Chapter 1.7, Problem 42E

To determine

An equation of the function whose graph described as the shape of $f(x)=x^3$ , but shifted $6$ units to the left, $6$ units down and then reflected in the $y-$ axis.

Answer to Problem 42E

  $g(x)={(-x+6)}^2-6$ is the required equation of the function.

Explanation of Solution

Given information :

Given function whose graph described as the shape of $f(x)=x^3$ , but shifted $6$ units to the left, $6$ units down and then reflected in the $y-$ axis.

Let $c$ be a positive real number. Vertical, horizontal shifts and reflections in the coordinate axes of the graph of $y=f(x)$ are represented as follows.

1. Horizontal shift $c$ units to the right: $h(x)=f(x-c)$
2. Horizontal shift $c$ units to the left: $h(x)=f(x+c)$
3. Vertical shift $c$ units up: $h(x)=f(x)+c$
4. Vertical shift $c$ units down: $h(x)=f(x)-c$
5. Reflection in the $x-$ axis: $h(x)=-f(x)$
6. Reflection in the $y-$ axis: $h(x)=f(-x)$

As given $f(x)$ shifted $6$ units to the left, $6$ units down then the resultant function transformed into the form of $h(x)=f(x+6)-6$ And then reflected in the $y-$ axis then $h(x)$ takes the form of $g(x)=f(-x+6)-6$

Since $f(x)=x^3\Rightarrow f(-x+6)={\left(-x+6\right)}^3$ .

Therefore, $g(x)={(-x+6)}^2-6$ is the required equation of the function.