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Precalculus

Practice Test

PrecalculusPractice Test
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Table of contents
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Functions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Practice Test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.

1.

x 2 9 + y 2 4 =1 x 2 9 + y 2 4 =1

2.

9 y 2 +16 x 2 −36y+32x−92=0 9 y 2 +16 x 2 −36y+32x−92=0

For the following exercises, sketch the graph, identifying the center, vertices, and foci.

3.

( x−3 ) 2 64 + ( y−2 ) 2 36 =1 ( x−3 ) 2 64 + ( y−2 ) 2 36 =1

4.

2 x 2 + y 2 +8x−6y−7=0 2 x 2 + y 2 +8x−6y−7=0

5.

Write the standard form equation of an ellipse with a center at ( 1,2 ), ( 1,2 ), vertex at ( 7,2 ), ( 7,2 ), and focus at ( 4,2 ). ( 4,2 ).

6.

A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.

7.

x 2 49 − y 2 81 =1 x 2 49 − y 2 81 =1

8.

16 y 2 −9 x 2 +128y+112=0 16 y 2 −9 x 2 +128y+112=0

For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

9.

( x−3 ) 2 25 − ( y+3 ) 2 1 =1 ( x−3 ) 2 25 − ( y+3 ) 2 1 =1

10.

y 2 − x 2 +4y−4x−18=0 y 2 − x 2 +4y−4x−18=0

11.

Write the standard form equation of a hyperbola with foci at ( 1,0 ) ( 1,0 ) and ( 1,6 ), ( 1,6 ), and a vertex at ( 1,2 ). ( 1,2 ).

For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

12.

y 2 +10x=0 y 2 +10x=0

13.

3 x 2 −12x−y+11=0 3 x 2 −12x−y+11=0

For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.

14.

( x−1 ) 2 =−4( y+3 ) ( x−1 ) 2 =−4( y+3 )

15.

y 2 +8x−8y+40=0 y 2 +8x−8y+40=0

16.

Write the equation of a parabola with a focus at ( 2,3 ) ( 2,3 ) and directrix y=−1. y=−1.

17.

A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

For the following exercises, determine which conic section is represented by the given equation, and then determine the angle θ θ that will eliminate the xy xy term.

18.

3 x 2 −2xy+3 y 2 =4 3 x 2 −2xy+3 y 2 =4

19.

x 2 +4xy+4 y 2 +6x−8y=0 x 2 +4xy+4 y 2 +6x−8y=0

For the following exercises, rewrite in the x ′ y ′ x ′ y ′ system without the x ′ y ′ x ′ y ′ term, and graph the rotated graph.

20.

11 x 2 +10 3 xy+ y 2 =4 11 x 2 +10 3 xy+ y 2 =4

21.

16 x 2 +24xy+9 y 2 −125x=0 16 x 2 +24xy+9 y 2 −125x=0

For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.

22.

r= 3 2−sinθ r= 3 2−sinθ

23.

r= 5 4+6cosθ r= 5 4+6cosθ

For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.

24.

r= 12 4−8sinθ r= 12 4−8sinθ

25.

r= 2 4+4sinθ r= 2 4+4sinθ

26.

Find a polar equation of the conic with focus at the origin, eccentricity of e=2, e=2, and directrix: x=3. x=3.

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