Question

determine the quadratic function whose graph is given below. the quadratic function which describes the given graph is f(x)= . (type an expression.)

Explanation:

Step1: Recall vertex - form of 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 $(-3,3)$, so $h=-3$ and $k = 3$. Then the function is $f(x)=a(x + 3)^2+3$.

Step2: Find the value of $a$

We know that the graph passes through the point $(0,-8)$. Substitute $x = 0$ and $y=-8$ into $f(x)=a(x + 3)^2+3$.
\[

$$\begin{align*} -8&=a(0 + 3)^2+3\\ -8&=9a+3\\ 9a&=-8 - 3\\ 9a&=-11\\ a&=-\frac{11}{9} \end{align*}$$

\]

Step3: Write the quadratic function

Substitute $a =-\frac{11}{9}$ into $f(x)=a(x + 3)^2+3$.
\[

$$\begin{align*} f(x)&=-\frac{11}{9}(x + 3)^2+3\\ &=-\frac{11}{9}(x^{2}+6x + 9)+3\\ &=-\frac{11}{9}x^{2}-\frac{22}{3}x-11 + 3\\ &=-\frac{11}{9}x^{2}-\frac{22}{3}x-8 \end{align*}$$

\]

Answer:

$-\frac{11}{9}x^{2}-\frac{22}{3}x - 8$