Chapter 3.5, Problem 1AGP

To determine

To find: Translation of given equation relating to graph of $f\left(x\right)=x$ .

Answer to Problem 1AGP

The graph of $g\left(x\right)=x+1$ is shift of $f\left(x\right)=x$ shifted vertically upward $1$ unit.

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $g\left(x\right)=x+1$

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,

  $f\left(x\right)=x$

  $g\left(x\right)=x+1$

  $\Rightarrow g\left(x\right)=f\left(x\right)+1$

Here, the constant value is $c=1$ , which is positive. Therefore, the graph $f\left(x\right)=x$ is moved by $1$ unit upwards.

Therefore, the graph of $g\left(x\right)$ is same as the graph of $f\left(x\right)$ which is $1$ unit ahead along y-axis.