Question

what is the equation of the quadratic function with a vertex at $(2,-25)$ and an $x$-intercept at $(7,0)$?
$\bigcirc$ $f(x) = (x - 2)(x - 7)$
$\bigcirc$ $f(x) = (x + 2)(x + 7)$
$\bigcirc$ $f(x) = (x - 3)(x + 7)$
$\bigcirc$ $f(x) = (x + 3)(x - 7)$

Explanation:

Step1: Use 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. Here, $(h,k)=(2,-25)$, so substitute these values:
$f(x)=a(x-2)^2-25$

Step2: Substitute x-intercept to find a

The x-intercept $(7,0)$ means $f(7)=0$. Substitute $x=7$ and $f(x)=0$ into the equation:
$0=a(7-2)^2-25$
Simplify the expression:
$0=a(5)^2-25$
$0=25a-25$
Solve for $a$:
$25a=25$
$a=1$

Step3: Expand to standard factored form

Substitute $a=1$ back into the vertex form, then expand and factor:
$f(x)=(x-2)^2-25$
This is a difference of squares: $f(x)=(x-2-5)(x-2+5)$
Simplify the terms:
$f(x)=(x-7)(x+3)$

Answer:

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