Chapter 3.5, Problem 48PPS

To determine

To find: The translation of given $f\left(x\right)$

Answer to Problem 48PPS

The $f\left(x-h\right)$ for $h<0$ is translation of graph of $f\left(x\right)$ moving $\left|h\right|$ units towards left along x-axis.

Explanation of Solution

Given information:

  $f\left(x\right)$

  $f\left(x-h\right)$ for $h<0$

A Translation is a movement of the graph either horizontally parallel to x-axis or vertically parallel to y-axis.

The graph of $y=f\left(x\right)$ where $f\left(x\right)=x$ is same as the graph of $y=x$ . Writing the equations as functions in the form of $f\left(x\right)$ is useful when applying translations and reflections to graphs.

If the function $y=f\left(x\right)$ is translated to $y=f\left(x+c\right)$ , then the graph of $f\left(x\right)$ is translated horizontally towards right if $c<0$ , the $f\left(x\right)$ is translated horizontally towards left if $c>0$ .

From the given information,

Let $g\left(x\right)$ be defined such that $g\left(x\right)=f\left(x-h\right)$

Since $h<0$

  $\Rightarrow -h>0$ ,therefore, the graph of $f\left(x\right)$ is translated horizontally towards left with $\left|h\right|$ units.

Therefore, the graph of $g\left(x\right)$ is same as the graph of $f\left(x\right)$ which is $\left|h\right|$ units back along x-axis.