The quadratic function given is [tex]\( g(x) = -5(x-3)^2 \)[/tex].
1. Finding the Vertex:
The general form of a quadratic function that reveals its vertex is [tex]\( g(x) = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex.
- In this function, [tex]\( g(x) = -5(x-3)^2 \)[/tex], we can see that [tex]\( h = 3 \)[/tex] and [tex]\( k = 0 \)[/tex].
Therefore, the vertex of the function [tex]\( g \)[/tex] is the point [tex]\((3, 0)\)[/tex].
2. Finding the [tex]\(x\)[/tex]-Intercept:
The [tex]\( x \)[/tex]-intercept is found by setting [tex]\( g(x) = 0 \)[/tex] and solving for [tex]\( x \)[/tex]:
[tex]\[
0 = -5(x-3)^2
\][/tex]
Dividing both sides by [tex]\(-5\)[/tex]:
[tex]\[
(x-3)^2 = 0
\][/tex]
Taking the square root of both sides:
[tex]\[
x-3 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 3
\][/tex]
Therefore, the [tex]\( x \)[/tex]-intercept of the function [tex]\( g \)[/tex] is the point [tex]\((3, 0)\)[/tex].
So, we have:
- The [tex]\( x \)[/tex]-intercept of function [tex]\( g \)[/tex] is [tex]\((3, 0)\)[/tex]
- The vertex of function [tex]\( g \)[/tex] is [tex]\((3, 0)\)[/tex]
