Chapter 9.5, Problem 66SR

To determine

Describe the transformations needed to obtain the graph of $g\left(x\right)=2x^2$ from the graph of $f\left(x\right)=4x^2$ .

Answer to Problem 66SR

  $g\left(x\right)=\frac{f\left(x\right)}{2}$

Explanation of Solution

Given information:

  $\begin{array}{l}g\left(x\right)=2x^2\\ f\left(x\right)=4x^2\end{array}$

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 $f\left(x\right)$ , divide by 2 in the function $f\left(x\right)$ .

Observe that when the unit C is dividing to the function, it will compressed the original $f\left(x\right)$ graph C unit in y- direction.

So, the graph of $g\left(x\right)$ can be obtained by shifting the graph of $f\left(x\right)$ compressed in y-direction by 1/2 units as shown below:

  $\begin{array}{l}f\left(x\right)=4x^2\\ f\left(x\right)=\frac{4x^2}{2}\\ f\left(x\right)=2x^2=g\left(x\right)\\ g\left(x\right)=\frac{f\left(x\right)}{2}\end{array}$

So, the graph of $g\left(x\right)$ can be obtained by dividing 2 to the function $f\left(x\right)$ that is $g\left(x\right)=\frac{f\left(x\right)}{2}$ .