Question

What is the vertex of `y=-3x^2 -4x-2`?

Answer 1

The vertex is `(-2/3, -2/3)`.

Explanation:

This equation is currently in standard form and you must convert it to vertex form to figure out the vertex.

Vertex form is usually written as `y=a(x-h)^2+k`, where the point (h,k) is the vertex.

To convert, we can use the process of completing the square.

First, we pull out the negative 3.

`y=-3(x^2+4/3x)-2`

In completing the square, you take half of the coefficient on the x term (4/3 here), square it, and add that into the problem. Since you are adding a value, you must also subtract the same value so as not to change the equation.

`y=-3(x^2+4/3x+4/9)-2+4/3`

Now it looks like I added in 4/9 and added 4/3, but you have to be careful. Because of the -3 in front of the parentheses, when I put in 4/9, it's really like I am subtracting 4/3. Thus, I must do the opposite to keep the equation the same, so I added 4/3 at the end.

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

I factored the binomial to simplify, and now I have the equation in the proper vertex form. The vertex is point (h,k) but because h is supposed to be subtracted from x, I need to flip the sign on the positive 2/3, giving us the point `(-2/3, -2/3)`.