Question

write an equation of the parabola in vertex form that passes through (13, 8) and has vertex (3, 2).
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.

Step2: Substitute vertex values

Given vertex $(3,2)$, so $h=3$, $k=2$. Substitute into the form:
$y = a(x-3)^2 + 2$

Step3: Solve for $a$ using given point

Substitute the point $(13,8)$ (so $x=13$, $y=8$) into the equation:
$8 = a(13-3)^2 + 2$
Simplify the expression inside the parentheses:
$8 = a(10)^2 + 2$
$8 = 100a + 2$
Subtract 2 from both sides:
$8-2 = 100a$
$6 = 100a$
Solve for $a$:
$a = \frac{6}{100} = \frac{3}{50}$

Step4: Write final equation

Substitute $a=\frac{3}{50}$ back into the vertex form:
$y = \frac{3}{50}(x-3)^2 + 2$

Answer:

$\frac{3}{50}(x-3)^2 + 2$