Chapter 13.7, Problem 39PPSE

Solution

To find: The equation of translated function which shifts $y=\cos \left(x\right)$ , $1.5$ units to the right.

The equation of translated function is written as $y=\cos \left(x-1.5\right)$ .

Given:

The parent function to translate $1.5$ units to the right is $y=\cos \left(x\right)$

Concept:

1. The horizontal translation of the function $f\left(x\right)$ is $g\left(x\right)=f\left(x-h\right)$ where $\left(h\right)$ is the amount of phase shift. If $h>0$ the phase shift is towards right and if $h<0$ the phase shift is towards left.
2. The vertical translation of the function $f\left(x\right)$ is $g\left(x\right)=f\left(x\right)+k$ where $\left(k\right)$ is the amount of midline the midline shift. If $k>0$ the midline shift is vertically upwards and if $k<0$ the midline shift is vertically downwards.

Calculation:

With the help of concept stated above using the clause $\left(a\right)$ , the parent function $y=\cos \left(x\right)$ is shifted $1.5$ units to the right which means $\left(h\right)=1.5$ . Therefore, the translated function is written as:

  $\begin{array}{c}y=\cos \left(x\right)\mathrm{....}\left(\text{Given}\right)\\ y=\cos \left(x-h\right)\mathrm{....}\left(\text{Translated function}\right)\\ y=\cos \left(x-\left(1.5\right)\right)\mathrm{....}\left(\text{Substitute }h=1.5\right)\\ y=\cos \left(x-1.5\right)\end{array}$

Conclusion:

The equation of translated function is written as $y=\cos \left(x-1.5\right)$ .