Intermediate Algebra (11th edition)

To find a quadratic function of the form

that models, or fits, these data, we choose three representative ordered pairs and use

them to write a system of three equations. Using

and

we substitute the x- and y-values from the ordered pairs into the quadratic form

to get three equations.

or (1)

or (2)

or (3)

We can find the values of a, b, and cby solving this system of three equations in three

variables using the methods of Section 4.2.From equation (1), Substitute

4973 for cin equations (2) and (3) to obtain two equations.

or (4)

or (5)

We can eliminate bfrom this system of equations in two variables by multiplying

equation (4) by and equation (5) by 2, and adding the results.

Divide by 120. Use a calculator.

We substitute for ain equation (4) or (5) to find that. Using the

values we have found for a, b, and c, our model is defined by

NOW TRY

NOTE In Example 6,if we had chosen three different ordered pairs of data, a

slightly different model would result. The quadratic regressionfeature on a graphing

calculator can also be used to generate the quadratic model that best fits given data.

See your owner’s manual for details.

y= -69.15x^2 + 863.6x+ 4973.

- 69.15 b=863.6

a=-69.15

120 a=- 8298

- 5

100 a+ 10 b+ 4973 =6694, 100 a+ 10 b= 1721

16 a+ 4 b+ 4973 =7321, 16 a+ 4 b= 2348

c= 4973.

a 11022 + b 1102 +c= 6694 100 a+ 10 b+ c= 6694

a 1422 +b 142 +c= 7321 16 a+ 4 b+ c= 7321

a 1022 +b 102 +c= 4973 c = 4973

y=ax^2 +bx +c

1 0, 4973 2 , 1 4, 7321 2 , 1 10, 6694 2 ,

y=ax^2 +bx +c

536 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

Years Since 1995

4000

5000

6000

7000

8000

0

268104

y

x

Births

U.S. HIGHER-ORDER

MULTIPLE BIRTHS

FIGURE 11

Years Since 1995

4000

5000

6000

7000

8000

0

268104

y

x

Births

U.S. HIGHER-ORDER

MULTIPLE BIRTHS

FIGURE 12

NOW TRY
EXERCISE 6
Using the points ,
, and , find
another quadratic model for
the data on higher-order
multiple births in Example 6.

1 4, 7321 2 1 8, 7663 2

1 0, 4973 2

NOW TRY ANSWER
6.y=-62.69x^2 +837.75x+ 4973