Practice Trigonometric Function Graphs in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

The graphs of \sin x, \cos x, and \tan x as functions of a real variable, characterized by amplitude, period, phase shift, and vertical shift.

If you track the y-coordinate of a point moving around the unit circle and plot it against the angle, you get the sine wave. It's the shape of ocean waves, sound waves, and alternating current. The general form y = a\sin(bx - c) + d lets you control four properties: how tall the wave is (a, amplitude), how fast it repeats (b, affecting period), where it starts (c, phase shift), and its vertical center (d, vertical shift).

Example 1

easy
Identify the amplitude, period, phase shift, and vertical shift of y=3\sin(2x-\pi)+1. Write in standard form y=a\sin(b(x-h))+k.

Example 2

hard
Write the equation of a cosine function with amplitude 4, period 6, phase shift right 1, and vertical shift down 2.

Example 3

easy
State the amplitude and period of each: (a) y=\sin(4x), (b) y=5\cos(x), (c) y=-2\sin\!\left(\frac{x}{3}\right).

Example 4

medium
A sound wave has the equation P(t)=0.002\sin(440\pi t) (pressure in Pa, time in seconds). Find the frequency (Hz) and explain the connection to the period.
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