Chapter 1.6, Problem 5E

Solution

To Describe:

The series of transformations that will change graph $y=x^2$ into the graph of given equation.

  $y={\left(100-x\right)}^2$ is the graph of $y=x^2$ shifted $100$ units to the left and then reflected about the $y$ -axis.

Given:

  $y={\left(100-x\right)}^2$ .

Concepts Used:

The graph of the $y=f\left(x+a\right);a>0$ is the graph of the equation $y=f\left(x\right)$ shifted left by $a$ units.

The graph of the $y=f\left(-x\right)$ is the graph of the equation $y=f\left(x\right)$ reflected about the $y$ -axis.

Calculations:

Rewrite $y={\left(100-x\right)}^2$ in terms of the function $f\left(x\right)=x^2$ .

  $\begin{array}{c}y={\left(100-x\right)}^2\\ =f\left(100-x\right)\\ =f\left(100+\left(-x\right)\right)\end{array}$

Thus, $y=f\left(100+\left(-x\right)\right)={\left(100-x\right)}^2$ is the graph of $y=f\left(x\right)=x^2$ first shifted $100$ units to the left, and then reflected about the $y$ -axis.

Conclusion:

  $y={\left(100-x\right)}^2$ is the graph of $y=x^2$ shifted $100$ units to the left and then reflected about the $y$ -axis.