Question

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

Answer 1

`color(red)(y = 1/4(x-1)^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 = ax^2 + bx + c`.

Your equation is

`4y = x^2-2x-31`

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

`y = 1/4x^2-1/2x-31/4`

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

Step 2. Move the constant to the other side.

`y+31/4 = 1/4x^2-1/2x`

Step 3. Factor out the coefficient `a`,

`y+31/4 = 1/4(x^2-2x)`

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

`(-2)^2/4 = 1`

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

`y+31/4 = 1/4(x^2-2x +1-1)`

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

`y+31/4 = 1/4((x-1)^2-1)`

Step 7. Distribute.

`y+31/4 = 1/4(x-1)^2-1/4×1`

Step 8. Multiply.

`y+31/4 = 1/4(x-1)^2-1/4`

Step 9. Isolate `y`.

`y = 1/4(x-1)^2-1/4-31/4`

Step 10. Combine like terms.

`y = 1/4(x-1)^2-32/4`

`y = 1/4(x-1)^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`).