Question

What is the vertex form of `y= (x + 10)(x – 4)`?

Answer 1

The vertex form for this equation is `y=(x+3)^2-49`

Explanation:

There are many ways to do this problem. Most people would expand this factored form to standard form and then complete the square to convert the standard form to the vertex form. THIS WOULD WORK, however there is a way to convert this directly to the vertex form. This is what I will demonstrate here.

An equation in factored form

`y=a(x-r_1)(x-r_2)`

has roots at `x=r_1` and `x=r_2`. The `x`-coordinate of the vertex, `x_v` must be equal the average of these two roots.

`x_v=(r_1+r_2)/2`

Here, `r_1=-10` and `r_2=4`, so

`x_v=(-10+4)/2=-3`

The `y`-coordinate of the vertex, `y_v` must be the value of `y` when `x=x_v`.

`y_v=(-3+10)(-3-4)=-49`

The general vertex form of a parabola whose vertex is at `(k, h)` is

`y=a(x-k)^2+h`.

Here, `a=1`, so the vertex form for this equation is

`y=(x+3)^2-49`.

We can see we get the same answer if we go the long way around by expanding and then completing the square.

`y=(x+10)(x-4)=x^2+6x-40=x^2+6x+9-49=(x+3)^2-49`