Question

How do you write `f(x) + 9 = 2(x - 1)^2` in standard form?

Answer 1

See a solution process below:

Explanation:

First, we need to expand the terms in parenthesis. This term is a special form of the quadratic.:

(a - b)^2 = a^2 - 2ab + b^2

Substituting `x` for `a` and `1` for `b` gives:

`f(x) + 9 = 2(x - 1)^2`

`f(x) + 9 = 2(x^2 - (2 * x * 1) + 1^2)`

`f(x) + 9 = 2(x^2 - 2x + 1)`

Next, we eliminate the parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis:

`f(x) + 9 = color(red)(2)(x^2 - 2x + 1)`

`f(x) + 9 = (color(red)(2) * x^2) - (color(red)(2) * 2x) + (color(red)(2) * 1)`

`f(x) + 9 = 2x^2 - 4x + 2`

Now, subtract `color(red)(9)` from each side of the function to complete the transformation to standard form:

`f(x) + 9 - color(red)(9) = 2x^2 - 4x + 2 - color(red)(9)`

`f(x) + 0 = 2x^2 - 4x - 7`

`f(x) = 2x^2 - 4x - 7`

Answer 2

`f(x)=2x^2-4x-7`

Explanation:

`"the standard form of a parabola is"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(f(x)=ax^2+bx+c;a!=0)color(white)(2/2)|)))`

`"expand the bracket using FOIL"`

`rArrf(x)+9=2(x-1)(x-1)`

`color(white)(rArrf(x)+9)=2(x^2-2x+1)`

`color(white)(rArrf(x)+9)=2x^2-4x+2`

`"subtract 9 from both sides"`

`rArrf(x)=2x^2-4x+2-9`

`color(white)(rArrf(x))=2x^2-4x-7larrcolor(red)" in standard form"`