Find the amplitude, period, and phase shift of y = 2 sin(r-)

The given function is y=2sin12x-π4 ⋯1 Consider the standard sin function with…

Answered: Find the amplitude, period, and phase…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Find the amplitude, period, and phase shift of y = 2 sin(r-)

Expert Solution

Step 1

The given function is

$y = 2 \sin (\frac{1}{2} x - \frac{π}{4}) \dots (1)$

Consider the standard sin function with amplitude, period and phase shift

$y = a \sin (b x - c) + d \dots (2)$

Where, $a$ is the amplitude of the function, $\frac{2 π}{|b|}$ is the period of the function and $\frac{c}{b}$ is the phase shift of the function

Now, compare (1) and (2) then

$a = 2, b = \frac{1}{2}, c = \frac{π}{4}, d = 0$

So, the amplitude of the function is 2.

Period of the function

$\begin{array}{rcl} \frac{2 π}{|b|} & = & \frac{2 π}{(\frac{1}{2})} \\ = & 2 π \times 2 \\ = & 4 π \end{array}$

Hence, the required period of the function is $4 π$ .

The phase shift of the function

$\begin{array}{rcl} \frac{c}{b} & = & \frac{\frac{π}{4}}{\frac{1}{2}} \\ = & \frac{π}{4} \times \frac{2}{1} \\ = & \frac{π}{2} \end{array}$

Hence, the required phase shift of the function is $\frac{π}{2}$

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