Sinusoidal Graphs Test Answer Key with Scoring – Precalculus

Question 1 Features of a Sinusoid (2 points)

(a) Amplitude 5; period 2π/4 = π/2.

(b) Factor 4x − π = 4(x − π/4): shift π/4 to the right; midline y = 2.

Scoring: 1 point per part.

Question 2 Equation from a Graph (2 points)

1. A = (2 − (−4))/2 = 3; midline D = (2 + (−4))/2 = −1; the period is 4 (from minimum to minimum), so B = π/2.
2. The graph rises through the midline at x = 1, so no reflection is needed: y = 3 sin(π/2 (x − 1)) − 1.

Scoring: 1 point for A, D and the period; 1 point for a correct equation (any equivalent form accepted).

Question 3 Tangent Features (2 points)

(a) Period π/3.

(b) 3x = ±π/2 gives x = π/6 and x = −π/6.

Scoring: 1 point per part.

Question 4 Write a Model from Extremes (3 points)

1. A = (14 + 6)/2 = 10; D = (14 − 6)/2 = 4.
2. Half period is 8 − 2 = 6, so the period is 12 and B = 2π/12 = π/6.
3. y = 10cos(π/6 (x − 2)) + 4.

Scoring: 1 point per part.

Question 5 Bobbing Buoy (3 points)

1. 2π ÷ (π/3) = 6 seconds.
2. Maximum 4 + 2.5 = 6.5 ft; minimum 4 − 2.5 = 1.5 ft.
3. h(1) = 2.5 sin(π/3) + 4 = 4 + 1.25√3 ≈ 6.17 ft.

Scoring: 1 point per part.

Question 6 Buoy at a Given Height (2 points)

2.5 sin(πt/3) = 1.25, so sin(πt/3) = 1/2. Then πt/3 = π/6 or 5π/6, giving t = 0.5 s and t = 2.5 s.

Scoring: 1 point for setting up sin = 1/2; 1 point for both times.

Question 7 Monthly Temperatures (3 points)

1. A = (94 − 38)/2 = 28, D = (94 + 38)/2 = 66, B = 2π/12 = π/6 and the maximum is at m = 7: T(m) = 28cos(π/6 (m − 7)) + 66.
2. T(3) = 28cos(−2π/3) + 66 = 28(−1/2) + 66 = 52°F.

Scoring: 1 point for A and D; 1 point for B and the shift; 1 point for the March value.

Question 8 Transformation Description (3 points)

1. Stretch vertically by 2 and reflect over the x-axis; stretch horizontally by a factor of 2; shift up 3.
2. Period 2π ÷ (1/2) = 4π.
3. Maximum 3 + 2 = 5, minimum 3 − 2 = 1: range [1, 5].

Scoring: 1 point per part.