Modeling Data With Quadratic Functions

Find the Quadratic Function That Models 3 Ordered Pairs

In a previous post about quadratic functions and the graphs, I discussed the standard form of a quadratic equation, f(x)=ax^2+bx+c. Because a parabola is symmetric, its equation can be found by solving a system of equations with three variables. I have only discussed solving equations with 2 variables previously.  If you know how to solve a system with 2 variables, then a system with 3 variables should be no problem. The process is rather simple. I will explain how to solve the system of three equations as well as a system of two equations in the examples.

Example 1 Find the Quadratic Function That Models 3 Ordered Pairs

Before discussing how to find the solution, there should be a discussion about what is to be found. Ultimately, the solution should be an equation in the standard form of a quadratic, f(x)=ax^2+bx+c. To find this equation or function involves multiple parts and each part consists of several steps. Many times we get so caught up in the details, we forget the overall picture.

The answer is: f(x)=x^2-6x+7. Yes, I just gave you the answer. Now, the question is:

Find the quadratic function that models the given ordered pairs: (2,-1), (3,-2), (1,2). Here is how you turn 3 ordered pairs into a quadratic function.

First you must create a system of three equations. You create this system of equations by substituting the ordered pairs into the standard form of a quadratic equation f(x)=ax^2+bx+c. Using the reflexive and substitution properties of equality, I use the equation ax^2+bx+c=y. This will make solve the system of equations easier.

Substituting the ordered pairs gives the following equations:

(2,-1) gives a(2^2)+b(2)+c=-1

(3,-2) gives a(3^2)+b(3)+c=-2

(1,2) gives a(1^2)+b(1)+c=2

Simplifying these equations, maintaining the order from above and labeling them as equations A, B and C results with:

4a+2b+c=-1 (A)

9a+3b+c=-2 (B)

a+b+c=2 (C)

Now that you have a system of three equations, it is good to review the overall goal, to find the coefficients of a quadratic function that models the ordered pairs given. Okay, on with solving the system of 3 equations. You need to use the 3 equations and the elimination method to create a new system of equations with only 2 variables. You have to choose two pairs of equations and eliminate a variable. For this example, I will subtract C from A to create a new equation D and I will subtract C from B to create equation E.

(A) 4a+2b+c=-1

\underline{-(C) a+b+c=2}

(D) 3a+b=-3

(B) 9a+3b+c=-2

\underline{-(C) a+b+c=2}

(E) 8a+2b=-4

Equations D and E form a new system of equations.

3a+b=-3 (D)

8a+2b=-4 (E)

To solve this system, it would be easiest to subtract with E from equation D multiplied by 2.

2(3a+b=-3) (D)

\underline{-(8a+2b=-4)} (E)

6a+2b=-6 (D)

\underline{-(8a+2b=-4)} (E)

-2a=-2  Divide both sides by -2 and


You now have the coefficient of the quadratic term. Using a = 1, you substitute into either equation D or E. For this problem, I used equation D.



3(1)+b=-3  Substitute 1 for a.

3+b=-3  Simplify.

b=-6  Subtract 3 from each side.

This is the coefficient of the linear term. Now use both a = 1 and b = -6 to find the value of the constant. You must use one of the original equations A, B or C. I used equation C because it is very simple.

a=1 and b=-6

a+b+c=2 (C)

1+(-6)+c=2  Substitute 1 for a and -6 for b.

-5+c=2  Simplify.

c=7  Add 2 to both sides.

Finally, we have all of the coefficients a = 1, b = -6 and c = 7 and can write the quadratic function that models the given points.

a=1, b=-6, c=7


f(x)=x^2-6x+7 This is the solution.

But how do you know it is correct? You should check your work by substituting the values a = 1, b = -6 and c = 7 into one of the equations A, B or C. Since I already used C I will use equation A.

a=1, b=-6, c=7

4a+2b+c=-1 (A)

4(1)+2(-6)+7=-1  Substitute.

4+(-12)+7=-1  Simplify.

-8+7=-1  Simplify.

-1=-1  Simplify and it checks.

This is just one example of finding the equation of a quadratic function from three points. Also, because the three original equations started with all three variables, it was a more complicated type.

Example Find the Quadratic Function That Models 3 Ordered Pairs For You To Do

Find the quadratic function that models the given ordered pairs:

(-2,-17), (-1,10), (5,-10).

Click here for the solution.

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