Free AP® Precalculus Practice Test (2026)

Question 1

For f(x) = −2x⁵ + 3x² − 1, which describes the end behavior of the graph?

Accepted Answer

As x → ∞, f(x) → −∞, and as x → −∞, f(x) → ∞.

Answer

As x → ∞, f(x) → ∞, and as x → −∞, f(x) → −∞.

Answer

As x → ∞ and as x → −∞, f(x) → −∞.

Answer

As x → ∞ and as x → −∞, f(x) → ∞.

Question 2

A geometric sequence has first term 5 and common ratio 3. What is its fourth term?

Accepted Answer

135

Answer

405

Answer

45

Answer

20

Question 3

What is the range of y = 3 sin(2(x − π/4)) + 1?

Accepted Answer

−2 ≤ y ≤ 4

Answer

−3 ≤ y ≤ 3

Answer

−1 ≤ y ≤ 1

Answer

−1 ≤ y ≤ 7

Question 4

Where does r(x) = (x² − 4)/(x² − x − 2) have a hole and a vertical asymptote?

Accepted Answer

A hole at x = 2 and a vertical asymptote at x = −1

Answer

Vertical asymptotes at x = 2 and x = −1

Answer

A hole at x = −1 and a vertical asymptote at x = 2

Answer

A hole at x = −2 and a vertical asymptote at x = −1

Question 5

What are all the solutions of log₂(x) + log₂(x − 2) = 3?

Accepted Answer

x = 4 only

Answer

x = −2 and x = 4

Answer

x = −2 only

Answer

x = 5 only

Question 6

What is the exact value of sin(7π/6)?

Accepted Answer

−1/2

Answer

1/2

Answer

−√3/2

Answer

√3/2

Question 7

What is the horizontal asymptote of the graph of y = (3x² + 1)/(6x² − x)?

Accepted Answer

y = 1/2

Answer

y = 0

Answer

y = 3

Answer

There is none.

Question 8

What is the inverse of f(x) = 3 + ln(x − 1)?

Accepted Answer

f⁻¹(x) = eˣ⁻³ + 1

Answer

f⁻¹(x) = eˣ⁺³ − 1

Answer

f⁻¹(x) = eˣ − 2

Answer

f⁻¹(x) = ln(x − 3) + 1

Question 9

What are all solutions of 2 cos x = −1 for 0 ≤ x < 2π?

Accepted Answer

x = 2π/3 and x = 4π/3

Answer

x = π/3 and x = 5π/3

Answer

x = 2π/3 and x = 5π/3

Answer

x = 4π/3 and x = 5π/3

Question 10

A polynomial with real coefficients has zeros at x = 2 and x = 3 − i. What is the least possible degree of the polynomial?

Accepted Answer

3

Answer

5

Answer

4

Answer

2

Question 11

For f(x) = x² + 1 and g(x) = 2x − 3, what is f(g(2))?

Accepted Answer

2

Answer

1

Answer

5

Answer

7

Question 12

What is the average rate of change of f(x) = x² − 3x on the interval [1, 4]?

Accepted Answer

2

Answer

1

Answer

5

Answer

6

Question 13

Which point in rectangular coordinates is the same as the polar point (r, θ) = (4, 5π/3)?

Accepted Answer

(2, −2√3)

Answer

(2√3, −2)

Answer

(−2, 2√3)

Answer

(2, 2√3)

Question 14

A town of 500 people doubles in population every 6 years. Which function models the population t years from now?

Accepted Answer

P(t) = 500 · 2^(t/6)

Answer

P(t) = 500 · 2^(6t)

Answer

P(t) = 500 · (1/2)^(t/6)

Answer

P(t) = 500 + 2(t/6)

Question 15

A function has these values: f(0) = 3, f(1) = 5, f(2) = 9, f(3) = 15, f(4) = 23. Which type of function fits these values exactly?

Accepted Answer

Quadratic

Answer

Linear

Answer

Exponential

Answer

Logarithmic

Question 16

What is the value of arcsin(sin(5π/6))?

Accepted Answer

π/6

Answer

5π/6

Answer

−π/6

Answer

7π/6

Question 17

An exponential function f(x) = a · bˣ has f(0) = 4 and f(2) = 36. What is f(3)?

Accepted Answer

108

Answer

52

Answer

72

Answer

324

Question 18

The graph of y = f(x) passes through the point (4, 5). Which point must be on the graph of y = 2f(x − 1) − 3?

Accepted Answer

(5, 7)

Answer

(3, 7)

Answer

(5, 4)

Answer

(4, 7)

Question 19

A Ferris wheel 40 m in diameter has its center 25 m above the ground and turns once every 8 minutes. A rider starts at the lowest point. Which function gives the rider’s height in meters t minutes later?

Accepted Answer

h(t) = 25 − 20 cos(πt/4)

Answer

h(t) = 25 + 20 cos(πt/4)

Answer

h(t) = 25 − 20 cos(8t)

Answer

h(t) = 20 − 25 cos(πt/4)

Question 20

Data plotted on a semi-log graph, with log y on the vertical axis and x on the horizontal axis, fall on a straight line. Which type of function best models y as a function of x?

Accepted Answer

Exponential

Answer

Linear

Answer

Quadratic

Answer

Logarithmic

Question 21

What is the solution set of (x − 1)(x + 2)² < 0?

Accepted Answer

x < 1, except x = −2

Answer

x < 1

Answer

−2 < x < 1

Answer

x > 1

Question 22

Which expression is equivalent to (1 − cos²x)/(sin x cos x) wherever both are defined?

Accepted Answer

tan x

Answer

cot x

Answer

sec x

Answer

sin x

Question 23

What is the value of log₃ 54 − log₃ 2?

Accepted Answer

3

Answer

log₃ 52

Answer

27

Answer

1

Question 24

Which polynomial has a graph that crosses the x-axis at x = −1 and touches the x-axis without crossing at x = 3?

Accepted Answer

f(x) = (x + 1)(x − 3)²

Answer

f(x) = (x + 1)²(x − 3)

Answer

f(x) = (x − 1)(x + 3)²

Answer

f(x) = (x + 1)(x − 3)

Question 25

For the polar function r = 1 + 2 sin θ on the interval π/2 < θ < π, what happens to points on the graph?

Accepted Answer

They get closer to the origin, because r is positive and decreasing.

Answer

They get farther from the origin, because r is positive and increasing.

Answer

They stay the same distance from the origin.

Answer

They are on the opposite side of the origin, because r is negative.

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