The equation of a parabola `P`

is `y = `

.`A`x^2 + `B`x + `C`

What are its vertex `(h, k)`

and its `y`

-intercept?

`(h, k) = (`

`H``,`

`K``)`

`y`

-intercept `=`

`C`

The `y`

-intercept is the point on the `y`

-axis where `x = 0`

.

If `x = 0`

, we have `y = `

, so the `A` \cdot 0^2 + `B` \cdot 0 + `C` = `C``y`

-intercept is

.`C`

The equation of a parabola with vertex `(h, k)`

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

.

We can rewrite the given equation as

, in order to get the form `A`x^2 + `A` \cdot `- 2 * H`x + `H * A * H` + `K``a(x - h)^2`

We factor out `A`

, giving `y = `

`A` ( x^2 + `2 * H`x + `H * H` ) + `K`

The equation in the parentheses is of the form `( a + b )^2`

, because ` ( x^2 + `

`2 * H` + `H * H` ) = ( x^2 + 2 \cdot `H` + `H`^2 )

Therefore, `y = `

.`A`( x - `H`)^2 + `K`

` y = `

.`A`(x - (`H`))^2 + `K`

Thus, the center `(h, k)`

is `(`

`H`, `K`)