Question

determine the quadratic function of the form $f(x)=a(x - h)^2 + k$ whose graph is given on the right.
$f(x)=\square$
(do not simplify.)

Explanation:

Step1: Identify the vertex \((h,k)\)

The vertex of the parabola is given as \((-2, 8)\) from the graph. So \(h = -2\) and \(k = 8\). The quadratic function is \(f(x)=a(x - (-2))^{2}+8=a(x + 2)^{2}+8\).

Step2: Find the value of \(a\)

We know the graph passes through the point \((0,0)\). Substitute \(x = 0\) and \(f(x)=0\) into the equation:
\[
0=a(0 + 2)^{2}+8
\]
\[
0 = 4a+8
\]
Subtract 8 from both sides:
\[
4a=-8
\]
Divide both sides by 4:
\[
a=-2
\]

Step3: Write the final function

Substitute \(a=-2\), \(h = -2\), and \(k = 8\) into the vertex form:
\(f(x)=-2(x + 2)^{2}+8\)

Answer:

\(f(x)=-2(x + 2)^{2}+8\)