How to Write Quadratic Equations Given a Vertex & Point

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

    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.

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