Question

write an equation of the parabola in vertex form.
an equation of the parabola is $y = \square$.

Explanation:

Step1: Recall vertex form

The vertex form of a parabola is $y=a(x-h)^2+k$, where $(h,k)$ is the vertex. From the graph, the vertex is $(4,-1)$, so substitute $h=4$, $k=-1$:
$y=a(x-4)^2-1$

Step2: Solve for $a$ using given point

Use the point $(8,3)$ on the parabola. Substitute $x=8$, $y=3$ into the equation:
$3=a(8-4)^2-1$
Simplify the equation:
$3=a(4)^2-1$
$3=16a-1$
Add 1 to both sides:
$4=16a$
Solve for $a$:
$a=\frac{4}{16}=\frac{1}{4}$

Step3: Substitute $a$ back into vertex form

Replace $a$ with $\frac{1}{4}$ in $y=a(x-4)^2-1$:
$y=\frac{1}{4}(x-4)^2-1$

Answer:

$\frac{1}{4}(x-4)^2-1$