How to Write Quadratic Equations Given a Vertex & Point

How to Write Quadratic Equations Given a Vertex & Point
••• ChristianChan/iStock/GettyImages

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

    The equation now reads as

    y = 5x^2 - 20x + 20 + 3

    which becomes

    y = 5x^2 - 20x + 23

    after combining like terms.


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

Related Articles

How to Solve for Both X & Y
How to Find the Slope of Linear Equations
How to Convert an Equation Into Vertex Form
How to Use Elimination to Solve the Linear Equation
How to Solve a Parabola
How to Find Quadratic Equations From a Table
How to Find Equations of Tangent Lines
How to Create Linear Equations
How to Find the Y-Intercept of a Circle
How to Convert Diameter to Square Centimeters
How to Factor Binomial Cubes
How to Convert From a Standard to a Vertex Form
Characteristics of Aquatic Plants
How to Convert Pounds Per Square Foot to PSI
How to Find the Cross Sectional Area of a Pipe
How to Graph and Find the Solution on a Calculator
The Foil Method With Fractions
How to Find the Line of Symmetry in a Quadratic Equation
How to Factorise a Quadratic Expression
How to Find Slope From an Equation

Dont Go!

We Have More Great Sciencing Articles!