calculus handbook algebra review

Calculus

Handbook

&

Algebra Review

3

Properties and Operations of Fractions

Let a, b, c, and d be real numbers, variables, or algebraic expressions such that 0b and 0d .

1. Equivalent Fractions: .a c

if and only if ad bcb d

2. Generate Equivalent Fractions: a ad

b bd

3. Rules of Signs:

a a a a aand

b b b b b

4. Add or Subtract (Like Denominators):

a c a c

b b b

5. Add or Subtract (Unlike Denominators):

a c ad bc

b d bd

6. Multiply Fractions: a c ac

b d bd

7. Divide Fractions: , 0a c a d ad

cb d b c bc

8. Simplify Complex Fractions: , 0ab

cd

a c adc

b d bc

Properties of Zero

Let a and b be real numbers, variables, or algebraic expressions:

1. 0 0a a and a a 2. 0 0a 3. 0

0, 0aa

4. 0

ais undefined 5.

0 is indeterminant form

0

6. Zero-Factor Property: 0, 0 0If ab then a or b .

4

Properties of Exponents Properties of Radicals

Let a and b be real numbers, variables, or algebraic

expressions, and let m and n be integers. (All

denominators and bases are non-zero.)

Let a and b be real numbers, variables, or algebraic

expressions, such that the indicated roots are real

numbers, and let m and n be positive integers.

0

1

1.

2.

3.

4.

5.

6. 1

1 17.

8.

9.

10.

m n m n

mm n

n

nm mn

m m m

m m

m

n

n

n

nn

m

n mn

m m

a a a

aa

a

a a

ab a b

a a

b b

a

aa a

a a

a a

a b

b a

nn

1.

2.

3. , 0

4.

5.

6. when n is even

a when n is odd

mn m n

n n n

n

nn

m n mn

nn

n n

a a

a b ab

a ab

bb

a a

a a

a a

a

5

Special Products

Let u and v be real numbers, variables, or algebraic expressions.

Sum and Difference of Same Terms

2 2u v u v u v

Square of a binomial

2 2 2

2 2 2

2

2

u v u uv v

u v u uv v

Cube of a Binomial

3 3 2 2 33 3u v u u v uv v

3 3 2 2 33 3u v u u v uv v

Quadratic Formula

If 2 , 0f x ax bx c a and 2 4 0b ac , then the real zeros of f are:

2 4

2

b b acx

a

Special Arithmetic Operations

1a b

b a

b aba

c c ab ac a b c

a b a b

c c c

ab acb c

a

ab a

c bc

bc

a ac

b

6

Factoring Special Polynomials

Difference of Two Squares

2 2u v u v u v

Perfect Square Trinomial

22 2

22 2

2

2

u uv v u v

u uv v u v

Sum or Difference of Two Cubes

3 3 2 2

3 3 2 2

u v u v u uv v

u v u v u uv v

Factoring By Grouping

3 2 2

2

acx adx bcx bd ax cx d b cx d

ax b cx d

Midpoint Formula

The midpoint of the segment joining the points 1 1,x y and 2 2,x y is:

1 2 1 2Midpoint ,2 2

x x y y

Distance Formula

The distance d between two points 1 1,x y and 2 2,x y in the plane is:

2 2

2 1 2 1d x x y y

7

Properties of Lines Slope

1. General Form: 0Ax By C

2. Vertical line: x a

3. Horizontal line: y b

4. Slope-Intercept Form: y mx b

5. Point-Slope Form: 1 1y y m x x

6. Parallel Lines: Slopes are equal

7. Perpendicular Lines: Slopes are opposite reciprocals

Finding Intercepts

1. To find x-intercepts, set y = 0 and solve the equation for x.

2. To find y intercepts, set x = 0 and solve the equation for y.

Tests for Symmetry

1. The graph of an equation is symmetric with respect to the y-axis if replacing x with –x

yields an equivalent equation.

2. The graph of an equation is symmetric with respect to the x-axis if replacing y with –y

yields an equivalent equation.

3. The graph of an equation is symmetric with respect to the origin if replacing x with –x

and y with –y yields an equivalent equation.

The slope of a non-

vertical line through

1 1 2 2, ,x y and x y is:

2 1

2 1

y ym

x x

where 1 2x x

8

Absolute Value Absolute Value Properties If a is a real number, then the

absolute value of a is: , 0

, 0

a if aa

a if a

Special Triangles

A right triangle with two sides of equal A right triangle with angles of 30°- 60°- 90°

lengths is a 45°- 45°- 90° triangle. has sides in the ratio of 1: 3 : 2 .

1. 0 2.

3. 4. , 0

a a a

aaab a b b

b b

x 2x 60

30

x 3

2x 45

45

x

x

9

Interval Notation

Notation Type of Interval Inequality Graph

,a b

Closed

a x b

,a b

Open

a x b

,a b

a x b

,a b

a x b

,a

x a

,a

Open

x a

,b

x b

,b

Open

x b

,

Entire Number Line

[ ] a b

( ) a b

( ] a b

[ ) a b

[ a

( a

] a

) a

10

Geometry Formulas

11

Trigonometry Identities, etc.

12

Process of Completing the Square

The process of completing the square has five major steps.

The summary below assumes that the equation being solved is in the variable x.

1. Use addition or subtraction to move the constant term to the right side and all other terms to the

left side.

2. Divide each term in the equation (on both sides) by the coefficient of the x2 term, unless the

coefficient is 1.

3. Determine the coefficient of the x term, divide it by two, square it, and add to both sides.

4. Factor the left side as a perfect square trinomial.

And if the object is to solve for x, then:

5. Take the square root of each side, and create two sub-problems from the result.

Process of Finding an Inverse

Solving for an inverse algebraically is a three step process:

1. Set the function = y

2. Swap the x and y variables

3. Solve for y

13

DO NOT MAKE THESE COMMON ALGEBRA MISTAKES!!!

Do Not Simplify Everything in Sight

Complicated fractions become less ugly as elements are simplified from both the numerator and

denominator: 0r

These simplifications can be performed ONLY when the numerator and denominator are factored.

To make sure that you simplify correctly you must:

1. Factor numerator and denominator first

2. Simplify only those factors common to both numerator and denominator

3. Accept the fact that many times no factorization is possible or at least none that

allow you to simplify a common factor.

DO NOT SIMPLIFY EVERYTHING IN SIGHT WITHOUT FACTORING FIRST AS THIS PERSON

DID IN THE EXAMPLE BELOW!

2 2

2 2

3 5 1 3 5 1 3 1 21

2 5 2 5 2 2

x x x x

x x x x

Parentheses Problems!

Change all signs when you distribute a negative sign: a x b a x b Incorrect

a x b a x b Correct

Squaring binomials results in a middle term: 2 2 2a b a b Incorrect

2 2 22a b a ab b Correct

Incorrect distribution involving powers: 5 5

3 2 3 6a b a b Incorrect

2 1 2 1 2

1 3 1 3 3

x x x x x

x x x x x

50 2 25 2

75 3 25 3

14

Coefficient is not glued to the variable.

Fraction Flubs!

Add correctly ; 1 1 1

a b a b

Multiplication:

1 1

5 5x

x

1 1 b a

a b ab

1 1 1

5 5x x

Exponent Errors!

Exponents apply to every FACTOR!

3 3

3 3

2 2

2 8

x x

x x

Use caution when moving from denominator to numerator:

1 11x y

x y

Leave this one alone.

2

2

2

2

14

4

44

xx

xx

Radical Radicals!

The radical applies to every factor inside: 16 16

16 4

x x

x x

OR

33

33

8 8

8 2

x x

x x

But when there are multiple terms inside: 2 2x y x y Leave this one alone. Cannot simplify.

15

Review Lesson 1

Vocabulary: Domain, Interval Notation

Examples:

What is meant by each number line shown below? Represent the domain of each number line two

ways: as an inequality and with correct interval notation.

Inequality: 7 1 3 7x or x

Interval Notation: [ , 1] (3,7]x

Inequality: 9 5 0x or x

Interval Notation: [ 9, 5) (0, )

Note: An open interval can look just like a point. For example, the interval (0, 10), which indicates real values from

0 to 10, not including 0 and 10, looks just like the point (0, 10), which is a point in the plane. We tell the difference

between them by context.

For example,

It doesn’t make any sense to say that ,x a b when (a, b) is a point (i.e. “x is in the point (a, b)” makes

no sense).

Nor does it make sense to say “the domain of f is the point (a, b)”. The domain of f(x) refers only to

x-values. (Even if the domain was just a single value of x, it still wouldn’t be a point.)

Likewise, you can’t say “the lines intersect at the interval (a, b)”. Lines intersect at a point, not an interval.

16

Domain and Range

Given a function y = f(x), the Domain of the function is the set of permissible inputs and the Range is the set of

resulting outputs. Domains can be found algebraically; ranges are often found algebraically and graphically.

Domains and Ranges are sets. Therefore, you must use proper set notation.

When finding the domain of a function, ask yourself what values can't be used. Your domain is everything else.

There are simple basic rules to consider:

- The domain of all polynomial functions is the set of real numbers R.

- Square root functions cannot contain a negative underneath the radical. Set the expression under the radical

greater than or equal to zero and solve for the variable. This will be your domain.

- Rational functions cannot have zeros in the denominator. Determine which values of the input cause the

denominator to equal zero, and set your domain to be everything else.

Examples:

3 2

23

2

( ) 6 5 11,

: ,

( ) 2 3 ,

: ( , ]

1( )

4

: ( , 2) ( 2,2) (2, )

f x x x x

Domain

g t t

Domain

ph p

p

Domain

.

Answers:

- Since f(x) is a polynomial, the domain of f(x) is all real numbers.

- Since g(t) is a square root, set the expression under the radical to greater than or equal to zero: 2 - 3t 0 2

3t 2/3 t. Confirm by graphing: you will see that the graph only exists to the left of 2/3 on the horizontal

axis.

- Since h(p) is a rational function, the denominator cannot equal zero. Set p2 - 4 = 0 and solve: p2 - 4 = 0 (p +

2)(p - 2) = 0 p = -2 or p = 2. These two p values need to be avoided, so the domain of h(p) is all real

numbers except p = -2 or 2.

17

Review Lesson 1 Homework Problems

Complete the following table by filling in every blank box.

Inequality Interval Notation Graph

1. 3 5x or x

2.

3.

4. 5 4x

5.

6.

7. 3 2 3x or x

8.

9.

10. 2 1 4x or x

18

Review Lesson 1 Homework – Continued

Determine the following domains. Use correct interval notation.

11. d(y) = y + 3 12. g(k) = 2k2 + 4k – 6

13. b(n) = 82 n 14. ttm 39)(

15. 42

5)(

x

xxu 16.

1

1)(

rrra

17. cc

cy3

2)(

2 18.

1

4)(

2

w

wwq

19. 3

)(

x

xxf 20. 82)( 2 vvvt

21. t

ttn

1)( 22.

4

4f x

x x

19

Review Lesson 2: Exponents, Radicals, Rational Exponents

Examples:

Use exponent properties to simplify:

A. 4 33 4ab ab B. 3

22xy C.

23

0 2

5

3

x

y z

1 1 4 3

2

3 4

12

a b

a b

3 3 2 3

3 6

2

8

x y

x y

2 3 2 6

2 0 2 2 2 4

5 25

3 9

x x

y z z

Rewrite each expression with positive exponents:

D. 1x E.

3 4

2

12

4

a b

a b

F.

223x

y

1

x

3 2 4 1124

55 5

5

33

a b

aa b

b

2

2

2 2

2 2 2 4

3

3 9

y

x

y y

x x

Convert to radical form: Convert to exponential form:

G.

2

7x H.

9

14y

I. 511 d J.

5 2

1

b

7 2x 914

1

y

511d 2

5

1

b

Simplify the radical by removing all possible factors:

K. 3 481x L.

3 640x M.

4

2

32a

b

3 3 327 3 3 3x x x x =3 6 2 38 5 2 5x x

4 2

2

16 2 4 2a a

b b

20

Review Lesson 2 – Examples - continued

When working with radicals it is often convenient to move the radical expression from the

denominator to the numerator or vice versa through a process called rationalizing.

Rationalize the denominator:

N. 1

2 O.

7

3x

1 2 2

22 2

7 3

3 3

7( 3)

3

x

x x

x

x

Rationalize the numerator:

P. 3

2

x

3 3 3

2 3 2 3

x x x

x x

Q. Here is an expression from calculus called the difference quotient. Notice that this expression

is undefined when 0h . Now rationalize the numerator and simplify. The simplified

function is not undefined when 0h . This procedure is frequently used in calculus.

x h x

h

1

( ) ( ) ( )

x h x x h x x h x h

h x h x h x h x h x h x x h x

21


Simplify each expression. Do not leave any negative exponents in your answers.

1. 4 25x x 2. 2

3x 3. 2

2 46 2x x

4.

5

3

3x

x 5.

222x

6. 3 1

2 32 4x x

7. 2 44 8x x 8.

33 4

5

x y

9.

42

2

5x

y

10.

4 8

3 35x x

11.

33 2

1

42

2

2

x

x

12.

4 2

3 3

1

3

x y

xy

13.

1

3 2

3

12

x x

x x

14.

1 5

2 2

3

2

5 5

5

x

x

15.

2 1

5 23 2x x

Simplify by removing all possible factors from the radical.

16. 8 17. 216x 18. 18

19. 316

27 20. 3

24

125 21.

3 754x

22. 9 4 53 144x y z 23. 5 93 32xy z 24. 3 4 63 56x y z

Convert to radical form: Convert to Exponential Form

25. 5

11x 26.

9

13n

27. 9 7m 28.

5 4

1

x

22

Review Lesson 2 Homework – continued

Rationalize either the denominator or the numerator.

29. 3

12 30.

1

2

x

31. 5

5 2 32.

4

x

x

33. 1

1x x 34.

2

3

x x

35. 2

5

x

x 36.

2

3 5

5

x

23

Review Lesson 3

Use the Quadratic Formula to find all real zeros. Give exact answers.

A. 2 3 9x x

23 3 4(1)( 9) 3 45 3 3 5

2(1) 2 2x

Completely factor:

B. 25 26 5x x C. 2 4 29 49x y z

=(x+5)(5x+1) =(3x-7y2z)(3x+7y2z)

D. 3 64y E.

6 3 9 38 27x y w z

=(y+4)(y2-4y+16) =(2x2y-3w3z)(4x4y2+6x2yw3z+9w6z2)

F. 2 2 3

7 3 2 1 3 2 1x x x x G. 5 2 38 6 12 9x x x

=(3x+2)(1-x)2[7(3x+2)+(1-x)] =2x2(4x3-3) -3(4x3-3)

=(3x+2)(1-x)2[21x+14+1-x] =(4x3-3)(2x2-3)

=(3x+2)(1-x)2(20x+15)

=5(3x+2)(1-x)2(4x+3)

H. 2 3 5 1

3 2 3 23 5 3 5x x x x I. 1 4 1 1

2 5 2 52 3 2 3x x x x

2 13 2

2 13 2

( 3) ( 5) [( 5) ( 3)]

8( 3) ( 5)

x x x x

x x

1152

11

52

( 2) ( 3) [( 3) ( 2)]

2 1

( 2) ( 3)

x x x x

x

x x

24

Synthetic Division Review:

J. Divide 4 210 2 4x x x by 3x

3 1 0 10 2 4

-3 9 3 -3

1 -3 -1 1 1 = 3 2 13 1

3x x x

x

Find all the real zeros of each polynomial.

K. 22 3y x x L. 4 64y x x

0=x(2x-3) 0=x(x3-64)

0=x(x-4)(x2+4x+16)

x=0, 3/2

x=0, 4

M. 3 22 6 3y x x x N. 2

1 8y x

0=x2(2x+1) + 3(2x+1) 0=x2+2x-7

0=(2x+1)(x2+3)

2 321 2 2

2

x

x=-1/2

O. 4 16y x P. 212 5 2y x x

0=(x2+4)(x2-4) 0=(4x+1)(3x-2)

0=(x2+4)(x+2)(x-2)

x=-1/4, 2/3

x=2, -2

Q. 2 3 2 2y x x x R. 238 20

4y x x

0=(x+2)(2x-3-1) 0=3/4x2+8x+20

0=(x+2)(2x-4) 0=3x2+32x+80

x=-2, 2 0=(x+2)(3x+30)

25

Factor and simplify

S. 2 1 1 6

3 5 3 51 3 1 3x x x x

1 13 5

1 13 5

1 13 5

15

13

( 1) ( 3) [( 1) ( 3)]

( 1) ( 3) [2 2]

2( 1) ( 3) ( 1)

2( 3) ( 1)

1

x x x x

x x x

x x x

x x

x

T. 2 11 1

2 21 13 32 23 2

3 2 1 3 6 3 2 2 1 3x x x x x

2 11 13 32 2

2132

2132

2132

2132

2 2

2 2

2 2 2

2 2

2

2

2 (3 2 ) (1 3 ) (3 2 ) (1 3 )

(3 2 ) (1 3 ) [2 (3 2 ) (1 3 )]

(3 2 ) (1 3 ) [6 4 1 3 ]

(3 2 ) (1 3 ) [ 6 1]

6 1

(3 2 ) (1 3 )

x x x x x

x x x x x

x x x x x

x x x x

x x

x x

26


Use the quadratic formula to find all real zeros. Give exact answers.

1. 28 2 1y x x 2. 2 6 1y x x

Completely factor each polynomial:

3. 481 x 4. 3 8x

5. 3 64x 6. 24 4 1x x

7. 23 5 2x x 8. 4 215 16x x

9. 29 12 4x x 10. 4 2 6x x

Use synthetic division to help factor the expression:

11. 3 23 6 2 1 ?x x x x

12. 3 7 6 3 ?x x x

13. 4 213 36 3 ?x x x

27

Review Lesson 3 Homework Problems - continued

Factor and simplify:

14. 1 1

2 22 1 2 2 1x x x

15. 2 35 42 21 3 1 2 5 1 1x x x x x

16. 3 2 1 1

4 3 3 42 3 3 2x x x x

17. 2 1

3 21 3 33

3 2 3 3 3 2 3 3x x x x

Find all of the real zeros of each polynomial.

18. 26 54y x 19. 3 24y x x

20. 2 32 4 2y x x x 21. 21 28

81 9y x x

22. 35 40y x 23. 3 2 4 4y x x x

24. 4 217 16y x x 25.

25 3 4 8 3 4 5 1y x x x

28

Review Lesson 4

Vocabulary: Slope, Parallel, Perpendicular, Linear, Intercepts

Find the x- and y-intercepts of each equation;

A. 23 1y x

X-int:

2

2

2 2

2

2

0 3 1

3 3

0 1

0 1

0 1

1

1

x

x

x

x

x

x

(1,0) (-1,0)

Y-int:

23 0 1

3 1

y

y

non reals

B.

29

2

xy

x

X-int:

2

2

2

2

90

2

92 0 2

2

0 9

9

3

x

x

xx x

x

x

x

x

(3,0) (-3,0)

Y-int:

29 0

0 2

94.5

2

y

y

(0, -4.5)

C. Find the slope of the line that passes through 2,5 5, 4and

4 5 93

5 2 3m

29

Write the equation of the line passing through:

D. 4,3 , 4, 4

78

72

12

7 18 2

4 3 7

4 4 8

3 (4)

3

m

y mx b

b

b

b

y x

E. 1 3 9 9

, , ,10 5 10 5

9 3 65 5 5 6

59 10110 10 10

9 6 95 5 10

9 275 25

1825

6 185 25

( )

m

y mx b

b

b

b

y x

Write an equation of the line through the given point and parallel to the given line.

F. 3, 2 , 7x y

7

7

1

x y

y x

m

Parallel slope = -1

2 1( 3)

1

1

y mx b

b

b

y x

G.

7 3, , 5 3 0

8 4x y

53

5 3 0

3 5

x y

y x

y x

Parallel slope = -5/3

3 5 74 3 8

3 354 24

5324

5 533 24

y mx b

b

b

b

y x

30

Write an equation of the line through the given point and perpendicular to the given line.

H. 3, 2 5 4 8x y

54

5 4 8

4 5 8

2

x y

y x

y x

Perpendicular slope=-4/5

45

25

4 25 5

2 (3)

y mx b

b

b

y x

I. 8,3 2 3 5x y

523 3

2 3 5

3 2 5

x y

y x

y x

Perpendicular slope=3/2

32

32

3 ( 8)

15

15

y mx b

b

b

y x

J. Find the equation of the y-axis.

x = 0

K. Write an equation of the horizontal line through 3, 5 .

y = -5

L. Write an equation of the line with y-intercept at -3 and perpendicular to all vertical lines.

y = -3

31


Find the x- and y-intercepts of each equation:

1. 3 4y x x 2.

2 2 3y x x

3. 2 29y x x 4.

2 4

4

xy

x

Find the slope of the line passing through the given points:

5. 2,1 , 4, 3 6. 3, 5 , 2, 5

7. 2 5 1 5

, , ,3 2 4 6

8. 3 5

, 5 , ,42 6

Write the equation of the line passing through the given points:

9. 4,3 , 0, 5 10. 2,3 , 2, 3

11. 1 2 5

,1 , ,3 3 6

12. 1

,2 , 6,22

32


Write the equations of the lines through the given point (a) parallel to the given line and (b)

perpendicular to the given line.

13. 2,1 4 2 3x y 14. 2 7

, 3 4 73 8

x y

15. 12, 3 4 0x 16. 1,0 3 0y

17. Write an equation of the vertical line with x-intercept at 3.

18. Write an equation of the horizontal line through 0, 5 .

19. Write an equation of the line with y-intercept at -10 and parallel to all horizontal lines.

20. Write an equation of the line with x-intercept at -5 and perpendicular to all horizontal lines.

33

Review Lesson 5 - Piecewise Functions

A. Graph:

2

1 1 1

( ) 2 1

1

x x

f x x

x x

C. Graph: 3 1

( )2 1

x xf x

x x

B. Graph:

1 1

( ) 11

x x

f xx

x

D. Graph:

2 2( )

1 2

x xf x

x


1. Graph:

2

1 0( )

0

x if xf x

x if x

2. Graph:

3

2 0

( ) 1 0

0

x x

g x x

x x

35

Review Lesson 5 Homework Problems – continued

3. Graph: 2 0

( ) 2 0

2 1 0

x x

h x x

x x

4. Graph:

2

1 1 0

2 2 0 1

( ) 2 1 2

1 2

2 4 2 3

x x

x x

k x x x

x

x x

36

Factoring Review

Solve.

1. 22 9 18 0x x 2. 2

6 4 5x x 3. 216 0x

4. 24 6 9 6x x x 5. 3 2

6 45 24x x x 6. 4 25 4 0x x

7. 3 22 3 12 8x x x 8. 3

8 18x x 9. 3 212 16 3 4x x x

37

Factor Completely.

10. 5 3 4 43( 2) (2 1) 12( 2) (2 1)x x x x 11.

1 12 22 3 4 42 (3 4) ( 5) 5 (3 4) ( 5)x x x x x x

12. 3 52 (2 7) 8 (7 2 )x y x xy x 13. 8 16x

14. 2n ny y 15.

3b bw w 16. 1 13 2n n nx x x

17. 3m mp p 18.

2 225( ) 16( )x y x y 19. 2 2x y ax ay

38

Algebra Extra Review Problems

Simplify. Use only positive exponents.

1. 3 2x x x 2.

22 3 3

4 2

x y xy

x y

3. 2

3 4x x 4. Evaluate: 1

303 2 , if 8x x x

Factor completely.

5. 22 50xy x 6. 2 49x

7. 23 5 2x x 8. 3 64x

Determine the domain of each function. Give answer in correct interval notation.

9. ( ) 2f x x 10.

2

2 2( )

4

x xg x

x

11. 2

( )5 20

xh x

x

12.

2

2xy

x x

39

Solve each of the following.

13. 34 16x x 14. 2 4 3x x

Find the slope.

15. (-4 , 2) and (-4 , 10) 16. (2 , 3) and (-1 , 3)

17. Explain the difference between a slope of zero and no slope.

18. Write an example equation for each type listed below.

Exponential: Rational:

Trigonometric: Quadratic:

calculus handbook algebra review

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