# How to Write Quadratic Equations Given a Vertex & Point

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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,

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

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

y = ax^2 + bx + c

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 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

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

results in

y = a(x - 2)^2 + 3.

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

y = a(x - 2)^2 + 3

results in

8 = a(3 - 2)^2 + 3 \text{ or } 8 = a(1)^2 + 3

which is 8 = ​a​ + 3.

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

8 - 3 = a - 3

which becomes ​a​ = 5.

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

y = a(x - 2)^2 + 3

results in

y = 5(x - 2)^2 + 3

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

x^2 - 4x + 4

which multiplied by 5 results in

5x^2 - 20x + 20