Question

select the correct answer.
what is the vertex of the quadratic function?
f(x) = (x - 8)(x - 4)
a. (4,8)
b. (-6,4)
c. (6,-4)
d. (-4,8)

Explanation:

Step1: Find the x - intercepts

For a quadratic function in factored form \(f(x)=(x - r_1)(x - r_2)\), the x - intercepts are at \(x = r_1\) and \(x=r_2\). Here, \(r_1 = 8\) and \(r_2=4\), so the x - intercepts are \(x = 8\) and \(x = 4\).

The x - coordinate of the vertex of a parabola (quadratic function) is the mid - point of the x - intercepts. The formula for the mid - point of two numbers \(a\) and \(b\) is \(\frac{a + b}{2}\). So, the x - coordinate of the vertex \(h=\frac{4 + 8}{2}=\frac{12}{2}=6\).

Step2: Find the y - coordinate of the vertex

To find the y - coordinate of the vertex, we substitute \(x = 6\) into the function \(f(x)=(x - 8)(x - 4)\).

\(f(6)=(6 - 8)(6 - 4)=(-2)\times(2)=-4\)

So the vertex of the quadratic function \(f(x)=(x - 8)(x - 4)\) is \((6,-4)\).

Answer:

C. \((6,-4)\)