Chapter 3.5, Problem 46PPS

To determine

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

Answer to Problem 46PPS

The $f\left(x\right)+k$ for $k<0$ is translation of graph of $f\left(x\right)$ moving $\left|k\right|$ units downwards along y-axis.

Explanation of Solution

Given information:

  $f\left(x\right)$

  $f\left(x\right)+k$ for $k<0$

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 $f\left(x\right)=x$ , then $f\left(x\right)+c=x+c$ i.e. we are adding the value $c$ to the whole function.

So, the addition of a constant $c$ to the whole function represents the vertical translation in the graph. If $c$ is positive, then the graph translates upwards. If $c$ is negative, then the graph translates downwards.

From the given information,

Let $g\left(x\right)$ be defined such that $g\left(x\right)=f\left(x\right)+k$ .

Since $k<0$ , the graph of $f\left(x\right)$ is translated vertically downwards with $\left|k\right|$ units.

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