Chapter 3.6, Problem 67E

To determine

when is the graph of $y=f\left(x\right)+w$ same as the graph of $y=f\left(x+w\right)$ for linear functions.

Answer to Problem 67E

When slope of the function $f\left(x\right)$ is 1.

Explanation of Solution

Concept used:

Vertical translation:

The graph of $g\left(x\right)=f\left(x\right)+k$ is a vertical translation of the graph of f .

The vertical shift is k units up if $k>0$ and k units down if $k<0$ .

Horizontal translation:

The graph of $g\left(x\right)=f\left(x-h\right)$ is a horizontal translation of the graph of f .

The horizontal shift is h units’ right if $h>0$ and h units left if $h<0$ .

Consider the functions $y=f\left(x\right)+w$ and $y=f\left(x+w\right)$ , these represents the vertical and horizontal translations of the graph of function $f\left(x\right)$ .

So, for the graph of $y=f\left(x\right)+w$ same as the graph of $y=f\left(x+w\right)$ , it requires that the vertical and the horizontal translations are same.

So, if the slope the function $f\left(x\right)$ is 1, then the change in y will be same as the change in x , which will result in the same vertical and the horizontal translations.

Thus, in that case the graph of $y=f\left(x\right)+w$ will be same as the graph of $y=f\left(x+w\right)$