`y = sin((pix)/2) + 1` Find the x-intercepts of the graph.

Textbook Question

Chapter 5, 5.3 - Problem 45 - Precalculus (3rd Edition, Ron Larson).
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lemjay | High School Teacher | (Level 3) Senior Educator

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`y= sin ((pix)/2) + 1`

Before we solve for the x-intercepts,  let's determine the period of this function.

Take note that if a trigonometric function has a form y= Asin(Bx + C) + D,  its period is:

`P e r i o d = (2pi)/B`

If we plug-in the value of B, we will get:

`P e r i o d = (2pi)/(pi/2) = ((2pi)/1)/(pi/2)=(2pi)/1* 2/pi=4`

Hence, the period of the given function is 4.

Let's solve now the x-intercepts. To solve, set y=0.

`y= sin ((pix)/2) + 1`

`0=sin((pix)/2) + 1`

`-1= sin ((pix)/2) `    

Take note that sine has a value of -1 at an angle  `(3pi)/2` .

`(3pi)/2 = (pix)/2`

Then, isolate the x.

`(3pi)/2*2/pi = (pix)/2*2/pi`

`3=x`

Since the period of the function is 4, therefore the x-intercepts are:

`x= 3 + 4n`

where n is any integer.

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