Chapter 4.7, Problem 25E

To determine

To write: a rule for $g$ to describe the indicated transformation of graph of $f$ .

Answer to Problem 25E

  $g\left(x\right)=-27x^3-18x^2+7$

Explanation of Solution

Given information:

Given function is,

  $f\left(x\right)=x^3+2x^2-9$

The indicated transformation is

Horizontal shrink by a factor of $\frac{1}{3}$ and a translation of 2 units up, followed by a reflection in the x -axis.

Concept used:

The graph of $g\left(x\right)=f\left(ax\right)$ can be found by horizontal shrunken the graph of $f\left(x\right)$ by a factor of $\frac{1}{a}$ units.

The graph of $g\left(x\right)=f\left(x\right)+k$ can be found by translating the graph of $f\left(x\right)$ by $k$ units up vertically.

The graph of $g\left(x\right)=-f\left(x\right)$ can be found by reflecting the graph of $f\left(x\right)$ about $x$ axis.

Calculation:

Consider the given function.

  $f\left(x\right)=x^3+2x^2-9$

Apply the transformation “Horizontal shrink by a factor of $\frac{1}{3}$ ”.

  $\begin{array}{c}f\left(3x\right)={\left(3x\right)}^3+2{\left(3x\right)}^2-9\\ =27x^3+18x^2-9\end{array}$

Now, apply the transformation “a translation of 2 units up” as shown:

  $\begin{array}{c}h\left(x\right)=f\left(3x\right)+2\\ =\left[{\left(3x\right)}^3+2{\left(3x\right)}^2-9\right]+2\\ =27x^3+18x^2-9+2\\ =27x^3+18x^2-7\end{array}$

Now, apply the transformation “reflection in the x -axis” as shown:

  $\begin{array}{c}g\left(x\right)=-h\left(x\right)\\ =-\left[27x^3+18x^2-7\right]\\ =-27x^3-18x^2+7\end{array}$