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Which function in vertex form is equivalent to f(x) = x2 + 8 – 16x?

f(x) = (x – 8)2 – 56
f(x) = (x – 4)2 + 0
f(x) = (x + 8)2 – 72
f(x) = (x + 4)2 – 32

1 answer

To convert the quadratic function \( f(x) = x^2 + 8 - 16x \) to vertex form, we need to complete the square.

  1. Start with the original function: \[ f(x) = x^2 - 16x + 8 \]

  2. To complete the square for the \( x^2 - 16x \) part, we take half of the coefficient of \( x \) (which is -16), square it, and add and subtract that value:

    • Half of -16 is -8, and squaring it gives \( (-8)^2 = 64 \).

  3. Rewrite the function, adding and subtracting 64: \[ f(x) = (x^2 - 16x + 64) - 64 + 8 \] \[ f(x) = (x - 8)^2 - 56 \]

Now we have the function in vertex form: \[ f(x) = (x - 8)^2 - 56 \]

Thus, the equivalent function in vertex form is: \[ \boxed{f(x) = (x - 8)^2 - 56} \]