Modeling With Quadratic Functions Worksheet - Section 2.4 Page 4
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8Writing Equations to Model Data
When data have equally-spaced inputs, you can analyze patterns in the differences of
the outputs to determine what type of function can be used to model the data. Linear
data have constant fi rst differences. Quadratic data have constant second differences.
The fi rst and second differences of f(x) = x
2
are shown below.
Equally-spaced x-values
−3
−2
−1
x
0
1
2
3
f(x)
9
4
1
0
1
4
9
−5
−3
−1
fi rst differences:
1
3
5
second differences:
2
2
2
2
2
Writing a Quadratic Equation Using Three Points
NASA can create a weightless environment by fl ying a plane in parabolic paths. The
Time, t
Height, h
table shows heights h (in feet) of a plane t seconds after starting the fl ight path. After
10
26,900
about 20.8 seconds, passengers begin to experience a weightless environment. Write
and evaluate a function to approximate the height at which this occurs.
15
29,025
20
30,600
SOLUTION
25
31,625
Step 1 The input values are equally spaced. So, analyze the differences in the outputs
30
32,100
to determine what type of function you can use to model the data.
35
32,025
h(10)
h(15)
h(20)
h(25)
h(30)
h(35)
h(40)
26,900
29,025
30,600
31,625
32,100
32,025
31,400
40
31,400
−75
−625
2125
1575
1025
475
−550
−550
−550
−550
−550
Because the second differences are constant, you can model the data with a
quadratic function.
Step 2 Write a quadratic function of the form h(t) = at
+ bt + c that models the
2
data. Use any three points (t, h) from the table to write a system of equations.
Use (10, 26,900): 100a + 10b + c = 26,900
Equation 1
Use (20, 30,600): 400a + 20b + c = 30,600
Equation 2
Use (30, 32,100): 900a + 30b + c = 32,100
Equation 3
Use the elimination method to solve the system.
Subtract Equation 1 from Equation 2.
300a + 10b = 3700
New Equation 1
800a + 20b = 5200
New Equation 2
Subtract Equation 1 from Equation 3.
Subtract 2 times new Equation 1
200a = −2200
from new Equation 2.
a = −11
Solve for a.
b = 700
Substitute into new Equation 1 to fi nd b.
c = 21,000
Substitute into Equation 1 to fi nd c.
The data can be modeled by the function h(t) = −11t
+ 700t + 21,000.
2
Step 3 Evaluate the function when t = 20.8.
h(20.8) = −11(20.8)
+ 700(20.8) + 21,000 = 30,800.96
2
P
assengers begin to experience a weightless environment at about 30,800 feet.
78
Chapter 2
Quadratic Functions
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