Chapter 1.6, Problem 4E

Solution

To Describe:

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

  $y={\left(x-3\right)}^2$ is the graph of $y=x^2$ shifted $3$ units to the right.

Given:

  $y={\left(x-3\right)}^2$ .

Concepts Used:

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

Calculations:

Rewrite $y={\left(x-3\right)}^2$ in terms of the function $f\left(x\right)=x^2$ .

  $\begin{array}{c}y={\left(x-3\right)}^2\\ =f\left(x-3\right)\end{array}$

Thus, $y=f\left(x-3\right)={\left(x-3\right)}^2$ is the graph of $y=f\left(x\right)=x^2$ shifted $3$ units to the right.

Conclusion:

  $y={\left(x-3\right)}^2$ is the graph of $y=x^2$ shifted $3$ units to the right.