Graphs of Sine and Cosine Test Answer Key with Scoring – Algebra 2

Question 1 Features of a sine function (3 points)

1. 3.
2. 2π/4 = π/2.
3. Midline y = 2, maximum 5, minimum −1.

Scoring: 1 point per part.

Question 2 Phase shift and range (2 points)

The shift is π/3 to the left. The midline is −1 and the amplitude is 2, so the range is −3 ≤ y ≤ 1.

Scoring: 1 point for the shift, 1 point for the range.

Question 3 Equation from a graph (3 points)

The maximum is 4 and the minimum is −2. Midline D = (4 − 2)/2 = 1, amplitude A = 3. Peaks at 0 and 4 give period 4, so B = 2π/4 = π/2. y = 3 cos(πx/2) + 1.

Scoring: 1 point each for D, A and B.

Question 4 Write a model (2 points)

Midline (11 − 3)/2 = 4, amplitude 7, B = 2π/8 = π/4. y = 7 cos(πx/4) + 4.

Scoring: 1 point for midline and amplitude, 1 point for B and the equation.

Question 5 Ferris wheel (3 points)

1. −15 cos 0 + 18 = 3 feet.
2. −15 cos(2π/3) + 18 = 7.5 + 18 = 25.5 feet.
3. The top is at 33 feet when cos(πt/15) = −1, so πt/15 = π and t = 15 seconds.

Scoring: 1 point per part.

Question 6 Tide depth (2 points)

High tide: 7 + 2 = 9 ft. Low tide: 7 − 2 = 5 ft. At t = 2: 2 cos(π/3) + 7 = 2(0.5) + 7 = 8 ft. 9 ft, 5 ft, 8 ft.

Scoring: 1 point for the two extremes, 1 point for d(2).

Question 7 Maximum points (2 points)

A maximum occurs when 3x = π/2 + 2πk. For k = 0: x = π/6. For k = 1: 3x = 5π/2, x = 5π/6. x = π/6 and 5π/6.

Scoring: 1 point for each maximum.

Question 8 Model from a table (2 points)

Midline 5 (the average of 8 and 2), amplitude 3, the cycle repeats at x = 8, so B = 2π/8 = π/4. y = 3 sin(πx/4) + 5. Check: at x = 2, 3 sin(π/2) + 5 = 8.

Scoring: 1 point for midline and amplitude, 1 point for B.

Question 9 Maximum of a reflected cosine (1 point)

The maximum is 3 + 2 = 5, and it occurs when cos x = −1. Maximum 5 at x = π.

Scoring: 1 point for both.