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Pre-Calculus

Marking Period 1:

Chapter 1: 1-6

Chaper 2:1-8

(27 days)

Marking Period 2:

Chapter 4:1-6, 8

(27 days)

Marking Period 3:

Chapter 4: 7

Chapter 5:1-3, 5

Chapter 6: 1-2, 6-7

(26 days)

Review: 4 Days

Note: This class really moves, you will really need to stick to this schedule. For marking period 1, you will start chapter 4 before the end of the marking period

Assignments for Chapter 1

Mini-Lecture 1.6 Day 1Transformations of Functions

Learning Objectives: Students will be able to….

Chapter PPg 129 1.6 1.7 1.8 1.10

Day 1

1-3,28-31,40-47,

74-78,86-87, 88-97

Day 2 1-51 EOO

Day 3 53-117 EOO

Day 4

1, 7, 13, 19 31, 35, 41, 47, 49, 55, 61, 63, 67,

69, 73, 75, 77

Day 5 2-49 EOO

Day 6 1, 3, 5, 7, 9, 11

Day 7 15,19, 21, 35, 37

Day 8 Review

Day 9 Test 1.6 – 1.10

Mrs. Badr

Precalculus

Phone

248-676-8320 ext.7049

Email

[email protected]

Web

www.classjump.com/mrsbadr

1. Recognize graphs of common functions.2. Use vertical shifts to graph functions.3. Use horizontal shifts to graph functions.4. Use reflections to graph functions.5. Use vertical stretching and shrinking to graph functions.6. Use horizontal stretching and shrinking to graph functions.7. Graph functions involving a sequence of transformations.

Mini-Lecture 1.6 Day 2Transformations of Functions

Learning Objectives: Students will be able to….1. Recognize graphs of common functions.2. Use vertical shifts to graph functions.3. Use horizontal shifts to graph functions.4. Use reflections to graph functions.5. Use vertical stretching and shrinking to graph functions.6. Use horizontal stretching and shrinking to graph functions.7. Graph functions involving a sequence of transformations.

Examples:

1. Begin by graphing ( )f x x . Then use transformations to graph 1( ) 22

f x x .

2. Begin by graphing ( )f x x . Then use transformations to graph ( ) 4f x x .3. Begin by graphing 3( )f x x . Then use transformations to graph 3( ) ( 1) 3f x x .4. Begin by graphing 2( )f x x . Then use transformations to graph 2( ) 2( 3) 1f x x .

Teaching Notes: Before shifting graphs of equations, practice shifting points first. One good type of exercise is to give the students the standard function and a word description of the transformed function, then have them find the equation of the transformed function. Emphasize the “Summary of Transformations Chart” in the book. Use a graphing utility calculator to show transformations. Begin with the standard function and show

one transformation at a time on the screen.

Answer: 1) 2)

x

y4

4

x

y

-4

4

3) 4)

x

y4

4

x

y

5

5

Mini-Lecture 1.7Combinations of Functions; Composite Functions

Learning Objectives: Students will be able to…

1. Find the domain of a function.2. Combine functions using the algebra of functions, specifying domains.3. Form composite functions.4. Determine domains for composite functions.5. Write functions as compositions.

Examples:

1. Find the domain of 1 1( )

2 3f x

x x

.

2. 2

3 2( )16

xf xx

and 2

5 4( )16

xg xx

, find f g , f g , fg ,

fg . Determine the

domain of each.3. 2( ) 3 4f x x x and ( ) 2g x x , find ( )f g x and determine its domain.

4. ( ) 4f x x and 2( ) 2 5g x x x , find all values of x that satisfy ( ) 4f g x .

Teaching Notes:

Review interval notation before discussing finding the domain of a function. Help students find domains of functions by recognizing categories of functions.

Category Domainpolynomial all real numbersrational set denominator 0radical set expression under the radical 0

Then write the domain in interval notation. Some students have trouble working with the notation f g x . Encourage them to rewrite this

notation as ( )f g x . Emphasize “Rules for Excluding Numbers from the Domain of ( ) ( ( )f g x f g x ” from the book.

Answer: 1) , 3 3, 2 2, ; 2) 2

8 216

xf gx

, , 4 4, 4 4, ;

2

2 616

xf g

x

, , 4 4, 4 4, ;2

4 2

15 2 832 256

x xfg

x x

, , 4 4, 4 4, ;

3 25 4

f xg x

, 4 4

, 4 4, , 4 4,5 5

; 3) 2( ) 2f g x x x , , ;

4) 3, 1

Mini-Lecture 1.8

Inverse Functions

Learning Objectives: Students will be able to…

1. Verify inverse functions.2. Find the inverse of a function.3. Use the horizontal line test to determine if a function has an inverse function.4. Use the graph of a one-to-one function to graph its inverse function.5. Find the inverse of a function and graph both functions on the same axes.

Examples:

1. 1( ) 42

f x x , find 1( )f x .

2. 2 3( )

1xf xx

, find 1( )f x .

3. ( ) 3 4f x x , ( ) 2 1g x x , 2( ) 2 5h x x x , evaluate (1)h g f without finding an equation for the function.

4. Using interval notation give the domain and range of f and 1f , if 3( ) 8f x x .

Teaching Notes:

Emphasize the steps given in the book for finding the inverse of a function. Showing tables with x- and y-values for the function and its inverse will help some students better understand the concept of an inverse. Using a graphing utility calculator to graph the function, its inverse, and y x in the same plane really helps students visualize the relationship between the function and its inverse.

Answer: 1) 1( ) 2 8f x x ; 2) 1 3( )

2x

f xx

; 3) 148; 4) Domain and range of ( )f x and 1( )f x are

, .

Mini-Lecture 1.10 Day 1Modeling with Functions

Learning Objectives: Students will be able to …

1. Construct functions from verbal descriptions.2. Construct functions from formulas.

Examples:

1. An open box is made from a square piece of cardboard 8 inches on a side by cutting identical squares from the corners and turning up the sides. Express the volume of the box, V, as a function of the length of the side of the square cut from each corner, x.

2. A coupon book for a bridge costs $24 per month. The toll for the bridge is normally $3.00, but it is reduced to $1.50 for people who have purchased the coupon book. Express the total monthly cost to use the bridge without a coupon book, f, as a function of the number of times in a month the bridge is crossed, x. Express the monthly cost to use the bridge with a coupon book, g, as a function of the number of times in a month the bridge is crossed, x. Determine the number of times in a month the bridge must be crossed so that the total monthly cost without the coupon book is the same as the total monthly cost with the coupon book ;

3. On a certain route. an airline carries 5000 passengers per month, each paying $80. A market survey indicates that for each $1 decrease in the ticket price, the airline will gain 50 passengers. Express the number of passengers per month, N, as a function of the ticket price, x. Express the monthly revenue for the route, R, as a function of the ticket price, x.

Answer: 1) 2 364 32 4V x x x ; Answer 2) 3.00 , 1.50 24f x g x , 16

Teaching Notes:

Emphasize the importance of these kinds of problems in calculus. Encourage students to learn the geometric formulas in Section P.8. Some students will attempt to “solve” their functions. Be sure that students understand the difference

between solving an equation and evaluating a function.

Mini-Lecture 1.10 Day 2Modeling with Functions

Learning Objectives: Students will be able to …

1. Construct functions from verbal descriptions.2. Construct functions from formulas.

Examples:

1. An open box is made from a square piece of cardboard 8 inches on a side by cutting identical squares from the corners and turning up the sides. Express the volume of the box, V, as a function of the length of the side of the square cut from each corner, x.

2. You have 70 yards of fencing to enclose a rectangular garden. Express the area of the garden, A, as a function of one of its dimensions, x.

3. You inherit $16,000 with the stipulation that for the first year the money must be placed in two investments expected to pay 6% and 8% annual interest. Express the expected interest, I, as a function of the amount of money invested at 6%, x.

Answer: 1) 2 364 32 4V x x x ; Answer: 2) Answer: 3)

Teaching Notes:

Emphasize the importance of these kinds of problems in calculus. Encourage students to learn the geometric formulas in Section P.8. Some students will attempt to “solve” their functions. Be sure that students understand the difference

between solving an equation and evaluating a function.

Precalculus Chapter 1 Name: ___________________________Review- Badr Hour: ______________1. Use the graph of f(x) to graph g(x). 2. Use the graph of f(x) to graph g(x).

3. Graph

4. Given the function f(x) = , write an equation so that f(x) is translated 2 units down, reflected over the x-axis, Stretched vertically by a factor 1/2, and translated right 1 unit.

5-8. Given and

a. Find

b. Find the domain of a

c. Find

d. Find the domain of c

9-10. Given that f(x) = and

,

a. Find .

b. Find the domain of .

11-12. Find and determine wheather the pair of functions given below are inverses of eachother.

and .

13-14. Given

a. Find the equation for

b. Graph in the same rectangular coordinate system.

Choose 115-16. A 400 room hotel can rent every one of its rooms at $120 per room. For each $1 increase in rent, two fewer rooms are rented. a. Express the number of rooms rented, N, as a function of the rent x.

b. Express the hotels revenue, R, as a function of the rent, x.

18-19. You inherit $10,000 with the stipulation that for the first year the money must be placed in two investments expected to earn 8% and 12% annual interest. a. Express the expected interest from both investments I, as a function of the amount of money invest in the 8%, x.

b. If the total interest for the year was $1,088, how much money was invested at each rate?

Choose 217. You have 600 yards of fencing to enclose a rectangular field. Express the area of the field, A, as a function of one of its dimensions, x.

20. The sum of two positive numbers is 86. Write a function that models the product of the two numbers in terms of one of the numbers, x.

21. A closed rectangular box with a square base has a volume of 800 cubic centimeters. Express the surface area of the box, A, as a function of the length of a side of its square base, x.

Precalculus Ch 1 Test Name: ___________________________Badr Hour: ______________1. Use the graph of f(x) to graph g(x). 2. Use the graph of f(x) to graph g(x).

3. Graph

4. Given the function f(x) = |x| , write an equation so that f(x) is translated 3 units down, reflected over the x-axis, shrunk vertically by a factor of 1/3, and translated left 2 units.

5-8. Given and

a. Find

b. Find the domain of a

c. Find

d. Find the domain of c

9-10. Given that f(x) = and

a. Find .

b. Find the domain of .

11-12. Find and determine wheather the pair of functions given below are inverses of eachother.

and .

13-14. Given

a. Find the equation for

b. Graph in the same rectangular coordinate system.

Choose 115-16. A 500 room hotel can rent every one of its rooms at $100 per room. For each $1 increase in rent, two fewer rooms are rented. a. Express the number of rooms rented, N, as a function of the rent x.

b. Express the hotels revenue, R, as a function of the rent, x.

18-19. You inherit $8,000 with the stipulation that for the first year the money must be placed in two investments expected to earn 6% and 8% annual interest. a. Express the expected interest from both investments I, as a function of the amount of money invest in the 6%, x.

b. If the total interest for the year was $586, how much money was invested at each rate?

Choose 219. You have 250 yards of fencing to enclose a rectangular field. Express the area of the field, A, as a function of one of its dimensions, x.

20. The sum of two positive numbers is 64. Write a function that models the product of the two numbers in terms of one of the numbers, x.

21. A closed rectangular box with a square base has a volume of 2700 cubic centimeters. Express the surface area of the box, A, as a function of the length of a side of its square base, x.

Assignments Sections 2.1-2.3Mrs. Badr- [email protected]

248-676-8320 ext. 7049www.classjump.com/mrsbadr

mailto:[email protected]

http://www.classjump.com/mrsbadr

2.1 2.2 2.3Day 10 1-53 EOODay 11 1-55 EOO

Day 12 1-41 EOODay 13 43-61 odd

Do all StepsDay 14 Practice Graphing WorksheetDay 15 ReviewDay 16 Quiz 2.1-2.3

Mini-Lecture 2.1Complex Numbers

Learning Objectives:

1. Add and subtract complex numbers.2. Multiply complex numbers.

3. Divide complex numbers.4. Perform operations with square roots of negative numbers.5. Solve quadratic equations with complex imaginary solutions.

Examples:

Perform the indicated operation.1. 6 3 2 2. 6 3 2 4 5i i 3. 2 23 1i i

4. Solve for x. 2 2 5x x

Teaching Notes:

Many times students will not remember to express square roots of negative numbers in terms of i before multiplying. In the beginning have students simplify as follows: 25 4 = 1 25 1 4 =

1 25 1 4 = 25 4i i = 2 100i = 10 Many students have problems with signs in complex number problems. Remind them often to be

careful with the signs when working with 2i . Emphasize that 2 2a bi a bi a b . Discourage the use of the FOIL method when multiplying a

complex number by its conjugate.

Answer: 1) 3 2 2 3i ; 2) 7 7i ; 3) 8 8i ; 4) 1 2i

Mini-Lecture 2.2Quadratic Functions

Learning Objectives:1. Recognize characteristics of parabolas.2. Graph parabolas.3. Determine a quadratic function’s minimum or maximum value.4. Solve problems involving a quadratic function’s minimum or maximum value.

x

y

5

5

Examples:1. Find the coordinates of the vertex for the parabola defined by the given quadratic function.

2( ) 3 5 4f x x x 2. Sketch the graph of the quadratic function. 2( ) 6 5f x x x 3. For the quadratic function, 2( ) 4 8f x x x ,

a) determine, without graphing, whether the function has a minimum value or a maximum value,b) find the minimum or maximum value and determine where it occurs.c) identify the function’s domain and its range.

4. Among all pairs of numbers whose sum is 50, find a pair whose product is as large as possible. What is the maximum product?

Teaching Notes: Remind students to review transformations of graphs before beginning to graph quadratic functions. Emphasize from the book “Graphing Quadratic Functions with Equations in Standard Form”. Stress the use of a from the standard form to determine the direction the parabola is opening before

beginning to graph it. Students need to recognize early on the benefits of knowing as much about a graph as possible before beginning to draw it.

In addition to the intercepts, encourage students to use symmetry to find additional points on the graph of a parabola.

Emphasize “Strategy for Solving Problems Involving Maximizing and Minimizing Quadratic Functions” in the book.

Many students will want to give the x-value found with 2bxa

as the maximum or minimum value of

the quadratic function. Emphasize that finding the maximum or minimum is a two-step process. First, find where it occurs, then find what it is.

Answer: 1) 5 23,

6 12 ;2)

3) a. minimum, b. minimum of -4 at 1, c. domain , , range 4, ; 4) (25, 25) , 625

Mini-Lecture 2.3 Day 1Polynomial Functions and Their Graphs

Learning Objectives:1. Identify polynomial functions.2. Recognize characteristics of graphs of polynomial functions.3. Determine end behavior.4. Use factoring to find zeros of polynomial functions.5. Identify zeros and their multiplicities.

6. Use the Intermediate Value Theorem.7. Understand the relationship between degree and turning points.8. Graph polynomial functions.

Examples:1. Use the Lead Coefficient Test to determine the end behavior of the graph of the given polynomial function. 5 3( ) 6 3 5f x x x x

2. Find the zeros of the given polynomial function and give their multiplicity. State whether the graph crosses or touches the x-axis or turns around at each zero.

3 2( ) 4 3 1f x x x

3. Find the zeros of the given polynomial function. 3 2( ) 3 16 48f x x x x 4. Graph the polynomial function. 3 2( ) 4 4f x x x x

Teaching Notes: Encourage students to use the Lead Coefficient test because it helps to know, in general, what the graph should look like before beginning to graph it. Remind the students when using a graphing utility calculator to select a viewing window that will show the end behavior of the graph.

When you begin to teach how to find the zeros of functions, advise students that they may need to review factoring by grouping.

Emphasize “Graphing a Polynomial Function” in the book.

Answer: 1) up on the right and down on the left; 2) zero of 3 with multiplicity 3 crosses, zero of -1 with multiplicity 2 touches and turns; 3) -3, -4, 4;

4)

x

y

5

5

Mini-Lecture 2.3 Day 2Polynomial Functions and Their Graphs

Learning Objectives:1. Identify polynomial functions.2. Recognize characteristics of graphs of polynomial functions.3. Determine end behavior.

4. Use factoring to find zeros of polynomial functions.5. Identify zeros and their multiplicities.6. Use the Intermediate Value Theorem.7. Understand the relationship between degree and turning points.8. Graph polynomial functions.

Example #41 p. 312 Use the equation:

A) Use the leading coefficient test to determine the graph’s end behavior.B) Find the x intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around,

at each intercept.C) Find the y intercept.D) Determine whether the graph has y-axis symmetry, origin symmetry, or neither.E) If necessary, find a few additional points on the graph. Use the maximum number of turning points to

check whether it is drawn correctly.

Answera: A)f(x) rises to the right and falls to the left. B{-2, 1, -1) f(x) crosses the asis at each. C) the y intercept is -2 D) Neither E)

Factor by grouping because the polynomial has 4 terms.

Set P(x) = 0 and solve.

The zeros of P(x) are -2, 2, and 1.

Teaching Notes: Encourage students to use the Lead Coefficient test because it helps to know, in general, what the graph should look like before beginning to graph it. Remind the students when using a graphing utility calculator to select a viewing window that will show the end behavior of the graph. When you begin to teach how to find the zeros of functions, advise students that they may need to

review factoring by grouping. Emphasize “Graphing a Polynomial Function” in the book.

Graphing Polynomials Practice Name: _________________________________

Example: 4423 xxxxP

a) Find the zeros by factoring.

b) Determine end behavior. As x , )(xP .

As x , )(xP .

c) Find the y intercept by plugging in zero for x. (0,4)

d) Plug in x values between the zeros to find local maxima.

e) Connect all the points in a smooth curve and you are done!

Graph each of the following below by following the steps above.

1. xxxxP 32 23 a)

b)

c)

d)

2. 234 32 xxxxP a)

Because the leading coefficient is positive and the degree is odd, the graph should go up on the right and down on the left.

Plug in -1 and 1.5 for x. Plot the resulting ordered pairs. (-1, 6) and (1.5, 0.875).

b)

c)

d)

3. 234 12 xxxxPa)

b)

c)

d)

4. 842 23 xxxxPa)

b)

c)

d)

5. 43 24 xxxPa)

b)

c)

d)

Pre-Calculus Name:_________________Section 2.1 to 2.3Quiz

Evaluate. Write answer in a + bi form.

1.) (7 - 3 i)(-2 – 5i) 2.)

Find all solutions to the equation. Use factoring, quadratic, etc. Write answer in a + bi form.

3.) 36x2 + 9 = 0 4.) x2 - 6x + 10 = 0

5-6 Fill in all of the requested information and draw a graph of the function.

5.) f(x) = x2 – 2x - 3 6.) f(x) = -(x – 1)2 + 4

Vertex:___________

Max/Min:_________

X-Int:____________

Y-Int:____________

Domain:_________

Range:___________

Vertex:___________

Max/Min:_________

X-Int:____________

Y-Int:____________

Domain:_________

Range:___________

7.) When a football is kicked, the height of the football, in feet, can be modeled by , where x is the horizontal distance, in feet, from the point of impact with the

kickers foot. What is the maximum height of the punt and how far from the point of impact does this occur? If the ball is not blocked, how far down the field will it go before hitting the field?

8.) p(x) = x2(x – 1)3(x + 2) 9.) p(x) = x3 + x2 – 4x – 4

10.) You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Assignments: Section 2.5-2.8Mrs. Badr – [email protected]


248-676-8320 ext. 7049www.classjump.com/mrsbadr

2.4 2.5 2.6 2.7 2.8

Day 17 1 – 45 EOO

Day 18 1-23 odd

Day 19 33-35, 53-60

Day 20 Graphing worksheet

Day 21 21-35 Odd,71-78

Day 22 1-57 EOO

Day 23 3-59 EOO

Day 24 1-10, 21-30

Day 25 Chapter Review pg 385 #11-29

Day 26 Go over Chapter Review

Day 27 Test 2.4-2.8 includes both calculator allowed and no calculator allowed portions

End of Marking Period 1

Mini-Lecture 2.4Dividing Polynomials; Remainder and Factor Theorems


1. Use long division to divide polynomials.

2. Use synthetic division to divide polynomials.3. Evaluate a polynomial using the Remainder Theorem.4. Use the Factor Theorem to solve a polynomial equation.

Examples:

1. Divide using long division. 5 4 2

3

6 9 4 63 2

x x x xx

2. Divide using synthetic division. 5 4 3 22 3 6

1x x x x x

x

3. If 4 3 2( ) 8 6 2 4 10f x x x x x , use synthetic division and the Remainder Theorem

to find 14

f

.

4. Solve the equation 3 2 912 1 02

x x x given that 12

is a root.

Teaching Notes:

Work a numeric long division problem next to an algebraic long division problem to emphasize the steps and so that students can see the similarities. Remind students repeatedly that when dividing using either long division or synthetic division, they

must put a zero in place of the missing power. Remind students repeatedly about the appropriate sign of c when using synthetic division to divide by x c . Remind students that they can check their answers for division problems by multiplying the quotient and the divisor and then adding any remainder.

Answer: 1) 22 3x x ; 2) 4 3 2 142 3 5 8

1x x x x

x

; 3)

1 45

4 4f

; 4) 1 1 2, ,

2 4 3

Mini-Lecture 2.5Zeros of Polynomial Functions


1. Use the Rational Zero Theorem to find possible rational zeros.

2. Find zeros of a polynomial function.3. Solve polynomial equations.4. Use the Linear Factorization Theorem to find polynomials with given zeros.5. Use Descartes’s Rule of Signs.

Examples:

1. Use the Rational Zero Theorem to list all of the possible zeros of 4 3 2( ) 5 3 2 4f x x x x x .2. For 3 26 11 6 0x x x ,

a) list all possible rational roots,b) use synthetic division to test the possible rational roots and find an actual root,c) use the quotient from part b) to find the remaining roots and solve the equation.

3. Find the nth-degree polynomial function with real coefficients satisfying the given conditions.

n=4; 13, , 24

i are zeros; (2) 144f

4. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for 4 3 2( ) 3 2 6 4f x x x x x .

Teaching Notes:

Remind students that when using the Rational Root Theorem the lead coefficient is q and the constant is p. Sometimes students think the order is alphabetical and that the lead coefficient is p because it comes first.

Emphasize “Properties of Polynomial Equations” in the book. Many students will forget that if c is the root, then x-c is the factor. They will try to use x+c. Sometimes students have a hard time understanding what is meant in Descartes’s Rule of Signs by “less than by a positive even integer”. Emphasize that whether they are dealing with positive or negative real zeros doesn’t matter. They will always subtract 2 from the maximum number of zeros until they arrive at 1 or zero.

Answer: 1) 1 4 2

1, , 4, , 2,5 5 5

; 2) 3 2, ,1

2 3 ; 3) 4 3 28 22 26 88 24 0x x x x ;

4) 3 or 1 positive real zeros, one negative real zero

Mini-Lecture 2.6Rational Functions and Their Graphs

Learning Objectives:1. Find the domain of rational functions.2. Use arrow notation.3. Identify vertical asymptotes.

4. Identify horizontal asymptotes.5. Use transformations to graph rational functions.6. Graph rational functions.7. Identify slant asymptotes.8. Solve applied problems involving rational functions.

Examples:

1. Find the domain of 2

2

5( )8 2 3

xg xx x

.

2. A company is planning to manufacture all-terrain vehicles (ATV’s). Fixed monthly cost will be $200,000 and it will cost $2500 to produce each ATV.

a) Write the cost function, C, of producing x, ATV’s.b) Write the average cost function,C , of producing x ATV’s.c) Find 1000C .d) What is the horizontal asymptote for the function, C ?

3. Find the horizontal asymptote of 212 5( )

3 4x xh x

x

.

4. The equation for f is given by the simplified expression that results after performing

the indicated operation. Write the equation for f and then graph the function.

211

111

x

x

Teaching Notes: Remind students to find the domain before reducing a rational function. When finding asymptotes, many students will just give 3 as the answer for the vertical asymptote, x=3,

or just give 4 for the horizontal asymptote, y=4. Emphasis that an asymptote is a line and must have the equation of a line.

Before discussing “cost” as a function, give some concrete examples with no variables and have students work them. Example: A chair manufacturer has a fixed monthly cost of $10,000 and it costs $200 to make each chair. How much does it cost to make 2000 chairs? Emphasize “Locating Horizontal Asymptotes” in the book. Students do not need to memorize the rules

in terms of n and m. They just need to know how to find the asymptote if the higher power is in the numerator or in the denominator, or if the degree of the numerator and the degree of the denominator are equal.

Emphasize “Strategies for Graphing Rational Functions” in the book.

Answer: 1)1 1 3 3

, , ,2 2 4 4

; 2) a. ( ) 200, 00 2500C x x ,

b) 200, 000 2500

( )x

C xx

; c)$2700; d) 2500y ; 3) y=4; 4)

2

2

2 1( )

x xf x

x x

Mini-Lecture 2.7Polynomial and Rational Inequalities


1. Solve polynomial inequalities.2. Solve rational inequalities.

3. Solve problems modeled by polynomial or rational inequalities. Examples:

1. Solve and express the solution in interval notation. 29 6 1 0x x

2. Solve and express the solution in interval notation. 1 2

0( 3)

x xx

3. Solve and express the solution in interval notation. 2 1 3x x

4. Find the domain of 2( ) 9 4f x x .

Teaching Notes:

Suggest that students review interval notation. Emphasize the “Procedure For Solving Polynomial Inequalities” in the book. Emphasize that with rational inequalities the solution must never cause division by zero. Caution them to check for the correct use of brackets or parentheses in solutions written in interval notation. Remind students that the first step in solving a rational inequality is to have zero on one side of the

inequality. Many students will try to solve a problem such as

1 32

xx

without subtracting three from both sides. Also, some will have trouble

combining with a common denominator. Remind them to review adding and subtracting rational expressions.

Answer: 1) ; 2) 2,1 3, ; 3) , 1 2, ; 4) 2 2, ,

3 3

Mini-Lecture 2.8Modeling Using Variation


1. Solve direct variation problems.2. Solve inverse variation problems.3. Solve combined variation problems.

4. Solve problems involving joint variation.

Examples:

1. Write an equation that expresses the relationship; then solve for y. Use k as the constant of variation.

x varies directly with y and inversely with the fourth power of z

2. y varies jointly as x and the square root of z and inversely with t cubed. 20y , when 2x , 4z , and 1t . What is y, when 3x , 9z , and 2t ?

3. The volume of a sphere varies directly with the cube of the radius. If the volume of a

sphere with a radius of 2 inches is 323

cubic inches, what would be the volume of a

sphere with a radius of 6 inches?

4. The force that it requires to stretch a spring varies directly with the distance that it is stretched. If a 10 pound force can stretch a spring 8 inches, how much force would it take to stretch the spring 12 inches?

Teaching Notes:

Remind students to always find the constant of variation first, if enough information is given in the problem.

Tell students to read each problem carefully. A common mistake is to read square root as squared. Emphasize “Solving Variation Problems” in the book. Emphasize the difference between joint variation and combined variation. Some students will get these confused.

Answer: 1) 4xz

yk

2)45

8y ; 3) 3288 in ; 4) 15 lbs.

Name: ______________________

Graphing Rational Functions: Pre-calculusDirections: Please graph each of the following functions carefully. Be sure to fill in all required information. REMEMBER: Graphs may cross horizontal or slant asymptotes, but should never EVER cross a vertical asymptote.

1. 4

32

2

xx

xf

2. 212 2

xx

xg

3. xx

xxxg

4243

2

2

x-intercepts:

y-intercepts:

x-intercepts:

y-intercepts:

x-intercepts:

y-intercepts:

x-intercepts:

y-intercepts:

4. 42

23

x

xxxr

5. 92

2

xx

xf

x-intercepts:

y-intercepts:

x-intercepts:

y-intercepts:

x-intercepts:

y-intercepts:

6. 183

952

xxx

xf

7. 7

232

3

xx

xxf

x-intercepts:

y-intercepts:

8. 82

42

xxx

xf

Precalculus Graphing Rational Functions: NO CALCULATOR Name: ___________________

Find each of the following pieces of information below. Use it to graph the function.

x-intercepts: __________________

y-intercepts: __________________

V.A: _________________________

H.A: _________________________

Slant Asymptote (if necessary):

x-intercepts: ___________________

y-intercepts: ___________________

V.A: __________________________

H.A: __________________________

Slant Asymptote (if necessary):

PreCalculus – Calculator Portion Name: _________________________

Solve each inequality.

1. 2 7 10x x

2. 3 2

01

x xx

3. 1 23

xx

4. B varies directly as A and inversely as the square of C. B = 7 when A = 9 and C = 6. Find B when A = 4 and C = 8.

5. The illumination provided by a car’s headlight varies inversely as the square of the distance from the headlight. A car’s headlight produces an illumination of 3.75 footcandles at a distance of 40 feet. What is the illumination when the distance is 50 feet?

Assignments: Sections 4.1-4.3

Mrs. Badr- [email protected] ext. 7049www.classjump.com/mrsbadr

4.1 4.2 4.3

Day 11-40

Skip 7-12

Day 241-69 odd71-76 all

Day 3Unit Circle: YOU MUST MEMORIZE THIS

Day 4 5-24

Day 5 25-38

Day 6 1-41 odd

Day 7 53-60 all

Day 8 Chapter Review 1-43 oddUnit Circle Test Attempt 1

Day 9Go over Chapter Review 1-43 odd

Day 10 Question & Answer Session for Test 4.1-4.3Unit Circle Test Attempt 2

Day 11Test 4.1-4.3

This test contains both calculator allowed and no calculator allowed portions

Unit Circle



Unit Circle Test

Place the degree angle measure of each angle in the dashed blanks inside the circle.Place the radian measure of each angle in the solid blanks inside the circle. Place the coordinates of each point in the ordered pairs outside of the circle.

PreCalculus 4.1 – 4.3 Test NO CALCULATOR Name: _______________________________________

Answer each question carefully. You MUST show work for full credit.

1. Given the point on the unit circle below, find all six trigonometric functions.

2. Given that , , Find the exact value of each of the remaining trig functions.

Use an identity to find the value of each expression.

3.

4.

Find the exact value of each expression below.

5.

6.

7.

8.

PreCalculus 4.1 – 4.3 Test CALCULATOR Name: _______________________________________

Answer each question carefully. You MUST show work (neatly) for full credit. Please circle your final answer(s).

Convert the following degrees to radians: Convert the following radians to degrees:

1. 325o 3.

2. -415o 4.

Find one positive and one negative coterminal angle. Your answer must be in the same form as the question.

5. 6.

Answer the following questions. Be sure to draw a picture if necessary.

7. When a six foot pole casts a four foot shadow, what is the angle of elevation of the sun? Round to the nearest degree.

Find the labeled side length x.

8 . 9.

16.6

10 x

x

Assignments: Sections 4.4 , 4.8Mrs. Badr – PreCalculus [email protected] ext. 7049www.classjump.com/mrsbadr

56o

35o


http://www.google.com/imgres?q=michigan+state+university+logo&um=1&hl=en&sa=N&biw=1440&bih=719&tbm=isch&tbnid=TqDGTlyIG2HOBM:&imgrefurl=http://campkesem.org/msu/&docid=y7iUc5PtiSBEBM&imgurl=http://campkesem.org/wp-content/themes/kesemtheme/images/logo-msu.jpg&w=800&h=1000&ei=KYeOUIbeO6eayQHt_oHgAw&zoom=1&iact=hc&vpx=816&vpy=248&dur=563&hovh=251&hovw=201&tx=117&ty=145&sig=117770838873514997370&page=1&tbnh=112&tbnw=88&start=0&ndsp=27&ved=1t:429,r:5,s:0,i:160%00%E5%A1%B9%EF%92%81%E1%B4%BB%E4%A1%BF%E2%B2%AF%E5%B6%82%E8%97%84%E6%8C%A7%00%00%EA%AE%A5

4.4 4.8

Day 12 #9-22, 23, 29, 33, 35-75 EOO

Day 13 4.4 Worksheet

Day 14 #1-15 Odd, 29, 31, 33, 35, 41, 45, 49, 51,53

Day 15 Pg 581 #47-67 Odd, 115-124

Day 16 Review

Day 17 Quiz 4.4, 4.8

4.4 Worksheet Name: ___________________________ Hour: ____

1. Given that 1312sin

, and is in Quadrant III, Find all 5 remaining trig functions.

sin cos tan sec csc cot

2. Given that 31tan

, and 0sin , Find all 5 remaining trig functions.


3. Given that 4csc , and is in Quadrant III, Find all 5 remaining trig functions.


#4-12: Find a reference angle for each of the following, and then evaluate using the reference angle.

4. 300sin 5. 2

9tan 6. 510sec

7. 240sec 8. 225sin 9. 6

35cos

10. 405tan11.

6tan

12. 3

13cot

#13-18: Evaluate each of the following:

13.2

3sin3

coscos3

sin 14. cos

6sin0cos

4sin

15.

65sin

411cos

65cos

411sin

16.

45sin

317cos

45cos

317sin

17.

65cos

415tan

23sin 18.

65cos

38tan

23sin

Assignments: Sections 4.5-4.6

Mrs. Badr- [email protected] ext. 7049www.classjump.com/mrsbadr



4.5 4.6Day 18 Amplitude and Vertical Shift

#1, 3, 5, 31, 32, 53-56Day 19 Period and Horizontal Shifts

#17-20, 35, 36, 43, 44Day 20 Graphing Sine and Cosine

worksheetDay 21 Graphing Tan and Cot

#5-11 odd, 17-23 oddDay 22 Graphing Sec and Cse

#29-43 oddDay 23 Graphing Tan, Cot, Sec, Cse

worksheetDay 24 Graphing Review Worksheet IDay 25 Graphing Review Worksheet IIDay 26 ReviewDay 27 Test-Graphing Trig Functions

Trigonometry Name_______________________________

WKS – Graphing Sine & Cosine Date___________________ Hour______

1. Consider the function:1cos3( ) 2

2 6y x

The amplitude is ________ The phase shift is ________ units up down left right

The period is ________ The vertical shift is _______ units up down left right

2. For 6sin(8 2 ) 3y x

Write the equation in standard form____________________________________



3. For sin ( )y a b x c d give a complete description of the translation that occurs based upon

a ________________________________________________________________________

b ________________________________________________________________________

c ________________________________________________________________________

d ________________________________________________________________________ Graph the basic function, either siny x or cosy x and then graph one complete cycle of the following functions.4-5. Sketch the graph of 2sin 1y x The amplitude is ________

The phase shift is ________ units up down left right

The period is ________

The vertical shift is _______ units up down left right

Neatly and accurately graph one cycle of the given function.

6. Sketch the graph of 1cos3

2y x

The amplitude is ________




7. Sketch the graph of cos(2 )2

y x





8. Sketch the graph of 2cos( ) 13

y x





9. Sketch the graph of 3 ( ) 14

y sin x





10. Sketch the graph of12sin( ) 12 2

y x






Trigonometry Name________________________________ Graphing TestTan, Cot, Sec, Csc Date____________________ Hour _______

1. For the function 12 tan( ) 12 4

y x state the following




2. For the function 4csc(2 ) 33

y x state the following




3. Graph one cycle of the basic function and then graph 1tan2

y x





4. Graph one cycle of the basic function and then graph tan(2 ) 2y x Write the equation in standard form____________________________________





5. Graph one cycle of the basic function and then graph cot 2y x The amplitude is ________




6. Graph one cycle of the basic function and then

graph cot( )2 8xy





The vertical shift is _______ units up down left right7. Graph one cycle of the basic function and then graph

1 csc 14

y x





8. Graph one cycle of the basic function and then graph

1 csc( )2 4

y x





9. Graph one cycle of the basic function and then graph sec 2 1y x The amplitude is ________




10. Graph one cycle of the basic function and then graph 2sec 1y x The amplitude is ________




PC 4:5-6 Name_______________________________

Graphing Review Worksheet I Date___________________ Hour______

1A. Consider the function: 2cos3( ) 13

y x .



2B. For 1 3sin(2 ) 32 2

y x




2-4. Sketch the graph of1 sin 12

y x





5-7. Sketch the graph of cos 2( )4

y x





8-10. Sketch the graph of tan(2 ) 2y x

Write the equation in standard form___________________________________





11-13. Sketch the graph of sec( )3

y x





14-16. Sketch the graph of cot 2y x ____________





17-20. Sketch the graph of 1sec 22

y x





PC 4:5-6 Name_______________________________ Graphing Review Worksheet II Date___________________ Hour______

1. Consider the function: 2cos( ) 13

y x .



2. For sin(2 ) 23

y x



The period is ________ The vertical shift is _______ units up down left rightGraph the basic function, either siny x or cosy x and then graph one complete cycle of the following functions.

4-5. Sketch the graph of1 sin 12

y x





Neatly and accurately graph one cycle of the given function.

6. Sketch the graph of cos(2 )2

y x

Write the equation in standard form_________________________





7. Sketch the graph of sec( )3

y x





8. Sketch the graph of csc( ) 14

y x





9. Sketch the graph of cot 2 1y x





10. Sketch the graph of tan(2 ) 2y x






PC 4(5-6) Test Name_______________________________

Date___________________ Hour______

1A. Consider the function: 34

8sin6

xy .



2B. For 48

4cos3

xy



2-4. Sketch the graph of 232

1sin

xy





5-7. Sketch the graph of 14

3cos2

xy





8-10. Sketch the graph of

2tan2

xy






xy

21cot3










17-20. Sketch the graph of xy 4sec3





Assignments: Sections 5.1 – 5.3Mrs. Badr – [email protected] ext. 7049www.classjump.com/mrsbadr



5.1 5.2 5.3

Day 1 #1 – 31 Odd

Day 2 #33 - 59 Odd

Day 3 #1 – 31 Odd

Day 4 #33 – 63 Odd

Day 5 Identities Practice 1 Worksheet

Day 6 Identities Practice 2 Worksheet

Day 7 #1 – 37 Odd

Day 8 #39 – 67 OddSkip 59, 61

Day 9 Review Worksheet

Day 10 Review

Day 11 Test: 5.1 – 5.3

Verifying Identities Worksheet: 5.1 – 5.2 Name: _________________________________

Practice I

1. 2 21 cot csc

2. sec cos sin tan 0

3. 2coscostan1 222 xxx

4.cos cot sin 2sin

cot

5. 2 2sin csc cos sin

6. xxx sincos)tan(

7. 2cossincossin 22 xxxx

8. 1cot1sin 22 xx

9. xxx 2tan1sec1sec

10. cossec

sectan 2

11.

tantantantan

sinsin

12. 22 coscossinsin

Name: _____________________________Hour: _____

Directions: Verify the identities below. All work must be shown and all steps must be correct in order to get the problem correct. You must get four identities correct to pass. Good luck.

1. 0seccossincot 2 xxxx

Identities Practice 2

2. 1sincotsec xxx

3. 1tansectansec aaaa

4. 1tan1cos 22 xx

5. xxx sin22cossin1 22

6. xx

xxcos

1csc

tancot

7. xx

xxx 22

222

tancos

sincscsin1

8.1csc

11csc

1tan2 2

xtt

9. xx

xx 22

22

costan1

sincos

10. x

xx

xcos

sin1sin1

cos

Trigonometry Practice ReviewName_________________________________ Hour :______

If 3cos5

when 2 , set up and solve for the following

1. tan 2. cos2

2. tan2

4. 2sin

5. SimplifyA) 2sin15 cos15o o B) cos 21 cos36 sin 21 sin 36o o o o

C) 22cos 87 1o D) sin 35 cos 27 cos35 sin 27o o o o

E) 2sin 34 cos34o o F)

sin 91 cos9

o

o

6. Express the following as a function of alone angles whose functional values cos( )

7. Write the expression sin12

as the difference of are known and then evaluate sin sin12

( - ) =

8. Verify: 2

2 tan sin 21 tan

x xx

9. Verfiy: cot csc cos 1 sinx x x x

10. Simplify tan tan

4 12

1 tan tan4 12

11. Verify 2 2sin( )sin( ) sin sin

Trigonometry: HomeworkName__________________________ Hour: _____

If 4cos5

when 02 and

15sin17

when

3 22 show all work as you find the exact value of

the following

sin cos( ) tan( )

cos

tan

tan

Trigonometry5:1-3 Test Name_________________________________ Hr___

If 4cos

5

when 2 , set up and solve for the following

1. tan 2. cos2

2. tan2

4. 2sin

5. SimplifyA) 2sin16 cos16o o B) cos 20 cos35 sin 20 sin 35o o o o

C) 22cos 66 1o D) sin 30 cos 28 cos30 sin 28o o o o

E) 2sin 32 cos32o o F)

sin191 cos19

o

o

6. Express the following as a function of alone whose functional values cos( )2

7. Write the expression cos 15o as the difference of angles are known and then evaluate\ cos 15o = cos( - ) =

8. Verify: xx

xxx 22

222

tancos

sincscsin1

9. Verfiy: xxx sin22cossin1 22

10. Simplify tan tan

12 4

1 tan tan12 4

BONUS: 11. Verify 2 2sin( )sin( ) sin sin

Assignments: Sections 4.7 and 5.5

Ms. Badr – [email protected] ext. 7049www.classjump.com/mrsbadr

4.7 5.5

Day 12 #1-61 ETO

Day 13 Worksheet

Day 14 #1 – 23 Odd

Day 15 #25 – 61 Odd

Day 16 # 63 – 95 Odd

Day 17 5-5 Worksheet

Day 18 Review

Day 19 Test Solving Trig Equations

PC 4-7 Worksheet Name: _________________________ Hour: ____

Report values in the following intervals:


tan x, x is any real number

Note: Your calculator will automatically do this so not need to check on part 1, but you do in parts 2, 3.

Part 1: Use a calculator to approximate the value of each expression correct to five decimal places, if defined.1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

Hint: the answer is not 7 since your calculator automatically uses appropriate intervals.

Hint: the answer is not -8.5 since your calculator will automatically use appropriate intervals.

Hint: the answer is not -8.5 since your calculator will automatically use appropriate intervals.

Part 2: Find the exact value without using a calculator. Make sure you report answers in the correct interval. Hints: #13-18 are unit circle values

13. 14. 15.

16. 17. 18.

Hint: For #19-21 are formula (be careful of interval), #24-27 require the use of the unit circle (be careful of intervals). #28-30 will require you to draw a right triangle.Report values in the following intervals:

tan x, x is any real number

19. 20. 21.

22. 23.

24.

25. 26. 27.

28. 29. 30.

Name: ______________________

Section 1: Find all values of in that satisfies each equation.

Trigonometry Review Worksheet: 5-5

1. 23

2sin

2. tan 2 3x

3. 3sin 10 7 4. 04sin12sin9 2

Section II: Find all values (in radians) that satisfy each equation.

4. 3sin = 23 5. 015sin2 x

Section III: Solve each equation for all real number answers in 2 ,0 .

6. 12sin3cos2cos3sin xxxx

7. 02sintan xx

8. 3sin 10 7

9. 10sin 6 02x

10. sin 2 sinx x

Solving Trig Equations Test – PreCalculus Name: _________________________________

No Calculator!Be sure to show ALL work for full credit.

Solve all equations on the interval 2,0 .

1. 21cos x 2. 2csc x 3.

33tan x

4. xx sin31sin5

5. xx cos29cos7

6. 12sin x

7. 034cos2 x

8. 01cos3cos2 2 xx

9. 03sin4 2 x

10. 03cos2cos2 xx

11. 01sincos2 2 xx

12. 22sin2coscos2sin xxxx

Assignments: Sections 6.1-6.2 and 6.6-6.7Ms. Badr – [email protected] ext. 7049www.classjump.com/mrsbadr


6.1 6.2 6.6 6.7

Day 20 1-15 odd

Day 21 17-29 odd

Day 22 1-28 odd

Day 2321-37 odd, 47, 49,

51 1-7 odd, 17-21 odd

Day 24 Work On Review Sheet

Day 25 Go over Review Sheet

Day 26 Quiz 6:1-3, 6-7

Review 6:1-2, 6-7 Name: _______________________________Hour:____

1. Given: .75.16,94,46 cCmAm Find the length of a.

2. Given: 8.56, 7, 9.04a b c Find the measure of angle C.

3-8. A. Solve the triangle, given: m< 54.4 , 6.24, 8.90oC b c

9-14. Solve the triangle, given: m A =119o , a = 10.2, b= 11.4

Find each specified vector or scalar. Given: u =2i-5j, v =-3i + 7j and w = -i -6j

15. v – w 16. 6v

17. 18.

19. Find the angle between vector v and vector w.

20. Find the vector v in terms of I and j whose magnitude is direction angle

Quiz 6:1-2, 6-7 Name: _______________________________Hour:____

1. Given m< 115.4 , 7.83, 15.42oC b a Find the length of side c.

2. Given: a b c 16 9 10, , . Find Bm

3-8. A. Solve the triangle, given: m< 32 7 37 5 28 6. , . , .a b

9-14. Solve the triangle, given: 120Am , a = 18, b= 3

Find each specified vector or scalar. Given: u =2i-5j, v =-3i + 7j and w = -i -6j

15. w- v 16. -7w

17. 18.

19. Find the angle between vector u and vector w.

20. Find the vector v in terms of I and j whose magnitude is and direction angle

Constructed Response No Calculator Allowed Calculator Allowed

Day 1 In class time to work on review packet, if more days available give additional day, if no time available they just have to work on it at home.

Day 2 Q & A

Day 3 Q & A Q & A

Day 4 Q & A

Day 5 Taken In Class the last day of the marking period before

Exam Day Multiple Choice Given During scheduled exam time

Pre-Calculus Name_______________

Semester 1 Final Exam Review

Constructed Response

1. Given ,

a. List all possible rational zeros

b. Using the graph, synthetic division, and factoring/quadratic formula, find all zeros of the function (you must sow work for each zero).

2. Graph the function

a. Period _______ b. amp___________c. phase shift _____________ d. vert shift __________

e. domain _____________ f. range ________________

3. Prove the following identity

4. Solve the following equation for all real numbers

MULITPLE CHOICE SECTION

Non-Calculator:

1. Find the domain of the function

2. Determine the domain of f+g, f-g, fg, and f/g of

3. Find when

4. Write an equation for the inverse function, when

5. What transformation(s) of , occur based on its parent function?

6. Graph

7. Divide and express the result in standard form

8. State whether the function crosses or turns around at each x-intercept

9. Divide and find the remainder

10. Use Descarte’s Rule of Signs to determine the amount of possible positive and negative zeros for

11. Find the vertical asymptote(s) of

12. Solve the rational inequality >0

13. State the correct value for , , , , , , etc.

14. Use the Pythagorean Identity to find , given and

15. Find the exact value of the expression

16. Be able to identify the graphs of each of the six trigonometric functions.

17. Determine the period of

18. Determine whether the following is True or False:


20. Use the identities for cos( x + y ) to evaluate

21. Find the exact value using identities for :

22. Find the exact value of , if lies in quadrant I, and lies in

quadrant IV

23. Find the exact value of , if lies in quadrant II

24. Solve for

25. Solve: for

Use the following vectors for 29-32: u = 2i – 5j, v = -3i + 7j

26. Find u +v

27. Find 4u – 2v

28. Find the magnitude

29. Find the dot product

PreCalculus Final Exam Review

Calculator Portion

1. You have 600 feet of fencing to enclose a rectangular plot that borders a river. If you do not fence the side along, the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

2. Convert to degrees.

3. Use a calculator to evaluate .

4. Find the arc length of the intercepted arc in a circle of radius 13 in. and central angle of 110o.

5. A building that is 21 meters tall casts a shadow 25 meters long. Find the angle of elevation of the sun to the nearest degree.

6. Solve the equation for all real numbers.

7. Given that , solve the triangle.

8. Given that , solve the triangle.

PreCalculus Midterm ExamConstructed ResponseSemester 1

1.) Given the function , a. List all possible rational zeros.

b. Using the graph, synthetic division, and factoring/quadratic formula, find ALL zeros of the function. You MUST show work for each zero.

2.) Graph the function: :

Amplitude: __________ phase shift: _______ Domain: ____________________________

Vertical shift: __________ Period: __________ Range: _____________________________

3.) Prove the following identity:

4.) Solve the following equation for all real numbers:

PreCalculus Midterm Exam

Calculator Portion – DO NOT WRITE ON THE EXAM!!

1. You have 400 feet of fencing to enclose a rectangular plot that borders a river. If you do not fence the side along, the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

a) 7,500 sq. ft. b) 10,000 sq. ft. c) 20,000 sq. ft. d) 30,000 sq. ft.

2. Convert radians to degrees.

a) -38o b) -76o c) -97o d) -105o

3. Use a calculator to evaluate to four decimal places.

a) 5.6713 b) -0.6181 c) 328.2796 d) 0.0030

4. Given a circle with radius 8 cm and a central angle of 70o, find the arc length of the intercepted arc. Round to the nearest hundredth.

a) 39.1 cm b) 9.77 cm c) 560 cm d) 3.11 cm

5. The angle of depression from the top of a 40 foot lighthouse to a sailboat in the distance is . How far is the sailboat from the base of the lighthouse (to the nearest foot)?

a) 3 miles b) 40 miles c) 572 miles d) 675 miles

6. Solve for all real numbers.

a) b) c) d)

7. In find all possible answers for If a = 15, b = 18, and .

a) 63.10o b) 53.42o c) 53.42o, 126.58o d) 63.1 o, 116.90o

8. In find all possible answers for b if c = 12, a = 9, and .

a) 5.6 b) 11.4 c) 17.9 d) 20.5

PreCalculus Final Exam

Semester 1 – Multiple Choice

NON-CALCULATOR PORTION - DO NOT WRITE ON THIS TEST!

9. Find the domain of the function

a) b) c) d)

10. Find the domain of .

a) b) c) d)

11. Find when

a) b) c) d)

12. Find an equation for the inverse function

a) b) c) d)

13. Which option reflects the correct transformation(s) of , based on its parent function?

a) Reflected over the x-axis, stretched vertically by 2, shifted right 3 units and up 4 units

b) Reflected over the x-axis, shrunk vertically by ½, shifted right 3 units and up 4 units

c) Reflected over the x-axis, stretched horizontally by 2, shifted left 3 units and up 4 units

d) Reflected over the x-axis, stretched horizontally by ½, shifted left 3 units and down 4 units

14. Which of the following is the correct graph of

a) b)

b) d)

15. Divide and express the result in standard form

a) b) c) d)

16. State whether the function crosses or turns around at each x-intercept

a) Crosses at x=0, turns around at x = -2 b) Turns around at x = 0, crosses at x = -2

c) Crosses at x = 0, turns around at x = 2 d) Turns around at x = 0, crosses at x = 2

17. Divide by (x-2) and find the remainder

a) b) c) d)

18. Use Descarte’s Rule of Signs to determine the amount of possible positive and negative zeros for

a) 2 positive, 1 negative b) 1 positive, 2 or 0 negative

c) 2 or 0 positive, 1 or 0 negative d) 2 or 0 positive, 1 negative

19. Find the vertical asymptote(s) of

a) b) c) d)

20. Solve the rational inequality

a) b) c) d)

21. State the correct value for

a) b) c) d)

22. Use the Pythagorean Identity to find , given and

a) b) c) d)

23. Find the exact value of the expression

a) b) c) d)

24. Which of the following is the graph of for ?

a) b) c).

25. Which of the following is the graph of for ?

a) b) c)

26. Evaluate

a) b) c) d)

27. Evaluate for

a) b) c) d)

28. Determine the period of

a) b) c) d)


a) TRUE b) FALSE


a) TRUE b) FALSE

31. Use the identities for to evaluate

a) b) c) d)

32. Find the exact value using identities for :

a) b) c) d) 1

33. Find the exact value of , if lies in quadrant IV, and lies in

quadrant III

a) b) c) d)

34. Find the exact value of , if lies in quadrant II

a) b) c) d)

35. Solve for

a) b) c) d)

36. Solve: for

a) b) c) d)

For questions 26-29, use the following:

v = 5i + 4j w = 6i -9j

37. Find v + w

a) i – 5j b) i + 5j c) 11i + 5j d) 11i – 5j

38. Find 3v – 2w

a) 3i+30j b) 3i – 6j c) 3i-30j d) 3i + 6j

39. Find the magnitude:

a) b) c) d)

40. Find the dot product:

a) 66 b) -21 c) -34 d) -6

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