To determine the [tex]$y$[/tex]-coordinate of the vertex of the parabola represented by [tex]\( g(x) \)[/tex], we follow these steps:
1. Express [tex]\( f(x) \)[/tex] in standard form:
[tex]\[ f(x) = (2 - x)(x + 4) \][/tex]
2. Expand [tex]\( f(x) \)[/tex]:
To simplify [tex]\( f(x) \)[/tex], we first perform the multiplication:
[tex]\[ f(x) = (2 - x)(x + 4) \][/tex]
Using distributive property:
[tex]\[ f(x) = 2x + 8 - x^2 - 4x \][/tex]
[tex]\[ f(x) = -x^2 - 2x + 8 \][/tex]
3. Express [tex]\( g(x) \)[/tex] in terms of [tex]\( f(x) \)[/tex]:
Given [tex]\( g(x) = f(x - 10) \)[/tex], we substitute [tex]\( x - 10 \)[/tex] in place of [tex]\( x \)[/tex] in the function [tex]\( f \)[/tex]:
[tex]\[ g(x) = f(x - 10) \][/tex]
Substituting [tex]\( x - 10 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = -(x - 10)^2 - 2(x - 10) + 8 \][/tex]
4. Simplify [tex]\( g(x) \)[/tex]:
Expand the expression:
[tex]\[ g(x) = -(x^2 - 20x + 100) - 2(x - 10) + 8 \][/tex]
[tex]\[ g(x) = -x^2 + 20x - 100 - 2x + 20 + 8 \][/tex]
Combine like terms:
[tex]\[ g(x) = -x^2 + 18x - 72 \][/tex]
5. Vertex form of a parabola:
The standard form of a quadratic function is [tex]\( ax^2 + bx + c \)[/tex]. For our function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -x^2 + 18x - 72 \][/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( b = 18 \)[/tex], and [tex]\( c = -72 \)[/tex].
6. Calculate the [tex]$y$[/tex]-coordinate of the vertex:
The formula for the [tex]$x$[/tex]-coordinate of the vertex in a parabola [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in values:
[tex]\[ x = -\frac{18}{2(-1)} = 9 \][/tex]
To find the [tex]$y$[/tex]-coordinate, substitute [tex]\( x = 9 \)[/tex] back into [tex]\( g(x) \)[/tex]:
[tex]\[ g(9) = -9^2 + 18 \cdot 9 - 72 \][/tex]
[tex]\[ g(9) = -81 + 162 - 72 \][/tex]
[tex]\[ g(9) = 81 - 72 \][/tex]
[tex]\[ g(9) = 9 \][/tex]
Therefore, the [tex]$y$[/tex]-coordinate of the vertex of the parabola represented by [tex]\( g(x) \)[/tex] is [tex]\(\boxed{9}\)[/tex].
