Answer :
Certainly! Let's go through this step by step to understand how to vertically compress the absolute value parent function, [tex]\( f(x) = |x| \)[/tex], by a factor of 5 and determine the new function.
### Step-by-Step Solution:
1. Understand the Parent Function:
The parent function given is [tex]\( f(x) = |x| \)[/tex]. This function produces a V-shaped graph where the slope of the lines are 1 and -1 on either side of the origin.
2. Vertical Compression:
Vertical compression involves scaling the y-values (outputs) of the function closer to the x-axis. This can be achieved by multiplying the function's output by a value less than 1. In this case, we want to compress the function by a factor of 5.
3. Apply the Vertical Compression:
To compress vertically by a factor of 5, we multiply the entire function [tex]\( f(x) \)[/tex] by [tex]\( \frac{1}{5} \)[/tex]. This means every y-value of the function [tex]\( f(x) \)[/tex] will be scaled down to one-fifth of its original value.
So, the transformation would be:
[tex]\[ g(x) = \frac{1}{5} \times |x| \][/tex]
4. Review the Options:
- A. [tex]\( g(x) = \frac{1}{5}|x| \)[/tex]
- B. [tex]\( g(x) = 5|x| \)[/tex]
- C. [tex]\( g(x) = |5x| \)[/tex]
- D. [tex]\( g(x) = |x - 5| \)[/tex]
After performing the vertical compression, the correct equation of the new function is:
[tex]\[ g(x) = \frac{1}{5}|x| \][/tex]
This corresponds to option A in the given choices.
### Conclusion:
The equation of the new function after vertically compressing [tex]\( f(x) = |x| \)[/tex] by a factor of 5 is:
[tex]\[ g(x) = \frac{1}{5}|x| \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\text{A. } g(x) = \frac{1}{5}|x|} \][/tex]
### Step-by-Step Solution:
1. Understand the Parent Function:
The parent function given is [tex]\( f(x) = |x| \)[/tex]. This function produces a V-shaped graph where the slope of the lines are 1 and -1 on either side of the origin.
2. Vertical Compression:
Vertical compression involves scaling the y-values (outputs) of the function closer to the x-axis. This can be achieved by multiplying the function's output by a value less than 1. In this case, we want to compress the function by a factor of 5.
3. Apply the Vertical Compression:
To compress vertically by a factor of 5, we multiply the entire function [tex]\( f(x) \)[/tex] by [tex]\( \frac{1}{5} \)[/tex]. This means every y-value of the function [tex]\( f(x) \)[/tex] will be scaled down to one-fifth of its original value.
So, the transformation would be:
[tex]\[ g(x) = \frac{1}{5} \times |x| \][/tex]
4. Review the Options:
- A. [tex]\( g(x) = \frac{1}{5}|x| \)[/tex]
- B. [tex]\( g(x) = 5|x| \)[/tex]
- C. [tex]\( g(x) = |5x| \)[/tex]
- D. [tex]\( g(x) = |x - 5| \)[/tex]
After performing the vertical compression, the correct equation of the new function is:
[tex]\[ g(x) = \frac{1}{5}|x| \][/tex]
This corresponds to option A in the given choices.
### Conclusion:
The equation of the new function after vertically compressing [tex]\( f(x) = |x| \)[/tex] by a factor of 5 is:
[tex]\[ g(x) = \frac{1}{5}|x| \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\text{A. } g(x) = \frac{1}{5}|x|} \][/tex]
