Chapter 1.6, Problem 7E

Solution

To Describe:

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

  $y={\left(x-1\right)}^2+3$ is the graph of $y=x^2$ shifted $1$ units to the right and $3$ units up.

Given:

  $y=x^2-100$ .

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 to the right by $a$ units.

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

Calculations:

Rewrite $y={\left(x-1\right)}^2+3$ in terms of the function $f\left(x\right)=x^2$ .

  $\begin{array}{c}y={\left(x-1\right)}^2+3\\ =f\left(x-1\right)+3\end{array}$

  $y=f\left(x-1\right)+3$ is the graph of $y=f\left(x\right)$ shifted $1$ units to the right and $3$ units up.

Conclusion:

  $y={\left(x-1\right)}^2+3$ is the graph of $y=x^2$ shifted $1$ units to the right and $3$ units up.