# 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.

### TL;DR (Too Long; Didn't Read)

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

### Cite This Article

#### MLA

Gartneer, Chance E.. "How To Write Quadratic Equations Given A Vertex & Point" *sciencing.com*, https://www.sciencing.com/write-equations-given-vertex-point-8541975/. 16 November 2020.

#### APA

Gartneer, Chance E.. (2020, November 16). How To Write Quadratic Equations Given A Vertex & Point. *sciencing.com*. Retrieved from https://www.sciencing.com/write-equations-given-vertex-point-8541975/

#### Chicago

Gartneer, Chance E.. How To Write Quadratic Equations Given A Vertex & Point last modified March 24, 2022. https://www.sciencing.com/write-equations-given-vertex-point-8541975/