Sure, let's tackle this problem step-by-step.
1. Start with the given function [tex]\( f(x) = (x-1)^2 \)[/tex].
2. Understand what a vertical stretch is:
- A vertical stretch of a function by a factor of [tex]\( a \)[/tex] means that every value of the function will be multiplied by [tex]\( a \)[/tex].
- In mathematical terms, if [tex]\( f(x) \)[/tex] is your original function and you want to stretch it vertically by a factor of [tex]\( a \)[/tex], the new function [tex]\( g(x) \)[/tex] will be [tex]\( g(x) = a \cdot f(x) \)[/tex].
3. Identify the stretch factor:
- In this problem, the function is being stretched by a factor of 2.
- This means we will multiply the original function by 2.
4. Apply the stretch:
- The original function is [tex]\( f(x) = (x-1)^2 \)[/tex].
- To apply the vertical stretch by a factor of 2, we multiply [tex]\( (x-1)^2 \)[/tex] by 2.
- Hence, the new function [tex]\( g(x) \)[/tex] is given by:
[tex]\[
g(x) = 2 \cdot (x-1)^2
\][/tex]
5. Write the final function:
- Therefore, the equation for [tex]\( g(x) \)[/tex] after the vertical stretch by a factor of 2 is:
[tex]\[
g(x) = 2 \cdot (x-1)^2
\][/tex]
So, the equation of [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = 2 \cdot (x-1)^2
\][/tex]
