Question

How do you write the function `f(x) = 4(x-2)^2+8` in standard form?

Answer 1

See a solution process below:

Explanation:

First expand the term with the exponent using this rule for quadratics:

`(color(red)(x) - color(blue)(y))^2 = color(red)(x)^2 - 2color(red)(x)color(blue)(y) + color(blue)(y)^2`

Therefore:

`f(x) = 4(color(red)(x) - color(blue)(2))^2 + 8`

`f(x) = 4(color(red)(x)^2 - (2 xx color(red)(x) xx color(blue)(2)) + color(blue)(2)^2) + 8`

`f(x) = 4(x^2 - 4x + 4) + 8`

Next, expand the terms in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis:

`f(x) = color(red)(4)(x^2 - 4x + 4) + 8`

`f(x) = (color(red)(4) xx x^2) - (color(red)(4) xx 4x) + (color(red)(4) xx 4) + 8`

`f(x) = 4x^2 - 16x + 16 + 8`

Now, combine like terms:

`f(x) = 4x^2 - 16x + (16 + 8)`

`f(x) = 4x^2 - 16x + 24`