Let [tex][tex]$f(x) = (x-3)^2$[/tex][/tex].

The function [tex][tex]$g(x)$[/tex][/tex] is a vertical stretch of [tex][tex]$f(x)$[/tex][/tex] by a factor of 2.

What is the equation of [tex][tex]$g(x)$[/tex][/tex]?

[tex][tex]$g(x) = $[/tex][/tex]



Answer :

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.