Question

this graph represents a quadratic function.
what is the value of $a$ in this functions equation?

Explanation:

Step1: Identify vertex form

The vertex form of a quadratic function is $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex. From the graph, the vertex is $(0,9)$, so $h=0$, $k=9$. Substitute into the formula:
$y = a(x-0)^2 + 9 = ax^2 + 9$

Step2: Use x-intercept to solve for a

The graph crosses the x-axis at $(3,0)$. Substitute $x=3$, $y=0$ into the equation:
$0 = a(3)^2 + 9$

Step3: Simplify and solve for a

Calculate $3^2=9$, then rearrange to isolate $a$:
$9a = -9$
$a = \frac{-9}{9} = -1$

Answer:

$-1$