Question

How do you convert `y=x^2-2x+3` in vertex form?

Answer 1

Please see the explanation.

Explanation:

The vertex form of a parabola that opens up or down is:

`y = a(x - h) + k`

where "a" is the same as the "a" in the standard form for a parabola that opens up or down:

`y = ax^2 + bx + c`

To convert to the vertex form, add 0 in the form of #ah^2 - ah^2 to the equation:

`y = ax^2 + bx + ah^2 - ah^2 + c`

Factor out "a" from the first 3 terms:

`y = a(x^2 + b/ax + h^2) - ah^2 + c`

Using the pattern (x - h)^2 = x^2 - 2hx + h^2, observe that middle term of the pattern equals the middle term of the equation:

`-2h = b/a`

Solve for h:

`h = -b/(2a)`

Substitute the left side of the pattern into the equation:

`y = a(x - h)^2 - ah^2 + c`

Substitute `-b/(2a)` for h:

`y = a(x - -b/(2a))^2 - a(-b/(2a))^2 + c`

In a problem with numbers, the last step is to combine the constant terms.

Given:

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

`a = 1`, therefore, we add 0 by adding `h^2 - h^2`

`y = x^2 - 2x + h^2 - h^2 + 3`

`h = -b/(2a) = -(-2)/(2(1)) = 1`

Substitute the left side of the pattern into the equation:

`y = (x - h)^2 - h^2 + 3`

Substitute 1 for h:

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

Combine the constant terms:

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