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

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

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

Answer

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: Use the x - intercept to find a

Since the x - intercept is at $(7,0)$, substitute $x = 7$ and $f(x)=0$ into the equation $0=a(7 - 2)^2-25$. First, simplify $(7 - 2)^2$: $(7 - 2)^2=25$, so the equation becomes $0 = 25a-25$. Add 25 to both sides: $25=25a$. Divide both sides by 25: $a = 1$. Then $f(x)=(x - 2)^2-25$. Expand $(x - 2)^2-25$: $f(x)=x^{2}-4x + 4-25=x^{2}-4x-21$. Factor $x^{2}-4x - 21=(x - 7)(x+3)$.

Answer:

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