its diagonal.
33. Circle the letters corresponding to true statements.
(a) A function could have two y-intercepts.
(b) This is a function: {(1, 2), (2, 1), (3, 1)}(c) f(x) = 2(x2 − 3)−1 goes through (2, 2)
(d) If W = 3t√
t2 − 1, then t is the dependent
variable.
(e) 5y − 2x = 7 is a linear function.
(f) If f(x) = x2 then f(2x) = 2x2.
34. Find the center, radius, area, and circumference of this circle: 9 − y2 = (x − 2)2.
35. Which of these formulas describe circles? If it is a circle, identify the center and radius.
(a) x2 = 9 − (y + 1)2
(b) 2(x + 1)2 + (y − 3)2 = 10
(c) (x − 3)2 − 3 = 2 − y2
(d) (x + 3)2 − (y − 9)2 = 4
36. Find the equation of the circle that is centered at (2, 7) and has a y-intercept at 4.
37. Belinda Carlisle’s pop 80’s hit “Circle in the Sand” was referring to a circle with diameter that stretchesfrom P (3, 0) to Q(−1, 6). Find the equation of this circle.
38. An astronaut in orbit h miles above the Earth weighs W = 180(
40004000+h
)2
pounds.
4 - Practice
(a) is the independent variable, and is the dependent variable.
(b) How much does he weigh on Earth?
(c) Suppose that the astronaut’s weight is 20 pounds. How many miles high is he?
39. Let f(x) = 3(x + 1)(4x − 1)(x2 − 9)
(a) Find the y-intercept.
(b) Find the x-intercept(s).
40. Find the equation of the line that connects the x and y intercepts of f(x) = 2 −√
x + 1.
41. If the point(−1
3 , y)
is on the unit circle in the third quadrant, what is y?
42. Brittany has a cell phone plan that charges $30 per month, plus 10 cents for every minute over a 500minute limit. Let x be the number of minutes, and f(x) the total charge.
(a) Assuming she talks for at least 500 minutes, write a formula for f(x)
(b) If she talked for 950 minutes, what was her bill?
(c) If her bill comes to $50, how many minutes did she talk that month?
43. Plot several points and sketch the graph of f(x) = |x2 − 4|.
44. If your fixed cost is $100, and you can make CN professor bobble-heads for $5 each and sell them for$8 each, write your profit as a function of units sold, x.
45. If f(x) = x2 + 3x, compute and simplify 2f(x) − f(2x).
46. Write the formula for a function that would have this graph.
10.80.60.40.2
0-0.2-0.4 x
543210-1
47. Sketch the graph of:
(a) f(t) = t3
(b) y = (x + 3)1/2
(c) f(x) = −x−1
(d) V = t1+t2
48. Given the graph of f(x), the graph of f(x + 4) − 2 has the same shape but shifted:
(a) right 2up 4
(b) left 2up 4
(c) left 4down 2
(d) right 4down 2
(e) down 2right 4
49. Is the function f(x) = x1+x2 even, odd, or neither? What does that say about graph’s symmetry?
50. Circle the letters corresponding to true statements.
(a) If f(−x) = f(x) then the graph is symmetricabout the origin.
(b) y = 3x2 is a fat parabola.
(c) f(x − 1) has the graph of f(x) shifted left
(d) f(x) = |x| is an odd function.
51. Let f(x) = 6 − 2|x + 1|
(a) Find the y-intercept.
5 - Practice
(b) Find the x-intercept(s).
(c) What is the domain of f?
(d) What is the range of f?
52. Find the domain:
(a) f(x) =√
3x + 5
(b) f(x) = (|x + 1| − 2)−2
(c) f(x) = 3√
x + 1
53. Find the range:
(a) y =√
x+23
(b) y = 4 − |x|(c) The top half of the unit circle.
(d) f(x) = 2(x − 3)3 + 7
(e) y = 2(x − 3)2 + 7
(f) f(x) = 1x2
54. If f(x) = x√x+3
, find the equation of the secant line between points where x = −2 and x = 6.
55. Find the equation of the secant line that connects the x and y intercepts of f(x) = (x − 3)(x2 + 4).
56. At which labeled point(s) is the graph
B
CD
E
F
A
(a) increasing
(b) decreasing
(c) concave up
(d) concave down
(e) inflection point
(f) local max
(g) local min
(h) global min
(i) global max
Also, sketch the tangent line to the graph at point B.
57. This table lists the total number of Facebook friends that Gertrude had over the last few months:
month 7 8 9 10 11friends 10 17 21 29 34
(a) What was the average rate of change (in friends per month) between months 7 and 11?
(b) Write a formula for this secant line.
(c) At this linear rate, predict how many friends she’ll have by month 14.
58. If h(t) = 200 − 16t2 is the height (in feet) of an object in free fall, find the average velocity (AROC)between time t = 1 and t = 3.
59. Describe in your own words the difference between average and instantaneous rates of change. Use acar’s speed to illustrate.
6 - Practice
60. Let f(x) = −3x2(2x − 5)(x2 + 4)3
(a) What is the degree of f?
(b) Find the roots
61. If p(x) = x2 − 1 and q(x) = 2x + 3, find and simplify:
(a) p + q
(b) p − 2q
(c) pq
(d) (p + 1)q
(e) p(x + 1)
62. Circle the letters corresponding to true statements.
(a) The domain of y =√
x2 + 1 is R
(b) y = 3 − (2 − x)2 is concave down.
(c) It is possible for a graph to have neither aglobal minimum nor a global maximum.
(d) Every global max is a local max.
(e) The degree of p(x) = (2x+3)5(1−x2) is five.
(f) f(x) = x2 + 2x +√
x is a polynomial.
(g) x = 2 is a root of p(x) = x3 − 5x2 + 12.
(h) f(x) = (x + 2)2 + 1 has no real roots.
(i) Every polynomial has a real root.
63. Factor y = 2x3 − 14x2 − 36x and list the roots.
64. Find the domain of f(x) = (9x − x3)−1.
65. Find the range of f(x) = x2 − 4x + 1.
66. Write y = 2x2 + 12x + 17 in standard form by completing the square. Then find the vertex.
67. Find the roots of
(a) p(x) = −3(x − 1)2 + 12. (b) y = 12x2 − 4x − 5
68. Find the vertex of
(a) y = 35 (x + 1)(x − 9) (b) y = 3 − 5(7 − x)2
69. Solve the equation (list all solutions):
(a) x2 + 6x = 3
(b) t2 = 2(5t + 12)
(c) x3 + 2x2 = 4x + 8
(d) (3 − x)−1 = xx+1
70. Write the quadratic that satisfies the given properties:
(a) vertex (2,−1), passes through (3, 1)
(b) passes through (−1, 0), (3, 0), and (0, 6)
(c) roots at x = −3 and x = 2, and goes thru (1, 6).
71. Consider the quadratic p(x) = 4 − (x + 1)2
(a) Write p(x) in general form.
(b) Write p(x) in factored form.
(c) Find the vertex.
(d) Find the roots.
(e) Find the y-intercept.
(f) Find the domain
(g) Find the range
72. Circle the letters corresponding to true statements.
7 - Practice
(a) y = 3x2 − 12x + 4 has its maximum valuewhen x = 2.
(b) (x + 1)2 = 2x has no real solutions.
(c) The maximum of x2 − 4x occurs at x = 2.
(d) The vertex of f(x) = x(x − 2) is at (0, 2).
(e) The parabola p(x) = (1 − x)2 is concave up.
73. Which of these is closest to a root of y = 2x2 − 7x + 4
(a) 0 (b) 1 (c) 34 (d) 2 (e) 7
2
74. If cost is 100 + 5P and demand is 175 − 10P , what price P should you charge to maximize profits?
75. Let f(x) = 8x − x2.
(a) Find the domain (b) Find the range (c) Is this functioneven/odd/neither?
76. Suppose that at price P , the demand for Cogswell Cogs is D = 30(P + 1)−1. The supply equation isS = 5(P − 4). What is the free-market equilibrium price?
77. Old MacDonald needs to build a fenced in chicken pen next to his barn. He has 60 feet of fence. Oneside of the rectangular pen is the barn, so the fence only needs to enclose the other three sides.
(a) Draw a picture and label the pen’s width (w) and length (ℓ).
(b) Find the dimensions that will maximize the area enclosed by the pen.
(c) How many square feet of space will the chickens have?
78. A military surveillance drone reached its target, flying with a 7 MPH tailwind. On the return trip, ithad a 9 MPH headwind. If the round trip took 8 hours, and the target was 120 miles away, what isthe airspeed of this aircraft?
79. A soccer ball is kicked so that it reaches a maximum height of 5 meters when it has traveled 10 metersin horizontal distance. The goal is 16 meters away and 3 meters high.
(a) Find the equation for the parabolic path of the soccer ball.
(b) Unfettered, how far away will the ball land?
(c) Would the ball go over the goal, or into it?
80. Given this circle: x2 + y2 = 6y − 5
(a) Find the center and radius.
(b) Find an equation for the bottom semi-circle, then sketch it.
(c) What is the domain?
(d) What is the range?
81. Solve the following:
8 - Practice
(a) x2 = 6x − 5
(b) x2 < 6x − 5
(c) x2 ≤ 6x − 5
(d) x2 ≥ 6x − 5
82. Write with a common denominator and then simplify completely:
x + 1
x− 3x−1
x + 3
83. Get a common denominator on the left, then solve the equation:
6 − 3x
x2 − 2x+
1 − x
x= 3
84. Find the horizontal asymptote:
(a) y = −3(1−2x)(5x−7)2(x2−9) (b) y = 2x+5
3x2−7x+1
85. If y = 2x2+5x−3x3+2
(a) Find the y-intercept.
(b) Find the root(s).
(c) Find the HA.
86. If y = 2x(x+1)x2−3x−4
(a) Find the HA.
(b) Find the VA.
87. Consider the rational function f(x) = 2x(6−2x)x2−13x+30 .
(a) Factor and simplify:
(b) Solve the equation f(x) = 16
(c) What is the horizontal asymptote?
(d) f(x) has vertical asymptote
88. Jared’s weight after t months on the Subway diet was
W =.9(t2 + 20000)
(.1t + 6)(.05t + 10)
(a) What did Jared weigh at the start?
(b) How much did he lose in the first 10 months?
(c) What will his weight level out at?
89. Which rational function would have this graph?
1
0.50
x
-0.5
-1
420-2-4
(a) x2
1+x (b) 2x1+x4 (c) 1
1+x2 (d) x−1x
90. While working construction in Alaska, Jen has determined that the local population of grizzly bears ison the rise according to the function:
P =100t + 63
2t + 7
9 - Practice
(a) What was the population when t = 0?
(b) Determine the long-term bear population.
(c) Sketch the bear population for t ≥ 0.
91. Find the equation of the secant line to y = x+7x2+5 between x = −4 and x = 2.
92. Circle the letters corresponding to true statements.
(a) The rational function f(x) = 6x−23x2+1 has horizontal asymptote y = 2.
(b) x = 1 is a root of p(x) = 3x7 − 2x4 + 5x2 − 8x + 2
(c) If the degree of the numerator is greater than the degree of the denominator, then the rationalfunction has asymptote y = 0.
(d) f(x) = (x−1)2
x+1 has vertical asymptote y = −1.
93. If f(x) = 9x,
(a) Find the y-intercept.
(b) Evaluate f(1.5).
(c) Find the equation of the secant line that connects the y-intercept with the point on the graphwhere x = 1.5.
94. Solve the equation 4x = 82x−1.
95. Find an exponential function with the following graph (the VA is y = 2 and it goes thru (1, 4) and(0, 10)):
x
2.521.5
y
1
16
12
0.5
8
4
00
96. Suppose the exponential function:y = 5bt + 1
goes through the point (2, 3). Then it also goes through the point
(a)(
4, 25
)
(b)(
4, 52
)
(c)(
4, 59
)
(d)(
4, 95
)
97. A room temperature (70 degrees) Pepsi is placed into the refrigerator (38 degrees). After one hour,the Pepsi has cooled to 50 degrees.
(a) Create an exponential model for the temperature of the Pepsi after t hours in the refrigerator.
(b) What will be the temperature of the Pepsi after two hours?
i. 30 ii. 38 iii. 40.5 iv. 42.5 v. 45.5