Question

write a quadratic function to model the graph to the right.

f(x) = \square

Explanation:

Step1: Identify the vertex form of a quadratic function

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. From the graph, the vertex is at \((0, 4)\) (since it's on the y - axis at \(y = 4\)). So \(h = 0\) and \(k = 4\). Substituting these values into the vertex form, we get \(f(x)=a(x - 0)^2+4=a x^{2}+4\).

Step2: Find the value of 'a'

We need to find a point on the parabola to determine the value of \(a\). Let's take the point \((2, 20)\) (from the graph, when \(x = 2\), \(y = 20\)). Substitute \(x = 2\) and \(y = 20\) into the equation \(f(x)=a x^{2}+4\):
\[

$$\begin{align*} 20&=a(2)^{2}+4\\ 20&=4a + 4\\ 4a&=20 - 4\\ 4a&=16\\ a&=\frac{16}{4}\\ a& = 4 \end{align*}$$

\]

Step3: Write the final quadratic function

Substitute \(a = 4\), \(h = 0\), and \(k = 4\) back into the vertex form. We get \(f(x)=4x^{2}+4\).

Answer:

\(4x^{2}+4\)