Chapter 1.6, Problem 18E

Solution

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

  $f{\left(x-1\right)}^2$

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

Given:

  $f{\left(x-1\right)}^2$

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

Calculation:

  $f{\left(x-1\right)}^2$

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 $y=f{\left(x-1+4\right)}^2=f{\left(x+3\right)}^2$ .

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

Graph:

  

Interpretation:

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