Just as a quadratic equation can map a parabola, the parabola's points can help write a corresponding quadratic equation. Parabolas have two equation forms – standard and vertex. In the vertex form,

the variables *h* and *k* are the coordinates of the parabola's vertex. In the standard form

a parabolic equation resembles a classic quadratic equation. With just two of the parabola's points, its vertex and one other, you can find a parabolic equation's vertex and standard forms and write the parabola algebraically.

## Substitute in Coordinates for the Vertex

## Substitute in Coordinates for the Point

## Solve for a

## Substitute a

## Convert to Standard Form

Set either form to zero and solve the equation to find the points where the parabola crosses the x-axis.

Substitute the vertex's coordinates for *h* and *k* in the vertex form. For an example, let the vertex be (2, 3). Substituting 2 for *h* and 3 for *k* into

results in

Substitute the point's coordinates for *x* and *y* in the equation. In this example, let the point be (3, 8). Substituting 3 for *x* and 8 for *y* in

results in

which is 8 = *a* + 3.

Solve the equation for *a*. In this example, solving for *a* results in

which becomes *a* = 5.

Substitute the value of *a* into the equation from Step 1. In this example, substituting *a* into

results in

Square the expression inside the parentheses, multiply the terms by *a*'s value and combine like terms to convert the equation to standard form. Concluding this example, squaring (*x* - 2) results in

which multiplied by 5 results in

The equation now reads as

which becomes

after combining like terms.