AP Precalculus practice questions

Q: Are these real AP Precalculus exam questions?

No. They are original practice questions written to match the topics and style of the AP Precalculus course. For released exam questions, see AP Central from the College Board.

Q: Is it free?

Yes. The practice questions and flashcards on this page are free, with no sign-up.

Q: Can I get more practice questions?

Yes. Paste your own notes or upload a textbook chapter into the quiz maker and it writes a new practice quiz from that material, with explanations for every answer.

Question 1

A polynomial has odd degree and a positive leading coefficient. What is its end behavior?

Accepted Answer

As x → ∞, f(x) → ∞; as x → −∞, f(x) → −∞

Answer

As x → ∞, f(x) → ∞; as x → −∞, f(x) → ∞

Answer

As x → ∞, f(x) → −∞; as x → −∞, f(x) → ∞

Answer

As x → ∞, f(x) → −∞; as x → −∞, f(x) → −∞

Question 2

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

Accepted Answer

x = −2

Answer

x = 2

Answer

x = 2 and x = −2

Answer

There is no vertical asymptote

Question 3

What is the horizontal asymptote of f(x) = (3x² + 1) / (x² − 5)?

Accepted Answer

y = 3

Answer

y = 0

Answer

y = 1/5

Answer

There is none

Question 4

For f(x) = (x − 1)²(x + 3), what does the graph do at x = 1?

Accepted Answer

It touches the x-axis and turns around.

Answer

It crosses the x-axis.

Answer

It has a vertical asymptote.

Answer

It has a hole.

Question 5

What is the average rate of change of f(x) = x² from x = 1 to x = 3?

Accepted Answer

4

Answer

2

Answer

8

Answer

6

Question 6

What is the maximum number of turning points a degree-4 polynomial can have?

Accepted Answer

3

Answer

4

Answer

2

Answer

5

Question 7

How is the graph of g(x) = f(x − 3) + 2 related to the graph of f?

Accepted Answer

Shifted 3 units right and 2 units up

Answer

Shifted 3 units left and 2 units up

Answer

Shifted 3 units right and 2 units down

Answer

Stretched by a factor of 3

Question 8

On an interval, f is increasing and its graph is concave up. What is true about the rate of change of f?

Accepted Answer

It is positive and increasing.

Answer

It is positive and decreasing.

Answer

It is negative and increasing.

Answer

It is constant.

Question 9

A population is modeled by P(t) = 200(1.05)ᵗ, with t in years. What does 1.05 tell you?

Accepted Answer

The population grows by 5% each year.

Answer

The population grows by 105 each year.

Answer

The population grows by 1.05% each year.

Answer

The starting population is 105.

Question 10

What is log₂(32)?

Accepted Answer

5

Answer

16

Answer

6

Answer

4

Question 11

Solve 3ˣ = 81.

Accepted Answer

x = 4

Answer

x = 3

Answer

x = 27

Answer

x = 9

Question 12

What is ln(e⁷)?

Accepted Answer

7

Answer

e⁷

Answer

1

Answer

7e

Question 13

Which expression is equal to log(a) + log(b)?

Accepted Answer

log(ab)

Answer

log(a + b)

Answer

log(a) · log(b)

Answer

log(a / b)

Question 14

A table shows outputs that are multiplied by the same factor each time the input increases by 1. Which model fits best?

Accepted Answer

Exponential

Answer

Linear

Answer

Quadratic

Answer

Logarithmic

Question 15

A substance has a half-life of 6 hours. If you start with 80 mg, how much remains after 18 hours?

Accepted Answer

10 mg

Answer

20 mg

Answer

40 mg

Answer

5 mg

Question 16

What is the inverse of f(x) = eˣ?

Accepted Answer

f⁻¹(x) = ln x

Answer

f⁻¹(x) = 1/eˣ

Answer

f⁻¹(x) = e⁻ˣ

Answer

f⁻¹(x) = log₁₀ x

Question 17

What is sin(π/6)?

Accepted Answer

1/2

Answer

√3/2

Answer

√2/2

Answer

1

Question 18

What is the period of y = sin(2x)?

Accepted Answer

π

Answer

2π

Answer

4π

Answer

π/2

Question 19

What is the amplitude of y = −3cos(x) + 1?

Accepted Answer

3

Answer

−3

Answer

1

Answer

4

Question 20

What is the midline of y = 4sin(x) − 2?

Accepted Answer

y = −2

Answer

y = 4

Answer

y = 2

Answer

y = 0

Question 21

What is tan(π/4)?

Accepted Answer

1

Answer

0

Answer

√2/2

Answer

Undefined

Question 22

What are the coordinates of the point on the unit circle at θ = π?

Accepted Answer

(−1, 0)

Answer

(0, 1)

Answer

(1, 0)

Answer

(0, −1)

Question 23

Convert the polar point (r, θ) = (2, π/2) to rectangular coordinates.

Accepted Answer

(0, 2)

Answer

(2, 0)

Answer

(0, −2)

Answer

(2, 2)

Question 24

Which identity is true for every angle θ?

Accepted Answer

sin²θ + cos²θ = 1

Answer

sin θ + cos θ = 1

Answer

tan²θ + 1 = sin²θ

Answer

sin θ · cos θ = 1

Average rate of change	(f(b) − f(a)) / (b − a), the slope of the secant line between two points.
End behavior of a polynomial	Set by the leading term: its degree (even or odd) and the sign of its coefficient.
Vertical asymptote vs. hole	A factor that cancels from numerator and denominator gives a hole; one left in the denominator gives a vertical asymptote.
Horizontal asymptote rules	Degree of top < bottom: y = 0. Equal degrees: ratio of leading coefficients. Top > bottom: none.
Multiplicity of a zero	Odd multiplicity: the graph crosses the x-axis. Even multiplicity: it touches and turns around.
Exponential vs. linear change	Linear: equal differences over equal intervals. Exponential: equal ratios over equal intervals.
Logarithm definition	log_b(x) = y means bʸ = x.
Log properties	log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n·log a.
Period of sin(bx)	2π / \|b\|
Unit circle	The point at angle θ is (cos θ, sin θ).
Polar to rectangular	x = r cos θ, y = r sin θ
Pythagorean identity	sin²θ + cos²θ = 1

AP Precalculus practice test

AP Precalculus key terms

About the AP Precalculus exam

Questions

Study your own notes