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

Nov 15, 2016

Explanation:

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

$y = a \left(x - h\right) + k$

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

$y = a {x}^{2} + b x + c$

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

$y = a {x}^{2} + b x + a {h}^{2} - a {h}^{2} + c$

Factor out "a" from the first 3 terms:

$y = a \left({x}^{2} + \frac{b}{a} x + {h}^{2}\right) - a {h}^{2} + c$

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:

$- 2 h = \frac{b}{a}$

Solve for h:

$h = - \frac{b}{2 a}$

Substitute the left side of the pattern into the equation:

$y = a {\left(x - h\right)}^{2} - a {h}^{2} + c$

Substitute $- \frac{b}{2 a}$ for h:

$y = a {\left(x - - \frac{b}{2 a}\right)}^{2} - a {\left(- \frac{b}{2 a}\right)}^{2} + c$

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

Given:

$y = {x}^{2} - 2 x + 3$

$a = 1$, therefore, we add 0 by adding ${h}^{2} - {h}^{2}$

$y = {x}^{2} - 2 x + {h}^{2} - {h}^{2} + 3$

$h = - \frac{b}{2 a} = - \frac{- 2}{2 \left(1\right)} = 1$

Substitute the left side of the pattern into the equation:

$y = {\left(x - h\right)}^{2} - {h}^{2} + 3$

Substitute 1 for h:

$y = {\left(x - 1\right)}^{2} - {1}^{2} + 3$

Combine the constant terms:

$y = {\left(x - 1\right)}^{2} + 2$