Chapter 11.2, Problem 22E

a.

To determine

To graph: thegiven functions on the same screen and describe the transformation of the functions.

Answer to Problem 22E

The graph of $y=-5^x$ and $y=5^{-x}$ is a reflection of $y=5^x$ across the x-axis and across the y-axis.

Explanation of Solution

Given:

The given functions are $y=5^x,y=-5^x,y=5^{-x},y=5^x+2,y=5^x-2,\text{ and }y={10}^x$ .

Calculation:

According to the question,

Draw the graph of the function $y=5^x,y=-5^x,y=5^{-x},y=5^x+2,y=5^x-2,\text{ and }y={10}^x$ on the same screen. And to compare $y=-5^x,y=5^{-x}$ to the parent graph $y=5^x$ .

The graph is given below:

  

Hence, thegraph of $y=-5^x$ is a reflection of $y=5^x$ across the x-axis. The graph of $y=5^{-x}$ is a reflection of $y=5^x$ across the y-axis

b.

To determine

To predict: thetransformation occurs with the graph of $y=5^x+2\text{ and }y=5^x-2$ .

Answer to Problem 22E

  $\begin{array}{l}y=5^x+2:\text{ shifted up 2 units}\\ y=5^x-2:\text{ shifted down 2 units}\end{array}$ .

Explanation of Solution

Given:

The given functions are $y=5^x,y=-5^x,y=5^{-x},y=5^x+2,y=5^x-2,\text{ and }y={10}^x$ .

Calculation:

From the graph in part (a),

The graph of $y=5^x+2$ is shifted up two units, while the graph of $y=5^x-2$ is shifted down two units.

Hence, the transformation occurs with the graph of $y=5^x+2\text{ and }y=5^x-2$ is $\begin{array}{l}y=5^x+2:\text{ shifted up 2 units}\\ y=5^x-2:\text{ shifted down 2 units}\end{array}$ .

c.

To determine

To compare: this function to graph of $y=5^x$ . Is graph of $y=5^{2x}$ is same as $y={10}^x$ .

Answer to Problem 22E

No.

Explanation of Solution

Given:

The functions are $y={10}^x$ can be written as $y={\left(2\cdot 5\right)}^x$ .

Calculation:

From the graph in part (a),

It is seen that the graph of $y={10}^x$ increases more quickly than $y=5^x$ .

Since, $5^{2x}={\left(5^2\right)}^x={25}^x\ne {10}^x$

Hence, it is concluded that both the graph of is not same.