How to Find Quadratic Equations From a Table

••• AndreyCherkasov/iStock/GettyImages

Given a quadratic equation, most algebra students could easily form a table of ordered pairs that describe the points on the parabola. However, some may not realize you can also perform the reverse operation to derive the equation from the points. This operation is more complex, but is vital to scientists and mathematicians who need to formulate the equation that describes a chart of experimental values.

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

Assuming you're given three points along a parabola, you can find the quadratic equation that represents that parabola by creating a system of three equations. Create the equations by substituting the ordered pair for each point into the general form of the quadratic equation, ax^2 + bx + c. Simplify each equation, then use the method of your choice to solve the system of equations for a, b and c. Finally, substitute the values you found for a, b and c into the general equation to generate the equation for your parabola.

    Select three ordered pairs from the table. For example, (1, 5), (2,11) and (3,19).

    Substitute the first pair of values into the general form of the quadratic equation: f(x) = ax^2 + bx + c. Solve for a. For example, 5 = a(1^2) + b(1) + c simplifies to a = -b - c + 5.

    Substitute the second ordered pair and the value of a into the general equation. Solve for b. For example, 11 = (-b - c + 5)(2^2) + b(2) + c simplifies to b = -1.5c + 4.5.

    Substitute the third ordered pair and the values of a and b into the general equation. Solve for c. For instance, 19 = -(-1.5c + 4.5) - c + 5 + (-1.5c + 4.5)(3) + c simplifies to c = 1.

    Substitute any ordered pair and the value of c into the general equation. Solve for a. For instance, you can substitute (1, 5) into the equation to yield 5 = a(1^2) + b(1) + 1, which simplifies to a = -b + 4.

    Substitute another ordered pair and the values of a and c into the general equation. Solve for b. For example, 11 = (-b + 4)(2^2) + b(2) + 1 simplifies to b = 3.

    Substitute the last ordered pair and the values of b and c into the general equation. Solve for a. The last ordered pair is (3, 19), which yields the equation: 19 = a(3^2) + 3(3) + 1. This simplifies to a = 1.

    Substitute the values of a, b and c into the general quadratic equation. The equation that describes the graph with points (1, 5), (2, 11) and (3, 19) is x^2 + 3x + 1.

Related Articles

How to Calculate the Slope of a Line of Best Fit
How to Determine the Y-Intercept of a Trend Line
How to Find Y Value for the Slope of a Line
How to Convert Graphs to Equations
How to Figure Out the Slope of a Line
How to Find an Equation Given a Table of Numbers
How to Write a Linear Regression Equation
How to Solve for Both X & Y
How to Create Linear Equations
How to Write Quadratic Equations Given a Vertex & Point
How to Find The Slope of a Line Given Two Points
How to Find the Inequalities From a Graph
How to Find the Vertices of an Ellipse
How to Solve a Parabola
To Calculate Arcsine, What Buttons Do You Press on...
How to Find X-Intercept & Y-Intercept
How to Calculate the Mean in a Probability Distribution
How to Calculate a Horizontal Tangent Line
How to Find the Slope of a Nonlinear Line
How to Find Equations of Tangent Lines