Question

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

Explanation:

Step1: Recall vertex form of quadratics

A quadratic with vertex $(h,k)$ is $f(x)=a(x-h)^2+k$

Step2: Plug in given vertex

Substitute $h=2, k=-25$: $f(x)=a(x-2)^2-25$

Step3: Use x-intercept to find $a$

Substitute $x=7, f(x)=0$: $0=a(7-2)^2-25$
Simplify: $0=25a-25$ → $25a=25$ → $a=1$

Step4: Expand to factored form

$f(x)=(x-2)^2-25$ is a difference of squares:
$f(x)=(x-2-5)(x-2+5)=(x-7)(x-2)$

Answer:

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