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

Explanation:

Step1: Recall vertex - form of quadratic

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 $(1,-4)$, so $h = 1$ and $k=-4$. Then $f(x)=a(x - 1)^2-4$.

Step2: Find the value of a

The parabola passes through the point $(2,-6)$. Substitute $x = 2$ and $y=-6$ into $f(x)=a(x - 1)^2-4$. We get $-6=a(2 - 1)^2-4$. Simplify the right - hand side: $-6=a\times1^2-4$, which is $-6=a - 4$. Add 4 to both sides of the equation: $a=-6 + 4=-2$.

Step3: Write the quadratic function

Substitute $a=-2$ into $f(x)=a(x - 1)^2-4$. So $f(x)=-2(x - 1)^2-4$. Expand it: $f(x)=-2(x^{2}-2x + 1)-4=-2x^{2}+4x-2-4=-2x^{2}+4x-6$.

Answer:

$f(x)=-2x^{2}+4x - 6$