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Final Exam Review Packet

Functions

For problems 1 – 6, solve for the domain algebraically. Then write the domain in interval notation.

1. [pic] 2. [pic]

3. [pic] 4. f(x) = x2 – 4x + 3

5. [pic] 6. [pic]

For problems 7 – 10,

A. Identify the parent function.

B. Determine the domain of the function (in interval notation).

C. Determine the range of the function (in interval notation).

D. Identify any intervals of increasing, decreasing, and constant behavior.

E. Identify any x-intercepts or y-intercepts.

F. Identify any maximum or minimum points.

G. Find f(−3).

H. Find f(x) = 2.

7. 8.

9. 10.

For problems 11 − 16, let f(x) = 2x2 + x − 3 and g(x) = x − 1. Find the following:

11. (f + g)(x) 12. (g − f)(x)

13. (f − g)(x) 14. (fg)(x)

15. [pic] 16. [pic]

For problems 17 − 26, let g(x) = 2x and h(x) = x2 + 4. Find the following:

17. [pic] 18. [pic]

19. [pic] 20. [pic]

21. [pic] 22. [pic]

23. [pic] 24. [pic]

25. [pic] 26. [pic]

Polynomials

For problems 1 – 4, use the Factor Theorem to determine if the given x-value is a root or linear factor of the polynomial.

1. x = 2 2. x = 5

f(x) = x5 + 5x4 − 5x3 − 25x2 + 4x + 20 f(x) = x5 + 5x4 − 5x3 − 25x2 + 4x + 20

3. x − 3 4. x + 4

f(x) = x6 − 30x4 + 129x2 − 100 f(x) = x6 − 30x4 + 129x2 − 100

For problems 5 − 9, use synthetic division to find the quotient.

5. (x5 + 5x4 − 5x3 − 25x2 + 4x + 20) ( (x + 5)

6. (7x6 − 4x5 + 11x4 − 2x3 + 6x2 − 14x + 10) ( (x + 6)

7. (x9 − 1) ( (x −1)

8. (x6 − 30x4 + 129x2 − 100) ( (x + 2)

9. (2x4 + 6x2 − 4) ( (x − 7)

For problems 10 − 13, solve for all the roots of the polynomial. Then write the polynomial as the product of linear factors.

10. f(x) = x3 + 2x2 − 4x − 8, x = −2

11. f(x) = x3 − 6x2 − 7x, x = −1

12. f(x) = 2x4 + 5x3 − 40x2 + 15x + 18, x = −6, 1

13. f(x) = x4 − 2x3 − 12x2 − 40x − 32, x = −2, 4

For problems 14 – 17, complete parts A − F.

14. x = 2 is a root of y = x3 − 3x2 + 4.

A. Identify the y-intercept

B. Is the equation even or odd?

C. Is the leading coefficient positive or negative?

D. Solve for the roots of the equation.

E. Complete the multiplicity table for the equation.

|Roots |Linear Factors |Multiplicity |Behavior at Root |

| | | | |

| | | | |

| | | | |

| | | | |

F. Sketch a graph of the equation.

Remember, good graphs have the following:

~ Labeled axes

~A labeled scale

~Arrowheads on the graph and axes

~A graph labeled with the equation

15. x = 1 and x = −2 are roots of y = x4 − x3 − 3x2 + 5x − 2.

A. Identify the y-intercept

B. Is the equation even or odd?

C. Is the leading coefficient positive or negative?

D. Solve for the roots of the equation.

E. Complete the multiplicity table for the equation.

|Roots |Linear Factors |Multiplicity |Behavior at Root |

| | | | |

| | | | |

| | | | |

| | | | |

F. Sketch a graph of the equation

Remember, good graphs have the following:

~ Labeled axes

~A labeled scale

~Arrowheads on the graph and axes

~A graph labeled with the equation

16. x = −2 is a root of y = −x3 − 2x2 + 4x + 8.

A. Identify the y-intercept

B. Is the equation even or odd?

C. Is the leading coefficient positive or negative?

D. Solve for the roots of the equation.

E. Complete the multiplicity table for the equation.

|Roots |Linear Factors |Multiplicity |Behavior at Root |

| | | | |

| | | | |

| | | | |

| | | | |

F. Sketch a graph of the equation.

Remember, good graphs have the following:

~ Labeled axes

~A labeled scale

~Arrowheads on the graph and axes

~A graph labeled with the equation

17. x = 0 and x = 3 are roots of y = −x4 + 6x3 − 9x2.

A. Identify the y-intercept

B. Is the equation even or odd?

C. Is the leading coefficient positive or negative?

D. Solve for the roots of the equation.

E. Complete the multiplicity table for the equation.

|Roots |Linear Factors |Multiplicity |Behavior at Root |

| | | | |

| | | | |

| | | | |

| | | | |

F. Sketch a graph of the equation.

Remember, good graphs have the following:

~ Labeled axes

~A labeled scale

~Arrowheads on the graph and axes

~A graph labeled with the equation

For problems 18 − 23, match the equation to the appropriate graph.

A. B.

C. D.

E. F.

18. f(x) = −x3 – 4x2 + 5 19. f(x) = −x4 + 6x3 − 9x2 + 2

20. f(x) = −2.1x5 − 2x4 + 3x3 +2x2 – 2 21. f(x) = x3 – 2x2 – 3x

22. f(x) = x5 + 6x3 + 9x 23. f(x) = x4 – 5x2 + 4

Rational Equations

For problems 1 – 10, solve for the discontinuities. An equation may have more than one discontinuity!!! State if the discontinuity is a vertical asymptote, a horizontal asymptote, or a hole.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. For the rational equation below, solve parts A – E.

[pic]

A. Solve for all the discontinuities D. Make a table of points

x-values y-values

B. Solve for the y-intercept

E. Sketch the graph of the rational equation

C. Solve for the x-intercept(s)

12. For the rational equation below, solve parts A – E.

[pic]

A. Solve for all the discontinuities D. Make a table of points

x-values y-values

B. Solve for the y-intercept

E. Sketch the graph of the rational equation

C. Solve for the x-intercept(s)

13. For the rational equation below, solve parts A – E.

[pic]

A. Solve for all the discontinuities D. Make a table of points

x-values y-values

B. Solve for the y-intercept

E. Sketch the graph of the rational equation

C. Solve for the x-intercept(s)

14. For the rational equation below, solve parts A – E.

[pic]

A. Solve for all the discontinuities D. Make a table of points

x-values y-values

B. Solve for the y-intercept

E. Sketch the graph of the rational equation

C. Solve for the x-intercept(s)

Trigonometry − Part I (Right Triangle Trigonometry and the Unit Circle)

For problems 1 − 7, solve for x. Round your answers to the nearest tenth.

1. 2.

3. 4.

5. Solve for the EXACT VALUE of x. 6. Solve for the EXACT VALUE of x.

7. Solve for the EXACT VALUE of x.

For problems 8 − 13, solve each triangle. (Find each angle measure and side length.) Round your answers to the nearest tenth.

8. 9.

10. 11.

12. 13.

For problems 14 − 17, determine if the given angles are coterminal. Explain why or why not.

14. 240°, 600° 15. 90°, 290°

16. [pic], [pic] 17. [pic], [pic]

For problems 18 − 25, solve for the exact value of sine, cosine, and tangent without using a calculator.

18. [pic] Reference Angle:

Special Triangle:

Coordinates:

Sine: Cosine: Tangent:

19. [pic] Reference Angle:

Special Triangle:

Coordinates:

Sine: Cosine: Tangent:

20. [pic] Reference Angle:

Special Triangle:

Coordinates:

Sine: Cosine: Tangent:

21. [pic] Reference Angle:

Special Triangle:

Coordinates:

Sine: Cosine: Tangent:

22. [pic] Reference Angle:

Special Triangle:

Coordinates:

Sine: Cosine: Tangent:

23. [pic] Reference Angle:

Special Triangle:

Coordinates:

Sine: Cosine: Tangent:

24. [pic] Reference Angle:

Special Triangle:

Coordinates:

Sine: Cosine: Tangent:

25. [pic] Reference Angle:

Special Triangle:

Coordinates:

Sine: Cosine: Tangent:

Trigonometry − Part II (Transformations and Graphs)

For problems 1 − 6,

A. Identify the parent equation of the graph.

B. Identify the transformations of the graph.

C. Calculate the amplitude and the period. (Show all your work for calculating the period!!!)

1. y = −2sin(t) + 6 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

For problems 7 – 10, identify the transformations of the following tangent graphs:

7. y = tan(t − [pic]) + 6 8. [pic]

9. [pic] 10. y = tan(−3t + 4[pic]) + 1

For problems 11 − 14, write an equation with the following characteristics:

11. A sine graph with a stretch of −2, a period shift of 4 and a vertical shift of −6.

12. A cosine graph with a compression of [pic], a period shift of −1, a phase shift of [pic], and a vertical shift of 8.

13. A cosine graph with a reflection over the x-axis, a period shift of [pic], a phase shift of [pic], and a vertical shift of 2.

14. A sine graph with a period shift of [pic], a phase shift of [pic], and a vertical shift of −10.

For problems 15 − 22,

A. Identify the transformations compared to the parent equation.

B. Graph the parent equation.

C. Make a table of values for the given function using the six key points (0, [pic], [pic], [pic], [pic]).

D. Graph the points in the table (from Part C).

E. Connect the points with a smooth line.

F. Describe the domain and the range of the function.

15. [pic] Remember, good graphs have:

~arrowheads on the ends of the graphs

~labeled equations

Domain of Transformed Graph:

Range of Transformed Graph:

16. [pic] Remember, good graphs have:

~arrowheads on the ends of the graphs

~labeled equations

Domain of Transformed Graph:

Range of Transformed Graph:

17. [pic] Remember, good graphs have:

~arrowheads on the ends of the graphs

~labeled equations

Domain of Transformed Graph:

Range of Transformed Graph:

18. [pic] Remember, good graphs have:

~arrowheads on the ends of the graphs

~labeled equations

Domain of Transformed Graph:

Range of Transformed Graph:

19. [pic] Remember, good graphs have:

~arrowheads on the ends of the graphs

~labeled equations

Domain of Transformed Graph:

Range of Transformed Graph:

20. [pic] Remember, good graphs have:

~arrowheads on the ends of the graphs

~labeled equations

Domain of Transformed Graph:

Range of Transformed Graph:

21. [pic] Remember, good graphs have:

~arrowheads on the ends of the graphs

~labeled equations

Domain of Transformed Graph:

Range of Transformed Graph:

22. [pic] Remember, good graphs have:

~arrowheads on the ends of the graphs

~labeled equations

Domain of Transformed Graph:

Range of Transformed Graph:

Trigonometric Identities

1. Simplify: [pic]

2. Simplify: [pic]

3. Simplify: [pic]

4. Prove: [pic]

5. Prove: [pic]

6. Prove: [pic]

7. Prove: [pic] (Hint: Think of the difference of squares.)

8. Prove: [pic]

-----------------------

57°

41°

x

4

10.8

x

x

47°

3

x

37°

10.3

[pic]

[pic]

6

22

x

x

[pic]

[pic]

x

4.5

5

53°

42°

[pic]

[pic]

8

17

28°

14

29.3

24°

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