Certainly! Let's break down how to vertically compress the function [tex]\( y = \sqrt{1 - x^2} \)[/tex] by a factor of 6.
### Step 1: Understanding Vertical Compression
When we compress a function vertically by a factor, we effectively reduce its height by that factor. If we have a function [tex]\( y = f(x) \)[/tex], and we want to compress it vertically by a factor of [tex]\( k \)[/tex], the new function will be [tex]\( y = \frac{1}{k} f(x) \)[/tex].
### Step 2: Original Function Description
The given function is:
[tex]\[ y = \sqrt{1 - x^2} \][/tex]
### Step 3: Apply Vertical Compression by a Factor of 6
Since the problem asks us to compress the function vertically by a factor of 6, we need to multiply the function by [tex]\( \frac{1}{6} \)[/tex]. Hence, our new function will be:
[tex]\[ y = \frac{1}{6} \sqrt{1 - x^2} \][/tex]
### Step 4: New Function Description
The new, vertically compressed function is:
[tex]\[ y = \frac{1}{6} \sqrt{1 - x^2} \][/tex]
### Summary
To summarize, the original function [tex]\( y = \sqrt{1 - x^2} \)[/tex] has been vertically compressed by a factor of 6 to become:
[tex]\[ y = \frac{1}{6} \sqrt{1 - x^2} \][/tex]
This is the detailed step-by-step solution to the problem.
