Answer: Choice B. The vertex is (6,-4)
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Work Shown:
Step 1 is to expand out (x-8)(x-4) using the FOIL rule or the box method or the distribution rule
(x-8)(x-4) = x(x-4)-8(x-4)
(x-8)(x-4) = x*x+x*(-4)-8*x-8*(-4)
(x-8)(x-4) = x^2-4x-8x+32
(x-8)(x-4) = x^2-12x+32
So (x-8)(x-4) turns into x^2-12x+32
x^2-12x+32 is the same as 1x^2+(-12x)+32 which is in the form ax^2+bx+c. We see that a = 1, b = -12, c = 32
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Use the values of a & b to find the value of h, which is the x coordinate of the vertex
h = -b/(2*a)
h = -(-12)/(2*1)
h = 12/2
h = 6
Then this is plugged back into the original function to find the y coordinate of the vertex. We can use either (x-8)(x-4) or x^2-12x+32 since they are equivalent expressions
k = y coordinate of vertex
k = f(h) = f(6) since h = 6
f(x) = (x-8)(x-4)
f(6) = (6-8)(6-4)
f(6) = (-2)(2)
f(6) = -4
note that
f(x) = x^2-12x+32
f(6) = (6)^2-12(6)+32
f(6) = 36-72+32
f(6) = -36+32
f(6) = -4
So we get the same result using either expression
So k = f(h) = f(6) = -4
Since h = 6 and k = -4, the vertex is (h,k) = (6,-4). So that's why the answer is choice B.
