Question

How do you write `y=3(x-1)^2+5` in standard form?

Answer 1

See a solution process below:

Explanation:

First, we need to expand the squared term using this rule:

`(color(red)(a) - color(blue)(b))^2 = color(red)(a)^2 - 2color(red)(a)color(blue)(b) + color(blue)(b)^2`

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

`y = 3(color(red)(x) - color(blue)(1))^2 + 5`

`y = 3(color(red)(x)^2 - [2 * color(red)(x) * color(blue)(1)] + color(blue)(1)^2) + 5`

`y = 3(x^2 - 2x + 1) + 5`

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

`y = color(red)(3)(x^2 - 2x + 1) + 5`

`y = (color(red)(3) xx x^2) - (color(red)(3) xx 2x) + (color(red)(3) xx 1) + 5`

`y = 3x^2 - 6x + 3 + 5`

Now, combine like terms:

`y = 3x^2 - 6x + (3 + 5)`

`y = 3x^2 - 6x + 8`

Answer 2

`y = 3x^2 + 6x + 8`

Explanation:

`y = 3(x - 1)^2 + 5`.
To convert the vertex form to standard form, we develop the vertex form:
`y = 3(x^2 - 2x + 1) + 5 = 3x^2 - 6x + 3 + 5`
`y = 3x^2 - 6x + 8`