find the equation of the quadratic function that has its vertex at (5, -2) and includes the point (6, -9)…

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. Given the vertex $(5,-2)$, we have $h = 5$ and $k=-2$. So the function is $f(x)=a(x - 5)^2-2$.

Step2: Substitute the given point to find $a$

Substitute the point $(x = 6,y=-9)$ into the equation $f(x)=a(x - 5)^2-2$. We get $-9=a(6 - 5)^2-2$. Simplify the right - hand side: $-9=a(1)^2-2$, which is $-9=a - 2$.

Step3: Solve for $a$

Add 2 to both sides of the equation $-9=a - 2$. We have $a=-9 + 2=-7$.

Answer:

$f(x)=-7(x - 5)^2-2$