Given the equation:

`\qquad y = `

`A_DISP`x^2 + `-2 * A * H`x + `A * H * H + K`

Find the parabola's vertex.

`\large{\left(\right.}`

`H``,\text{ }`

`K``\large{\left.\right)}`

a point, like

`(-1, 2)`

When the equation is rewritten in vertex form like this, the vertex is the point `(h, k)`

:

`\qquad `

`formatGroup( GROUP1, [ 0 ] )`

We can rewrite the equation in vertex form by completing the square. First, move the constant term to the left side of the equation:

`\qquad `

`formatGroup( COMP_SQR1, [ 0, 1 ] )`

Next, we can factor out a

from the right side:`A`

`\qquad `

`formatGroup( COMP_SQR2, [ 0 ] )`

We can complete the square by taking half of the coefficient of our x term, squaring it, and adding it to both sides of the equation.
The coefficient of our x term is

, so half of it would be `-2 * H`

,
and squaring that gives us `-H`

. Because we're adding the `H * H`

inside the parentheses on the right where it's being multiplied by `H * H`

, we need to add `A`

to the left side to make sure we're adding the same thing to both sides.
`A * H * H`

`\qquad `

`formatGroup( COMP_SQR3, [ 0, 1, 2 ] )`

Now we can rewrite the expression in parentheses as a squared term:

`\qquad `

`formatGroup( COMP_SQR4, [ 0 ] )`

Move the constant term to the right side of the equation. Now the equation is in vertex form:

`\qquad `

`formatGroup( COMP_SQR5, [ 0 ] )`

Now that the equation is written in vertex form, the vertex is the point `(h, k)`

:

`\qquad `

`formatGroup( GROUP1, [ 0 ] )`

The vertex is `(`

. Be sure to pay attention to the signs when interpreting an equation in vertex form.
`H`, `K`)