Chapter 1.6, Problem 8E

Solution

To Describe:

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

  $y={\left(x+50\right)}^2-279$ is the graph of $y=x^2$ shifted $50$ units to the left and $279$ units down.

Given:

  $y={\left(x+50\right)}^2-279$ .

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 left 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 down by $a$ units.

Calculations:

Rewrite $y={\left(x+50\right)}^2-279$ in terms of the function $f\left(x\right)=x^2$ .

  $\begin{array}{c}y={\left(x+50\right)}^2-279\\ =f\left(x+50\right)-279\end{array}$

  $y=f\left(x+50\right)-279$ is the graph of $y=f\left(x\right)$ shifted $50$ units to the left and $279$ units down.

Conclusion:

  $y={\left(x+50\right)}^2-279$ is the graph of $y=x^2$ shifted $50$ units to the left and $279$ units down.