Question

What is the vertex form of `y=x^2-2x+6`?

Answer 1

In vertex form, the parabola's equation is `y=(x-1)^2+5`.

Explanation:

To convert a parabola in standard form to vertex form, you have to make a squared binomial term (i.e. `(x-1)^2` or `(x+6)^2`).

These squared binomial terms -- take `(x-1)^2`, for example -- (almost) always expand to have `x^2`, `x`, and constant terms. `(x-1)^2` expands to be `x^2-2x+1`.

In our parabola:

`y=x^2-2x+6`

We have a part that looks similar to the expression we wrote before: `x^2-2x+1`. If we rewrite our parabola, we can "undo" this squared binomial term, like this:

`y=x^2-2x+6`

`color(white)y=color(red)(x^2-2x+1)+5`

`color(white)y=color(red)((x-1)^2)+5`

This is our parabola in vertex form. Here's its graph:

graph{(x-1)^2+5 [-12, 13.7, 0, 13.12]}