Modeling Data

Applications with three unknowns usually require solving a system of three equations. If we know three points on the graph, we can find the equation of a parabola in the form LaTeX: y=ax^2+bx+c by solving a system of three equations with three variables.

Example

Find the equation of the parabola LaTeX: y=ax^2+bx+c that passes through the points (2, 4), (−1, 1), and (−2, 5).

Solution

Since the three points lie on the graph of the given equation LaTeX: y=ax^2+bx+c they must satisfy the equation.

Substituting each ordered pair into the equation gives three equations with three unknowns.

$LaTeX: 4=a\left(2\right)^2+b\left(2\right)+c,\:or\:4=4a+2b+c$

$LaTeX: 1=a\left(-1\right)^2+b\left(-1\right)+c,\:or\:1=a-b+c$

$LaTeX: 5=a\left(-2\right)^2+b\left(-2\right)+c,\:or\:5=4a-2b+c$

To solve this system, first eliminate c using the first two equations.

LaTeX: 4=4a+2b+c

LaTeX: -1=-a+b-c

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LaTeX: 3=3a+3b

Now, use the second and third equations to eliminate the same unknown, c.

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LaTeX: -4=-3a+b

We can now solve the system of equations in two unknowns we have created. We will solve this system by eliminating a.

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$LaTeX: b=-\frac{1}{4}$

Find a by substituting $LaTeX: b=-\frac{1}{4}$ into equation .

$LaTeX: 3=3a+3\left(-\frac{1}{4}\right)$

Multiplying both sides by $LaTeX: \frac{1}{3}$ and solving for LaTeX: a gives us:

$LaTeX: 1=a-\frac{1}{4}$

$LaTeX: a=\frac{5}{4}$

Finally, find c by substituting $LaTeX: a=\frac{5}{4}$ and $LaTeX: b=-\frac{1}{4}$ into .

$LaTeX: 1=\frac{5}{4}-\left(-\frac{1}{4}\right)+c$

$LaTeX: 1=\frac{6}{4}+c$

$LaTeX: -\frac{1}{2}=c$

The required equation is $LaTeX: y=\frac{5}{4}x^2-\frac{1}{4}x-\frac{1}{2}$ , or LaTeX: y=1.25x^2-0.25x-0.5 .