Chapter 9.5, Problem 68SR

To determine

Describe the transformations needed to obtain the graph of $g\left(x\right)=x^2+3$ from the graph of $f\left(x\right)=x^2-6$ .

Answer to Problem 68SR

  $g\left(x\right)=f\left(x\right)+9$

Explanation of Solution

Given information:

  $\begin{array}{l}g\left(x\right)=x^2+3\\ f\left(x\right)=x^2-6\end{array}$

Concept used:

Transformation of functions: For any positive integer a , we have

• $y=f\left(x+a\right)$ is the graph of $y=f\left(x\right)$ shifted a units left.
• $y=f\left(x-a\right)$ is the graph of $y=f\left(x\right)$ shifted a units right.
• $y=f\left(x\right)+a$ is the graph of $y=f\left(x\right)$ shifted a units up.
• $y=f\left(x\right)-a$ is the graph of $y=f\left(x\right)$ shifted a units down.
• $y=af\left(x\right)$ is the graph of $y=f\left(x\right)$ stretches a units in y direction if $a>1$ and compresses a units in y direction if $0<a<1$
• $y=f\left(ax\right)$ is the graph of $y=f\left(x\right)$ stretches a units in x direction if $0<a<1$ and compresses a units in x direction if $a>1$ .
• $y=-f\left(x\right)$ is the graph of $y=f\left(x\right)$ reflected about x axis.
• $y=f\left(-x\right)$ is the graph of $y=f\left(x\right)$ reflected about y axis.

Calculation:

In order to get the graph of $g\left(x\right)$ from the graph of $f\left(x\right)$ , add 9 in the function $f\left(x\right)$ .

Observe that when the unit C is adding to the function, it will move the original $f\left(x\right)$ graph C unit up.

So, the graph of $g\left(x\right)$ can be obtained by shifting the graph of $f\left(x\right)$ in upward direction by 9 units as shown below:

  $\begin{array}{l}f\left(x\right)=x^2-6\\ f\left(x\right)=x^2-6+9\\ f\left(x\right)=x^2+3=g\left(x\right)\\ g\left(x\right)=f\left(x\right)+9\end{array}$

So, the graph of $g\left(x\right)$ can be obtained by adding 9 to the function $f\left(x\right)$ that is $g\left(x\right)=f\left(x\right)+9$ .