Chapter 9.6, Problem 34PPS

To determine

Describe the transformations needed to obtain the graph of $y=2^x$ from the graph of $y=2^x+6$ .

Answer to Problem 34PPS

  $y=y+6$

Explanation of Solution

Given information:

  $\begin{array}{l}y=2^x\\ y=2^x+6\end{array}$

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 $y=2^x$ from the graph of $y=2^x+6$ , add 6 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 6 units as shown below:

  $\begin{array}{l}y=2^x\\ y=2^x+6\end{array}$

So, the graph of $y=2^x+6$ can be obtained by adding 6 to the function $y=2^x$ that is $y=y+6$ .