Question

the graph of function g is a parabola with the vertex located at (5, 9). the parabola also passes through the points (7, 1) and (3, 1). select an equation in vertex form for the function.
a. $g(x) = 2(x - 5)^2 + 9$
b. $g(x) = 2(x - 5)^2 - 9$
c. $g(x) = -2(x - 5)^2 + 9$
d. $g(x) = -2(x + 5)^2 + 9$

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is $g(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex.

Step2: Plug in vertex coordinates

Given vertex $(5,9)$, substitute $h=5$, $k=9$:
$g(x)=a(x-5)^2+9$

Step3: Solve for $a$ using given point

Use point $(7,1)$: substitute $x=7$, $g(x)=1$
$1=a(7-5)^2+9$
$1=a(2)^2+9$
$1=4a+9$
$4a=1-9=-8$
$a=\frac{-8}{4}=-2$

Step4: Write final equation

Substitute $a=-2$ back into vertex form:
$g(x)=-2(x-5)^2+9$

Answer:

C. $g(x) = -2(x - 5)^2 + 9$