To find the equation of [tex]\( g(x) \)[/tex] based on the given function [tex]\( f(x) = (x - 3)^2 \)[/tex] and the information about the vertical stretch, we will follow these steps:
1. Identify the given function: [tex]\( f(x) = (x - 3)^2 \)[/tex]
2. Vertical stretch factor: The stretch factor is given as 2. This means the output of the function [tex]\( f(x) \)[/tex] will be multiplied by 2 to give the new function [tex]\( g(x) \)[/tex].
3. Apply the vertical stretch to the function: To incorporate the vertical stretch factor into [tex]\( f(x) \)[/tex], we multiply the entire function by the stretch factor.
So, [tex]\( g(x) = 2 \cdot f(x) \)[/tex].
4. Substitute [tex]\( f(x) \)[/tex] into the equation: Replace [tex]\( f(x) \)[/tex] with [tex]\((x - 3)^2\)[/tex].
Therefore, [tex]\( g(x) = 2 \cdot (x - 3)^2 \)[/tex].
So, the equation of [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = 2(x - 3)^2 \][/tex]
To confirm, let's evaluate [tex]\( g(x) \)[/tex] at some specific points, ensuring our transformation is correct:
- For [tex]\( x = 0 \)[/tex]:
[tex]\( g(0) = 2(0 - 3)^2 = 2 \cdot 9 = 18 \)[/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\( g(1) = 2(1 - 3)^2 = 2 \cdot 4 = 8 \)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\( g(2) = 2(2 - 3)^2 = 2 \cdot 1 = 2 \)[/tex]
These evaluations confirm that our derived equation [tex]\( g(x) = 2(x - 3)^2 \)[/tex] gives the correct function values as expected.
