Chapter 1.6, Problem 45E

Solution

To Explain: The given basic graph and a sequence of transformation which used to produce given graph function.

Given:

  $y={\left(3x\right)}^2-4$

Graph:

  

Interpretation:

The parent function is the simplest form of the type of function given.

  $y=x^2$

Simplify the term $y={\left(3x\right)}^2-4$ :

Apply the product rule to $3x$ :

  $\begin{array}{l}y=3^2{x}^2-4\\ y=9x^2-4\end{array}$

Assume that $y=x^2$ is $f\left(x\right)=x^2$ and $y={\left(3x\right)}^2-4$ is $g\left(x\right)=9x^2-4$

  $\begin{array}{c}f\left(x\right)=x^2\\ g\left(x\right)=9x^2-4\end{array}$

The transformation being described is from $f\left(x\right)=x^2$ to $g\left(x\right)=9x^2-4$ .

  $f\left(x\right)=x^2\to g\left(x\right)=9x^2-4$

Compare and list the transformations:

Parent Function: $y=x^2$

Horizontal Shrink: Shrinked by a factor of $\frac{1}{3}$ .

Vertical Shift: Down $4$ units

Reflection about the $x-$ axis: None

Reflection about the $y-$ axis: None