Finding Quadratic Equations from a Table

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.

Step 1

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

Step 2

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.

Step 3

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.

Step 4

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](3^2) + (-1.5c + 4.5)(3) + c simplifies to c = 1.

Step 5

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.

Step 6

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.

Step 7

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.

Step 8

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.