Question

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

Answer 1

`y=-4(x+1/8)^2-47/16`

Explanation:

Begin by grouping the terms involving `x` together.

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

Factor out `-4` from the `x` terms.

`y=-4(x^2+1/4x)-3`

Complete the square. Using the formula `(b/2)^2` we get `((-1/4)/2)^2=(-1/8)^2=1/64`.

We now know that to complete the square by adding `1/64` within the parentheses. Because we are adding `1/64`, we must also subtract the amount by which it changed the problem.

`y=-4(x^2+1/4x+1/6464/)-3+1/16`

Since the `1/16` is within the parentheses, it is multiplied by `-4`, meaning overall, it changes the problem by `-1/16`. To undo this change, we add `1/16` outside the parentheses.

Now that we completed the square, the terms involving `x` can be factored like so:

`y=-4(x+1/8)^2-47/16`

The equation is now written in vertex form.