find the equation of the quadratic function f whose graph is shown below. f(x)=□

Explanation:

Step1: Recall quadratic - 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. From the graph, the vertex is $(3, - 3)$, so $h = 3$ and $k=-3$. Then $f(x)=a(x - 3)^2-3$.

Step2: Find the value of $a$

We know that the point $(2,-5)$ lies on the parabola. Substitute $x = 2$ and $y=-5$ into $f(x)=a(x - 3)^2-3$. [ \begin{align*} -5&=a(2 - 3)^2-3\ -5&=a(-1)^2-3\ -5&=a - 3\ a&=-2 \end{align*} ]

Step3: Write the quadratic function

Substitute $a=-2$ into $f(x)=a(x - 3)^2-3$. We get $f(x)=-2(x - 3)^2-3$. Expand it: [ \begin{align*} f(x)&=-2(x^{2}-6x + 9)-3\ &=-2x^{2}+12x-18 - 3\ &=-2x^{2}+12x-21 \end{align*} ]

Answer:

$f(x)=-2(x - 3)^2-3$ (or $f(x)=-2x^{2}+12x - 21$)