what is the equation of the quadratic function with a vertex at (2, - 25) and an x - intercept at (7,0)?\n○…

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. Here, $h = 2$ and $k=-25$, so $f(x)=a(x - 2)^2-25$.

Step2: Substitute the x - intercept into the function

Since the x - intercept is at $(7,0)$, substitute $x = 7$ and $f(x)=0$ into $f(x)=a(x - 2)^2-25$. We get $0=a(7 - 2)^2-25$.

Step3: Solve for a

First, simplify the equation: $0=a(5)^2-25$, which is $0 = 25a-25$. Add 25 to both sides: $25=25a$. Then divide both sides by 25, so $a = 1$.

Step4: Expand the vertex - form

$f(x)=(x - 2)^2-25=x^{2}-4x + 4-25=x^{2}-4x-21$. Factor the quadratic function: $x^{2}-4x - 21=(x + 3)(x - 7)$.

Answer:

D. $f(x)=(x + 3)(x - 7)$