Teacher answer key · Grades 10–12 · about 25 minutes

Answer key: Sine & Cosine Graphs Worksheet

For teachers. This page is hidden from search engines and not linked from student pages, but it is not password-protected, so please don’t share the link with students. Student worksheet: Sine & Cosine Graphs Worksheet.

Goal: See how A, B, h and k change the graph of y = A sin(B(x − h)) + k, and use amplitude, period and midline to model a repeating situation.

Connects to Common Core HSF-BF.B.3 (Identify the effect on a graph of replacing f(x) by f(x) + k, k f(x), f(kx) and f(x + k).); HSF-TF.B.5 (Choose trigonometric functions to model periodic phenomena with a specified amplitude, frequency and midline.)

Completed results table

Values are the readouts of the tool (Sine & cosine graph explorer) for each setting.

Setting	Amplitude	Period	Midline	Range
A = 1, B = 1, k = 0 (start)	1	6.283 rad	y = 0	[-1, 1]
A = 2	2	6.283 rad	y = 0	[-2, 2]
B = 2	1	3.142 rad	y = 0	[-1, 1]
B = 0.5	1	12.566 rad	y = 0	[-1, 1]
k = 1	1	6.283 rad	y = 1	[0, 2]
A = 3, k = −1	3	6.283 rad	y = -1	[-4, 2]

Answers

What did changing A do to the graph? Which other readings changed?
A stretches the wave vertically. With A = 2 the peaks are at 2 and the troughs at −2, so the range became [−2, 2]. The period and the midline did not change.
Use period = 2π ÷ B to explain the periods you recorded for B = 2 and B = 0.5.
For B = 2: 2π ÷ 2 = π ≈ 3.142 radians, so the wave repeats twice as often. For B = 0.5: 2π ÷ 0.5 = 4π ≈ 12.566 radians, so it repeats half as often.
For A = 3 and k = −1, explain how to find the range from the amplitude and the midline.
The wave goes 3 above and 3 below the midline y = −1, so it runs from −1 − 3 = −4 up to −1 + 3 = 2. The range is [−4, 2].
Find a phase shift h that makes the sine graph look exactly like the cosine graph. Test your answer.
h = −π/2 ≈ −1.571, which shifts the sine wave π/2 to the left, because sin(x + π/2) = cos x. Type −1.571 into the phase shift box and compare it with the cosine wave.
A city’s temperature over one day ranges from 10 °C to 30 °C. Write a model y = A sin(B(x − h)) + k for it, with x in hours and a period of 24 hours.
The midline is k = (30 + 10) ÷ 2 = 20 and the amplitude is A = (30 − 10) ÷ 2 = 10. For a 24-hour period, B = 2π ÷ 24 = π/12 ≈ 0.262. So y = 10 sin(π/12 (x − h)) + 20, with h chosen so the temperature rises through the midline at the right time of day.
Which of A, B, h and k can never change the period of the graph? Explain.
A, h and k. A stretches the wave vertically, h slides it sideways and k moves it up or down. Only B squeezes or stretches it horizontally, so only B changes the period 2π ÷ B.

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