• 80 questions

ALEKS Calculus Placement Test Practice

Eighty unofficial precalculus and trigonometry questions for students aiming at a calculus placement.

If f(x) = x² − 2x, what is f(a + 1)?

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

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Unofficial practice questions, not affiliated with McGraw Hill. Bank checked October 8, 2026.

Next step Next step: calculus practiceAll ten ALEKS topic areas

All 80 questions

Functions and graphs 20

  • A function is defined by f(x) = |x − 3| when x < 1 and f(x) = 1/(x + 1) when x ≥ 1. What is f(−2) + f(1)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Sets, Relations, and Functions
  • The graph of y = ax³ passes through the point (2, −4). What is a? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Graphs and Transformations
  • The graph of y = x³ is shifted 2 units to the left. What is the y-intercept of the new graph? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Graphs and Transformations
  • The graph of y = √x is reflected over the y-axis and then stretched vertically by a factor of 2. Which equation is the new graph? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Graphs and Transformations
  • If f(x) = 1/(x − 2) and g(x) = 4/x, what is f(g(x)) in simplest form? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Combining Functions; Composite Functions; Inverse Functions
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Functions and graphs, continued

  • If f(x) = x² + 1 and g(x) = 3x − 2, what is f(g(x))? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Combining Functions; Composite Functions; Inverse Functions
  • If f(x) = x² − 2x, what is f(a + 1)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Sets, Relations, and Functions
  • If f(x) = x² − 9 and g(x) = x + 3, what is (f/g)(x) for x ≠ −3? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Combining Functions; Composite Functions; Inverse Functions
  • If g(x) = √(x + 7)/(x − 1), what is g(9)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Sets, Relations, and Functions
  • On which interval is f(x) = x² − 4x + 1 decreasing? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Graphs and Transformations
  • The point (2, −3) is on the graph of y = f(x). Which point must be on the graph of y = f(x − 4) + 1? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Graphs and Transformations
  • What are the x-intercepts of the graph of y = |x − 1| − 3? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Graphs and Transformations
  • What is the inverse of f(x) = (x + 1)/(x − 2)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Combining Functions; Composite Functions; Inverse Functions
  • What is the maximum value of f(x) = −x² + 6x − 4? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Graphs and Transformations
  • What is the range of f(x) = (x − 2)² + 5? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Sets, Relations, and Functions
  • What is the union of the intervals [−4, 1) and (−1, 6]? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Sets, Relations, and Functions
  • What is the y-intercept of the graph of f(x) = (x − 2)(x + 1)(x + 4)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Polynomial Functions
  • Where does the graph of f(x) = (x − 1)²(x + 3) touch the x-axis without crossing it? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Polynomial Functions
  • Which interval notation describes the set of x with x ≤ −2 or x > 3? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Sets, Relations, and Functions
  • Which is the vertex form of y = 2x² − 8x + 3? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Functions and Graphs, Graphs and Transformations

Rational expressions 17

  • Divide 2x³ − 3x² + 4x − 5 by x − 2. What are the quotient and the remainder? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify (2x² − 8)/(x² − 4x + 4). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify (8a³b)/(3c) ÷ (4ab²)/(9c²). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify (x² − 1)/(x + 4) ÷ (x + 1)/(x² + 4x). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify (x² − 3x)/(9 − x²). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify (x² − xy)/(x² − y²). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify 2/(x − 1) + 3/(x + 2). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify 5/(6x) − 1/(4x). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify the complex fraction (1 + 1/x)/(1 − 1/x²). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Simplify x/(x² − 4) − 1/(x + 2). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • Solve (x + 1)/(x − 2) = (x + 3)/(x + 1). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Equations
  • Solve 2/x + 1/(x − 1) = 2. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Equations
  • Solve x/(x − 3) = 2 + 3/(x − 3). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Equations
  • What are the asymptotes of the graph of f(x) = (2x + 1)/(x − 3)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Functions
  • What is the domain of f(x) = (x + 1)/(x² − x − 6)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Expressions
  • What is the x-intercept of the graph of f(x) = (2x − 6)/(x + 1)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Rational Functions
  • y varies directly with the square of x, and y = 18 when x = 3. What is y when x = 5? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Rational Expressions, Applications of Rational Expressions

Radical expressions 15

  • At which point does the graph of y = √(x − 4) + 1 start? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Functions
  • Rationalize the denominator of 4/(3 − √5). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Expressions
  • Simplify (27x⁶)^(−2/3), where x is positive. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Higher Roots and Rational Exponents
  • Simplify (x^(2/3) · x^(1/2))/x^(1/6), where x is positive. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Higher Roots and Rational Exponents
  • Simplify 3√8 − √50 + √18. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Expressions
  • Simplify √(6x) · √(15x³), where x is positive. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Expressions
  • Simplify √(50x⁴y³), where x and y are positive. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Expressions
  • Simplify √2/√6 and rationalize the denominator. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Expressions
  • Simplify √3(√6 + 2√3). McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Expressions
  • Simplify ∛54. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Expressions
  • Solve (x − 2)³ = −64. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Equations
  • Solve √(3x + 1) = x − 1. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Equations
  • Solve √(x + 7) − √(x − 1) = 2. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Equations
  • What is 16^(−3/4)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Higher Roots and Rational Exponents
  • What is the domain of f(x) = √(10 − 2x)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Radical Expressions, Radical Functions

Exponentials and logarithms 12

  • A population of 400 triples every 5 years. After how many years does it reach 10,800? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Applications with Exponential Equations
  • Expand log₃ (9x²/y), where x and y are positive. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Properties of Logarithms
  • Solve 2 log₅ x − 1 = 3. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Logarithmic and Exponential Equations
  • Solve 3 ln x + 2 = 14. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Logarithmic and Exponential Equations
  • Solve log₃ x + log₃ (x − 6) = 3. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Logarithmic and Exponential Equations
  • What is e^(2 ln 3)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Properties of Logarithms
  • What is log₄ 8? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Properties of Logarithms
  • What is the domain of f(x) = log₂ (x − 3)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Exponential and Logarithmic Functions
  • What is the range of f(x) = 3^x − 4? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Exponential and Logarithmic Functions
  • Which expression equals log₅ 12? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Properties of Logarithms
  • Which is the exact solution of 3^(x + 1) = 7? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Logarithmic and Exponential Equations
  • Which line is the vertical asymptote of f(x) = log₃ (x + 5) − 2? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Exponentials and Logarithms, Exponential and Logarithmic Functions

Trigonometry 16

  • A triangle has sides of 5 and 8 with a 60° angle between them. How long is the third side? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Right Triangle Trigonometry
  • Convert 5π/12 radians to degrees. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Angles on the Unit Circle
  • If sin θ = 3/5 and θ is in the second quadrant, what is tan θ? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Unit Circle Trigonometry
  • In triangle ABC, angle A is 30°, angle B is 45°, and side a, opposite A, is 6. How long is side b? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Right Triangle Trigonometry
  • The legs of a right triangle are 5 and 5√3. How large is the angle opposite the leg of length 5? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Right Triangle Trigonometry
  • Simplify sec x − sin x tan x. McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Trigonometric Identities
  • What is arccos(−1/2)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Inverse Trigonometric Functions
  • What is cos(4π/3)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Unit Circle Trigonometry
  • What is tan(5π/6)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Unit Circle Trigonometry
  • What is the period of y = 4 cos(x/3)? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Graphing Trigonometric Functions
  • What is the reference angle of 7π/6? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Angles on the Unit Circle
  • Which angle between 0 and 2π is coterminal with −5π/3? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Angles on the Unit Circle
  • Which point on the unit circle belongs to the angle 5π/4? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Unit Circle Trigonometry
  • Which values of x in the interval [0, 2π) solve 2 cos x + 1 = 0? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Trigonometric Equations
  • Which values of x in the interval [0, 2π) solve 2 cos² x = 1 + sin x? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Trigonometric Equations
  • Which values of x in the interval [0, 2π) solve tan x = −1? McGraw Hill ALEKS, curriculum of the course Prep for Calculus 1: Trigonometry, Trigonometric Equations

How this list was compiled: Our own questions. Topic areas: McGraw Hill publishes the curriculum of its ALEKS course Prep for Calculus 1 at https://www.aleks.com/about_aleks/course_products?cmscache=detailed&detailed=ghighedmath35_prepcalculus#ghighedmath35_prepcalculus (281 topics in ten areas; read 2026-10-08). Facts about the Placement Assessment: McGraw Hill, ALEKS Placement, Preparation and Learning, https://www.mheducation.com/highered/digital-products/aleksppl.html (read 2026-10-08). Every answer is worked out from the numbers in the question, and the printed choices are read back and compared with it. Licence: Own work. Questions, answer choices and explanations were written for this site. Nothing is copied or reworded from an ALEKS assessment, an ALEKS learning module or another practice test.

How it works

McGraw Hill describes one ALEKS Placement Assessment for courses from Basic Math to Calculus, not a separate calculus test. This ALEKS precalculus practice test covers the upper areas in 80 questions of our own. Unofficial; not affiliated with McGraw Hill.

Read the full guide

Five areas, 80 questions

The areas are five of the ten in McGraw Hill's published curriculum for the ALEKS course Prep for Calculus 1, read October 8, 2026: functions and graphs, rational expressions, radical expressions, exponentials and logarithms, and trigonometry. Each gets questions in proportion to its topics there: 20, 17, 15, 12 and 16.

What is left out

Arithmetic, linear equations, factoring and geometry are on the ALEKS math placement practice page. Limits are left out too: that curriculum lists limits and continuity among extra topics that are not in the course by default.

Reading the notation

Powers are written x²; other exponents use a caret, as in 16^(−3/4). log₃ is a base-3 logarithm and ln the natural one. An angle with no degree sign is in radians.

  • Learn sine and cosine at π/6, π/4 and π/3. Special angles in the other quadrants reuse those values with a sign.
  • Check each answer to a radical, rational or logarithmic equation in the original: three questions here turn on a rejected answer.

Questions and answers

Is there an ALEKS calculus placement test?

Not a separate one. The one Placement Assessment is open-response, McGraw Hill's page says: 25 questions or less across 314 topics.

What math is on the ALEKS test for calculus?

McGraw Hill's ALEKS PPL page lists no topics for the assessment. Its published curriculum for the course Prep for Calculus 1 covers real numbers, equations, polynomials, lines and geometry as well as the five areas practised here.

Can you use a calculator on the ALEKS placement test?

Louisiana State University's ALEKS page says ALEKS PPL provides an online calculator when needed and that any other calculator is prohibited. Every question on this page can be worked by hand.

What ALEKS score do you need for calculus?

McGraw Hill's page names no score. The number comes from the school: look on its own placement page for the score its calculus course asks for.

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