To determine the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) = 4 f(x) \)[/tex], we need to analyze what happens to the [tex]\( y \)[/tex]-intercept of the original function [tex]\( f(x) \)[/tex] when it is scaled by a factor of 4.
1. Identify the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]:
Given that the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is at [tex]\( (0, 2) \)[/tex], it means that when [tex]\( x = 0 \)[/tex] in the function [tex]\( f(x) \)[/tex], the value of [tex]\( f(x) \)[/tex] is 2. Mathematically, this is written as:
[tex]\[
f(0) = 2
\][/tex]
2. Determine the effect of scaling on [tex]\( f(x) \)[/tex]:
The function [tex]\( g(x) \)[/tex] is defined as [tex]\( g(x) = 4 f(x) \)[/tex]. To find the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex], we need to evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 4 f(0)
\][/tex]
3. Substitute the value of [tex]\( f(0) \)[/tex]:
We know from step 1 that [tex]\( f(0) = 2 \)[/tex]. Substituting this into the expression for [tex]\( g(0) \)[/tex], we get:
[tex]\[
g(0) = 4 \times 2 = 8
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is at the point [tex]\((0, 8)\)[/tex].
So, the point that represents the [tex]\( y \)[/tex]-intercept of the function [tex]\( g \)[/tex] is:
[tex]\[
(0, 8)
\][/tex]
Thus, the correct answer is:
[tex]\[
(0, 8)
\][/tex]
