Question

write a quadratic function to model the graph to the right. f(x) =

Explanation:

Step1: Recall quadratic - function vertex form

The vertex - form of a quadratic function is $f(x)=a(x - h)^2 + k$, where $(h,k)$ is the vertex of the parabola.

Step2: Identify the vertex from the graph

From the graph, the vertex of the parabola is $(3,2)$. So, $h = 3$ and $k = 2$. Then the function is $f(x)=a(x - 3)^2+2$.

Step3: Find the value of $a$

We can use another point on the graph, say the $y$ - intercept. The $y$ - intercept is the point where $x = 0$. From the graph, when $x = 0$, $y=11$. Substitute $x = 0$ and $y = 11$ into $f(x)=a(x - 3)^2+2$:
\[

$$\begin{align*} 11&=a(0 - 3)^2+2\\ 11&=9a+2\\ 9a&=9\\ a&=1 \end{align*}$$

\]

Step4: Write the quadratic function

Substitute $a = 1$ into $f(x)=a(x - 3)^2+2$. We get $f(x)=(x - 3)^2+2=x^{2}-6x + 9+2=x^{2}-6x+11$.

Answer:

$f(x)=x^{2}-6x + 11$