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.
What to do
- Open the sine & cosine graph explorer and keep “Sine” selected. Start with “Amplitude A” = 1, “Frequency factor B” = 1, “Phase shift h (radians)” = 0 and “Vertical shift k” = 0.
- Change the values to match each row of the table, and record the amplitude, period, midline and range. Go back to the starting values between rows.
- Finally, press “Cosine” and compare the cosine wave with the sine wave.
Record your results
| Setting | Amplitude | Period | Midline | Range |
|---|---|---|---|---|
| A = 1, B = 1, k = 0 (start) | ||||
| A = 2 | ||||
| B = 2 | ||||
| B = 0.5 | ||||
| k = 1 | ||||
| A = 3, k = −1 |
Questions
What did changing A do to the graph? Which other readings changed?
Use period = 2π ÷ B to explain the periods you recorded for B = 2 and B = 0.5.
For A = 3 and k = −1, explain how to find the range from the amplitude and the midline.
Find a phase shift h that makes the sine graph look exactly like the cosine graph. Test your answer.
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.
Which of A, B, h and k can never change the period of the graph? Explain.
Free from Geometry Lab · geometrylab.net · 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.
Teaching with this? Find the answer key and more activities on the Geometry Lab for teachers page.