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Quizzes > High School Quizzes > Mathematics

AP Precalc Unit 2 Practice Quiz

Ace your review, master unit essentials

Difficulty: Moderate
Grade: Grade 12
Study OutcomesCheat Sheet
Paper art representing a trivia quiz for high school algebra students

What is the solution to the equation 2x + 3 = 11?
4
3
5
8
By isolating x, subtract 3 from both sides to obtain 2x = 8, and then dividing by 2 gives x = 4. This makes 4 the correct answer.
Which function represents a vertical shift upward by 5 units of a given function f(x)?
f(x) + 5
f(x) - 5
5f(x)
f(x)/5
Adding 5 to the function f(x) translates its graph upward by 5 units. The other options represent different transformations.
What is the factored form of the quadratic expression x² - 9?
(x - 3)(x + 3)
(x - 9)(x + 1)
(x + 3)²
(x - 3)²
x² - 9 is a difference of squares and factors neatly into (x - 3)(x + 3). This is a standard factoring formula.
If f(x) = 3x, what is the value of f(4)?
12
7
4
3
Substituting 4 for x in f(x) = 3x results in 3(4) = 12. This straightforward evaluation confirms the result.
What is the domain of the function f(x) = √(x - 2)?
x ≥ 2
x ≤ 2
x > 2
All real numbers
The expression inside the square root must be non-negative, so x - 2 ≥ 0, which simplifies to x ≥ 2. This is why the domain is x ≥ 2.
What are the solutions to the quadratic equation x² - 5x + 6 = 0?
x = 2, 3
x = -2, -3
x = 1, 6
x = 0, 5
Factoring the quadratic gives (x - 2)(x - 3) = 0, which implies x = 2 or x = 3. This method confirms the correct solutions.
What is the inverse of the function f(x) = 2x + 5?
(x - 5)/2
(x + 5)/2
2x - 5
2x + 5
To find the inverse, swap x and y and solve for y. This process yields y = (x - 5)/2, which is the correct inverse function.
Determine the vertex of the quadratic function f(x) = x² - 4x + 3.
(2, -1)
(2, 3)
(-2, 3)
(-2, -1)
The vertex formula, -b/(2a) for the x-coordinate, gives 2 and substituting back yields -1 for the y-coordinate. Therefore, the vertex is (2, -1).
For the function f(x) = 1/(x - 1), what is the vertical asymptote?
x = 1
y = 1
x = 0
y = 0
The vertical asymptote occurs where the denominator equals zero. Setting x - 1 = 0 reveals that x = 1 is the vertical asymptote.
Which of the following is an incorrect logarithmic identity?
log_b(m + n) = log_b(m) + log_b(n)
log_b(mn) = log_b(m) + log_b(n)
log_b(m/n) = log_b(m) - log_b(n)
log_b(m^n) = n log_b(m)
The identity log_b(m + n) = log_b(m) + log_b(n) is false because logarithms do not distribute over addition. The other identities are standard logarithmic properties.
Find the sum of the arithmetic series: 3, 7, 11, ... , 51.
351
378
364
350
This arithmetic series starts at 3 with a common difference of 4 and contains 13 terms. Using the formula for the sum of an arithmetic series confirms that the sum is 351.
What is the composition (f ∘ g)(x) if f(x) = x² and g(x) = x + 3?
x² + 6x + 9
x² + 3
x + 9
2x² + 3
Substituting g(x) into f gives f(x + 3) = (x + 3)², which expands to x² + 6x + 9. This is the correctly composed function.
Solve for x in the exponential equation 2^(x - 1) = 8.
4
3
8
2
Express 8 as 2^3 so that the equation becomes 2^(x - 1) = 2^3. Equating the exponents yields x - 1 = 3, hence x = 4.
What is the remainder when the polynomial 2x³ - 3x² + x - 5 is divided by (x - 1)?
-5
1
5
-1
According to the Remainder Theorem, the remainder is found by evaluating the polynomial at x = 1, which produces -5. This confirms the correct remainder.
Which transformation does the function f(x) = -2(x - 3)² + 4 represent?
Reflection over the x-axis, vertical stretch of 2, shift right by 3, and shift up by 4
Reflection over the y-axis, vertical stretch of 2, shift left by 3, and shift down by 4
Reflection over the x-axis, vertical stretch of 2, shift left by 3, and shift up by 4
Vertical stretch of 2, shift right by 3, and shift up by 4
The coefficient -2 indicates a reflection over the x-axis and a vertical stretch by a factor of 2. The term (x - 3) shifts the graph right by 3, and the +4 shifts it upward by 4.
Given the functions f(x) = √(2x + 3) and g(x) = x² - 1, what is (f ∘ g)(x)?
√(2x² + 1)
x² - 1
√(2x - 1)
2x² + 1
Substituting g(x) into f(x) gives f(g(x)) = √(2(x² - 1) + 3) = √(2x² - 2 + 3) = √(2x² + 1). This is the correct composition.
Solve for x: log₃(x² - 4) = 2.
x = √13 or x = -√13
x = √13
x = -√13
x = 3
Converting the logarithmic equation to its exponential form yields x² - 4 = 9, so x² = 13. Both x = √13 and x = -√13 satisfy the equation while keeping the logarithm's argument positive.
Find the composition (g ∘ f)(x) if f(x) = ln(x) and g(x) = e^(2x).
x²
2x
ln(x²)
e^(2x)
Calculating (g ∘ f)(x) involves computing g(ln(x)) = e^(2ln(x)) which simplifies to (e^(ln(x)))² = x². This is the correct composite function.
Determine the solution set for the equation |2x - 3| = 5.
x = 4 or x = -1
x = 4
x = -1
x = 1
The absolute value equation splits into two cases: 2x - 3 = 5 and 2x - 3 = -5, leading to x = 4 and x = -1 respectively. Both solutions satisfy the original equation.
If f(x) = x³ - x² - x + 1, what value of x satisfies f(x) = 0?
x = 1
x = -1
x = 0
x = 2
Substituting x = 1 into f(x) yields 1 - 1 - 1 + 1 = 0, confirming that x = 1 is a root of the equation. This is verified through direct substitution.
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Study Outcomes

  1. Analyze and simplify complex algebraic expressions.
  2. Apply exponent and radical rules to solve equations.
  3. Interpret and graph linear, quadratic, and polynomial functions.
  4. Synthesize strategies for solving systems of equations and inequalities.
  5. Evaluate transformations and behaviors of algebraic functions.
  6. Formulate and test models based on real-world algebraic scenarios.

AP Precalc Unit 2 Review Cheat Sheet

  1. Master the Function Concept - Think of a function as a trusty vending machine: each button (input) dispenses exactly one snack (output). Grasping this idea unlocks your ability to tackle more advanced math topics with confidence. Flipped Math: Functions & Limits
  2. Conquer Limits - Limits reveal how functions behave as inputs get really close to specific values, almost like peeking through a tiny keyhole to see what's coming. Learning limit rules helps you analyze continuity and prepare for calculus. OpenStax: Key Concepts in Limits
  3. Graph Domain & Range - Plotting a function on a coordinate plane shows you all the inputs (domain) and outputs (range) it can take. Being able to read these sets from a graph is like mapping out every possible move in a game. Flipped Math: Domain & Range Guide
  4. Embrace Continuity - A continuous function flows without breaks, holes, or jumps - imagine driving on a perfectly smooth road. Recognizing continuity is key for understanding limits and ensuring your graphs behave nicely. OpenStax: Continuity Explained
  5. Solve Linear Equations - Linear equations like ax + b = 0 are the first puzzles you crack in algebra, teaching you to isolate variables step by step. Mastering these skills builds a solid foundation for more complex problem solving. OpenStax: Linear Equations
  6. Dive into Polynomial Properties - Polynomials have cool characteristics like end behavior and real zeros that determine their shape and roots. Understanding these properties makes graphing them feel like assembling a fun Lego set. OpenStax: Polynomial Key Concepts
  7. Learn Law of Sines & Cosines - For triangles that aren't right angles, these laws are your secret weapons to find missing sides or angles. They turn tricky trigonometry problems into straightforward calculations. OpenStax: Law of Sines & Cosines
  8. Understand the Unit Circle - The unit circle maps every angle to a point on a circle of radius one, defining sine and cosine values beautifully. Knowing this circle by heart is like carrying a trigonometry cheat sheet in your mind. OpenStax: The Unit Circle
  9. Explore Polar Coordinates - Polar coordinates use a distance and an angle to locate points, offering a fresh perspective compared to the usual x-y system. This approach shines when you graph spirals, circles, and other curves. OpenStax: Polar Coordinates
  10. Practice Polynomial Division - Long division and synthetic division are your go-to methods to break down complex polynomials into simpler pieces. These techniques help you factor expressions and solve higher‑degree equations with ease. OpenStax: Dividing Polynomials
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