Question

the graph of function $g$ is a parabola with the vertex located at $(5, 9)$. the parabola also passes through the point $(3, 1)$. which in vertex form for this 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

The vertex form of a parabola is $g(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex.
Given vertex $(5,9)$, substitute $h=5$, $k=9$:
$g(x)=a(x-5)^2+9$

Step2: Solve for $a$ using given point

Substitute the point $(3,1)$ (where $x=3$, $g(x)=1$) into the equation:
$1=a(3-5)^2+9$
Simplify the expression inside the parentheses:
$1=a(-2)^2+9$
Calculate $(-2)^2$:
$1=4a+9$
Subtract 9 from both sides:
$1-9=4a$
$-8=4a$
Divide both sides by 4:
$a=\frac{-8}{4}=-2$

Step3: Write final vertex form

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

Answer:

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