How To Write Quadratic Equations Given A Vertex & Point

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.

1. Substitute in Coordinates for the Vertex

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

\(y = a(x – 2)^2 + 3.\)

2. Substitute in Coordinates for the Point

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.

3. Solve for a

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

\(8 – 3 = a – 3\)

which becomes ​a​ = 5.

4. Substitute a

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

\(y = 5(x – 2)^2 + 3\)

5. Convert to Standard Form

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

\(5x^2 – 20x + 20\)

The equation now reads as

\(y = 5x^2 – 20x + 20 + 3\)

which becomes

\(y = 5x^2 – 20x + 23\)

after combining like terms.