Chapter 1.6, Problem 19E

Solution

To transform the graph of $f$ into the graph of $g$ .

  $f\left(x\right)={\left(x-2\right)}^3$

  $g\left(x\right)=-{\left(x+2\right)}^3$

Given:

  $f\left(x\right)={\left(x-2\right)}^3$

  $g\left(x\right)=-{\left(x+2\right)}^3$

Calculation:

  $f\left(x\right)={\left(x-2\right)}^3$

The graph of $g\left(x\right)$ can be obtained from the graph of $f\left(x\right)$ by following the sequence of transformation:

  $\left(a\right)$ A horizontal transformation to the left by $4$ units to get $y=f{\left(x+4\right)}^3=f{\left(x-2+4\right)}^3=f{\left(x+2\right)}^3$ .

  $\left(b\right)$ Reflection across $x$ -axis to get $y=-f{\left(x+2\right)}^3=g\left(x\right)$ .

Graph:

  

Interpretation:

The graph of $f\left(x\right)$ is transformed into $g\left(x\right)$ .