Trigonometric Function Graphs Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
State the amplitude and period of each: (a) y=sin⁡(4x)y=\sin(4x), (b) y=5cos⁡(x)y=5\cos(x), (c) y=−2sin⁡ ⁣(x3)y=-2\sin\!\left(\frac{x}{3}\right).

Solution

  1. 1
    (a) a=1a=1, b=4b=4: amplitude 11, period 2π/4=π/22\pi/4=\pi/2.
  2. 2
    (b) a=5a=5, b=1b=1: amplitude 55, period 2π2\pi.
  3. 3
    (c) ∣a∣=2|a|=2, b=1/3b=1/3: amplitude 22, period 2π/(1/3)=6π2\pi/(1/3)=6\pi.

Answer

(a) amp=11, period=π/2\pi/2; (b) amp=55, period=2π2\pi; (c) amp=22, period=6π6\pi
Amplitude is ∣a∣|a| (always positive); period is 2π/∣b∣2\pi/|b| for sine and cosine. Increasing bb shortens the period; decreasing bb lengthens it. The sign of aa causes reflection but does not affect amplitude.

About Trigonometric Function Graphs

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

Learn more about Trigonometric Function Graphs →

More Trigonometric Function Graphs Examples

Example 1 easy

Identify the amplitude, period, phase shift, and vertical shift of [formula]. Write in standard form

Example 2 hard

Write the equation of a cosine function with amplitude [formula], period [formula], phase shift righ

Example 4 medium

A sound wave has the equation [formula] (pressure in Pa, time in seconds). Find the frequency (Hz) a

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