# How do you write 4y= x^2 - 2x - 31 into vertex form?

Jul 29, 2015

$\textcolor{red}{y = \frac{1}{4} {\left(x - 1\right)}^{2} - 8}$

#### Explanation:

The vertex form of a quadratic is given by y = a(x – h)^2 + k, where ($h , k$) is the vertex.

The "$a$" in the vertex form is the same "$a$" as in $y = a {x}^{2} + b x + c$.

$4 y = {x}^{2} - 2 x - 31$

Step 1. Divide both sides by the coefficient of $y$.

$y = \frac{1}{4} {x}^{2} - \frac{1}{2} x - \frac{31}{4}$

We convert to the "vertex form" by completing the square.

Step 2. Move the constant to the other side.

$y + \frac{31}{4} = \frac{1}{4} {x}^{2} - \frac{1}{2} x$

Step 3. Factor out the coefficient $a$,

$y + \frac{31}{4} = \frac{1}{4} \left({x}^{2} - 2 x\right)$

Step 4. Square the new coefficient of $x$ and divide by 4.

${\left(- 2\right)}^{2} / 4 = 1$

Step 5. Add and subtract this value inside the parentheses..

$y + \frac{31}{4} = \frac{1}{4} \left({x}^{2} - 2 x + 1 - 1\right)$

Step 6. Express the right hand side as a square.

$y + \frac{31}{4} = \frac{1}{4} \left({\left(x - 1\right)}^{2} - 1\right)$

Step 7. Distribute.

y+31/4 = 1/4(x-1)^2-1/4×1

Step 8. Multiply.

$y + \frac{31}{4} = \frac{1}{4} {\left(x - 1\right)}^{2} - \frac{1}{4}$

Step 9. Isolate $y$.

$y = \frac{1}{4} {\left(x - 1\right)}^{2} - \frac{1}{4} - \frac{31}{4}$

Step 10. Combine like terms.

$y = \frac{1}{4} {\left(x - 1\right)}^{2} - \frac{32}{4}$

$y = \frac{1}{4} {\left(x - 1\right)}^{2} - 8$

The equation is now in vertex form.

y = a(x – h)^2 + k, where ($h , k$) is the vertex.

$h = 1$ and $k = - 8$, so the vertex is at ($1 , - 8$).