Determine the quadratic function whose graph is given in the figure.

Given figure of graph:

We want to determine the quadratic function whose graph is given in the figure.

From the graph we can read two points:

• the vertex: V(4, - 8)
• the intersection with the y-axis: A(0, 24)

Since we know that the vertex of the parabola we will use the vertex form of the quadratic function:

$f\left(x\right)=a{(x-h)}^2+k$

where h is x coordinate of vertex (h-x_V) and k is y coordinate of vertex k = y_V.

By including the vertex V(4, - 8) in the upper formula (where h = 4 and k = - 8) we get:

$f\left(x\right)=a{(x-4)}^2-8$

In order to determine the leading coefficient a we will use another known condition (intersection with y-axis):

If A(0 , 24) then

$$\begin{gathered} f\left(0\right)=a{(0-4)}^2-8 \\ 24=16a-8 \\ 16a=32 \\ a=2 \end{gathered}$$

The equation of the quadratic function is given by :

$$\begin{gathered} f\left(x\right)=2{(x-4)}^2-8 \\ f\left(x\right)=2(x^2-8x+16)-8 \\ f\left(x\right)=2x^2-16x+32-8 \\ f\left(x\right)=2x^2-16x+24 \end{gathered}$$