Chapter 4.7, Problem 23E

To determine

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

Answer to Problem 23E

  $g\left(x\right)=-x^3+9x^2-27x+21$

Explanation of Solution

Given information:

Given function is,

  $f\left(x\right)=x^3-6$

The indicated transformation is

Translation of 3 units left, followed by a reflection in the y -axis.

Concept used:

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

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

Calculation:

Consider the given function.

  $f\left(x\right)=x^3-6$

Apply the transformation “Translation of 3 units left”.

  $\begin{array}{c}f\left(x+3\right)={\left(x+3\right)}^3-6\\ =x^3+9x^2+27x+27-6\\ =x^3+9x^2+27x+21\end{array}$

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

  $\begin{array}{c}g\left(x\right)={\left(-x\right)}^3+9{\left(-x\right)}^2+27\left(-x\right)+21\\ =-x^3+9x^2-27x+21\end{array}$