find the equation of the quadratic function $f$ whose graph is shown below.

Explanation:

Step1: Identify the vertex from the graph

The vertex $(h, k)$ is given as $(4, -4)$.

Step2: Use the vertex form of a quadratic

The vertex form is $f(x) = a(x - h)^2 + k$. Substituting the vertex: $$f(x) = a(x - 4)^2 - 4$$

Step3: Substitute the second point to find $a$

Use the point $(6, -12)$ for $x$ and $f(x)$: $$-12 = a(6 - 4)^2 - 4$$

Step4: Solve for the leading coefficient $a$

Simplify the equation: $$-12 = a(2)^2 - 4 \implies -12 = 4a - 4 \implies -8 = 4a \implies a = -2$$

Step5: Write the final equation

Substitute $a = -2$ back into the vertex form: $$f(x) = -2(x - 4)^2 - 4$$

Answer:

$f(x) = -2(x - 4)^2 - 4$