Chapter 7.6, Problem 66HP

a.

To determine

To write the function according to the given transformation

Answer to Problem 66HP

  $y=2{(5)}^{-x}$

Explanation of Solution

Given:

The parent function: $f(x)=5^x$

We have, $y=f(x)=5^x$

Now, taking reflection about y- axis, which means,

  $y=f(-x)=5^{-x}$

Again, vertically compressing the obtained function by a factor of 2

  $y=2f(-x)=2{(5)}^{-x}$

Conclusion:

Hence, the transformed function is: $y=2{(5)}^{-x}$ .

b.

To determine

To write the function according to the given transformation

Answer to Problem 66HP

  $y=0.5{(5)}^{x+2}+8$

Explanation of Solution

The parent function: $f(x)=5^x$

We have, $y=f(x)=5^x$

Now, horizontally stretching the parent function by a factor of 0.5

  $y=0.5f(x)=0.5{(5)}^x$

Again, translating the obtained function to the left, which means,

  $y=0.5f(x+2)=0.5{(5)}^{x+2}$

Again, translating the obtained function to the up, which means,

  $y=0.5f(x+2)+8=0.5{(5)}^{x+2}+8$

Conclusion:

Hence, the transformed function is: $y=0.5{(5)}^{x+2}+8$ .

c.

To determine

To write the function according to the given transformation

Answer to Problem 66HP

  $y=-2{(5)}^{x-2}$

Explanation of Solution

The parent function: $f(x)=5^x$

We have, $y=f(x)=5^x$

Now, taking reflection about x- axis, which means,

  $y=-f(x)=-5^x$

Again, vertically compressing the obtained function by a factor of 2

  $y=-2f(x)=-2{(5)}^x$

Now, shifting the obtained function to right by 2 units, which means,

  $y=-2f(x-2)=-2{(5)}^{x-2}$

Conclusion:

Hence the transformed function is: $y=-2{(5)}^{x-2}$ .

d.

To determine

To write the function according to the given transformation

Answer to Problem 66HP

  $y=5{(5)}^{x-2}-8$

Explanation of Solution

The parent function: $f(x)=5^x$

We have, $y=f(x)=5^x$

Now, horizontally compressing the parent function by a factor of 5, which means,

  $y=5f(x)=5{(5)}^x$

Again, translating the obtained function to the right, which means,

  $y=5f(x-2)=5{(5)}^{x-2}$

Again, translating the obtained function to the down, which means,

  $y=5f(x-2)-8=5{(5)}^{x-2}-8$

Conclusion:

Hence, the transformed function is: $y=5{(5)}^{x-2}-8$ .