Question

short answer
all work supporting you solution must be shown.

1. given the vertex is $(-1, -4)$ and a point on the graph is $(4, 46)$. determine the equation of the parabola in the form $y = a(x - p)^2 + q$ 2 marks

Explanation:

Step1: Substitute vertex into vertex form

The vertex form is $y=a(x-p)^2+q$, where $(p,q)$ is the vertex. For vertex $(-1,-4)$, $p=-1$, $q=-4$.
$y=a(x-(-1))^2+(-4)$
Simplify to get: $y=a(x+1)^2-4$

Step2: Substitute point to solve for $a$

Use the point $(4,46)$: substitute $x=4$, $y=46$ into the equation.
$46=a(4+1)^2-4$
Calculate $(4+1)^2=25$:
$46=25a-4$
Add 4 to both sides:
$46+4=25a$
$50=25a$
Solve for $a$: $a=\frac{50}{25}=2$

Step3: Write final equation

Substitute $a=2$ back into the vertex form.
$y=2(x+1)^2-4$

Answer:

$y=2(x+1)^2-4$