AP Prep — Precalculus Practice

Question 1 of 10

MCQU1Topic 1.4Medium No calc Diagram

The graph of f(x) = (x − 1)(x + 2)(x − 4) has how many turning points (points where it changes direction)?

A2

B1

C0

D3

Explanation

This is a degree-3 polynomial with three distinct real zeros, so the graph must rise and fall to pass through all three, producing exactly 2 turning points — the maximum n − 1 = 2 for degree 3.

Question 2 of 10

MCQU2Topic 2.3Easy No calc

For f(x) = 7(2)ˣ, what is f(0), and what does it represent in the model?

A14, the initial value

B0, the starting point

C7, the initial value

D2, the growth factor

Explanation

Any nonzero base raised to the 0 power is 1, so f(0) = 7·2⁰ = 7·1 = 7. In an exponential model a·bˣ, the coefficient a is the initial value (the output when x = 0).

Question 3 of 10

MCQU2Topic 2.9Medium No calc

What is the value of log₂(32)?

A4

B6

C16

D5

Explanation

log₂(32) asks for the exponent on 2 that gives 32. Since 2⁵ = 32, the value is 5. The answer 16 is 32/2, and 4 would give only 2⁴ = 16.

Question 4 of 10

MCQU1Topic 1.11Medium No calc

When the polynomial p(x) = 2x³ − 3x² + 4x − 5 is divided by (x − 2), what is the remainder?

A11

B0

C−5

D7

Explanation

By the Remainder Theorem the remainder equals p(2) = 2(8) − 3(4) + 4(2) − 5 = 16 − 12 + 8 − 5 = 7. A remainder of 0 would mean (x − 2) is a factor, and −5 is just the constant term.

Question 5 of 10

MCQU2Topic 2.1Easy No calc

Which statement best describes the sequence 3, 6, 12, 24, …?

AGeometric with common ratio 3

BArithmetic with common difference 2

CArithmetic with common difference 3

DGeometric with common ratio 2

Explanation

Each term is found by multiplying the previous term by the same factor: 6/3 = 12/6 = 24/12 = 2. A constant ratio (not a constant difference) makes the sequence geometric with common ratio 2.

Question 6 of 10

MCQU3Topic 3.5Medium No calc

A sinusoidal function oscillates between a minimum of y = 2 and a maximum of y = 8. What are its amplitude and midline?

AAmplitude 6, midline y = 4

BAmplitude 3, midline y = 4

CAmplitude 5, midline y = 3

DAmplitude 3, midline y = 5

Explanation

The amplitude is half the peak-to-trough distance: (8 − 2)/2 = 3. The midline is the average of the max and min: (8 + 2)/2 = 5. Using the full distance 6 for amplitude is a common error.

Question 7 of 10

MCQU1Topic 1.6Easy No calc

Consider the polynomial function f(x) = -3x⁴ + 5x³ - x + 2. Which statement best describes the end behavior of the graph of f?

AAs x → -∞, f(x) → -∞; as x → +∞, f(x) → -∞.

BAs x → -∞, f(x) → +∞; as x → +∞, f(x) → +∞.

CAs x → -∞, f(x) → +∞; as x → +∞, f(x) → -∞.

DAs x → -∞, f(x) → -∞; as x → +∞, f(x) → +∞.

Explanation

End behavior is governed by the leading term, -3x⁴. Because the degree 4 is even, both ends of the graph go the same direction; because the leading coefficient -3 is negative, both ends fall. So f(x) → -∞ as x → -∞ and as x → +∞. Lower-degree terms do not affect end behavior.

Question 8 of 10

MCQU3Topic 3.4Medium No calc Word Diagram

The graph of a sinusoidal function is shown below. Which equation represents it?

Ay = 4cos(π/3 x) + 3

By = 4sin(π/3 x) + 3

Cy = 3cos(π/3 x) + 4

Dy = 4cos(π/6 x) + 3

Explanation

Amplitude = (7 − (−1))/2 = 4 and midline = (7 + (−1))/2 = 3. Consecutive maxima are one period apart, so the period is 6 and b = 2π/6 = π/3. Since a maximum occurs at x = 0, an unshifted cosine fits: y = 4cos(π/3 x) + 3. The choice with b = π/6 uses 2π/12 and treats the period as 12; but the maxima are 6 apart, so the period is 6.

Question 9 of 10

MCQU3Topic 3.3Easy No calc

What is the exact value of cos(π/3)?

A√2/2

B√3/2

C1

D1/2

Explanation

On the unit circle, the angle π/3 (60°) has cosine 1/2. The value √3/2 is the sine of π/3, so the two are easy to swap.

Question 10 of 10

MCQU3Topic 3.2Medium No calc Word Diagram

The terminal ray of an angle θ in standard position passes through the point (−8, 15). What are sin θ and cos θ?

Asin θ = -8/17, cos θ = 15/17

Bsin θ = 15/17, cos θ = -8/17

Csin θ = 15/8, cos θ = -8/15

Dsin θ = 15/17, cos θ = 8/17

Explanation

The distance from the origin is r = √((−8)² + 15²) = √(64 + 225) = √289 = 17. By definition sin θ = y/r = 15/17 and cos θ = x/r = −8/17. The choice with sin = −8/17 swaps the roles of x and y; sine uses the y-coordinate (15) and cosine uses the x-coordinate (−8).

Free Response 1 · Section II

FRQModeling a Periodic ContextU3 No calc

The figure shows the graph of h, the height (in meters, relative to a fixed reference) of a marked point on a slowly rotating wheel, as a function of time t in seconds. No calculator is allowed.

(a) From the graph, state the amplitude, the midline, and the period of h.

(b) Write an equation for h(t) using a cosine function.

(c) Find all values of t in 0 ≤ t ≤ 8 at which h(t) equals the midline value.

(d) Explain what the amplitude and the midline represent physically for the marked point.

Free response is self-scored — work it out, then reveal the model answer and scoring checklist to compare.

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