4-3
Content Standards
Modeling With
Quadratic Functions
F.IF.5 Relate the domain of a function to its graph
and, where applicable, to the quantitative relationship
it describes.
Also F.IF.4
Objective To model data with quadratic functions
Try making a sketch
of the path of the
ball based on what
you know about
projectile motion.
You and a friend are tossing a
ball back and forth. You each toss
oss
and catch the ball at waist level,,
3 feet high. What equation, in
standard form, models the path of
the ball? Explain your reasoning.
6 ft
10 ft
MATHEMATICAL
PRACTICES When you know the vertex and a point on a parabola, you can use vertex form to write
an equation of the parabola. If you do not know the vertex, you can use standard form
and any three points of the parabola to find an equation.
Essential Understanding Three noncollinear points, no two of which are in line
vertically, are on the graph of exactly one quadratic function.
Problem 1 Writing an Equation of a Parabola
How do you use the
3 given points?
Use them to write a
system of 3 equations.
Solve the system to get
a, b, and c.
A parabola contains the points (0, 0), (21, 22), and (1, 6). What is the
equation of this parabola in standard form?
Substitute the (x, y) values into y 5 ax2 1 bx 1 c to write a system of equations.
Use (21, 22).
Use (0, 0).
y5
ax2
1 bx 1 c
y5
ax2
Use (1, 6).
1 bx 1 c
y 5 ax2 1 bx 1 c
0 5 a(0)2 1 b(0) 1 c
22 5 a(21)2 1 b(21) 1 c
6 5 a(1)2 1 b(1) 1 c
05c
22 5 a 2 b 1 c
65a1b1c
Since c 5 0, the resulting system has two variables. e
a 2 b 5 22
a1b56
Use elimination.
a 2 and b 4.
Substitute a 5 2, b 5 4, and c 5 0 into standard form: y 5 2x2 1 4x 1 0.
y 5 2x2 1 4x is the equation of the parabola that contains the given points.
Got It? 1. What is the equation of a parabola containing the points
(0, 0), (1, 22), and (21, 24)?
Lesson 4-3 Modeling With Quadratic Functions
0209_hsm11a2se_cc_0403.indd 209
209
3/28/11 8:43:58 PM
STEM
Physics Campers at an aerospace camp launch rockets
on the last day of camp. The path of Rocket 1 is modeled
by the equation h 5 216t2 1 150t 1 1 where t is time in
seconds and h is the distance from the ground. The path of
Rocket 2 is modeled by the graph at the right. Which rocket
flew higher?
What property of the
quadratic tells you
how high the rocket
flew?
The parabolas model the
rockets’ paths, so the
maximums of each
parabola describe how
high the rockets flew.
Find the maximum height of each rocket by using the models
of their paths.
Height (ft)
Problem 2 Comparing Quadratic Models
800
y
700
600
500
400
300
200
100
0
x
2 4 6 8 10 12 14 16
Time (s)
Rocket 1
The maximum height of Rocket 1 is at the vertex of the parabola.
b
b
22a
, f Q22a
R
Use the vertex formula.
150
150
, f Q2
R
2(216)
2(216)
2
a 5 216, b 5 150
(4.7, 352.6)
Simplify.
The maximum height of Rocket 1 is 352.6 feet.
Rocket 2
The maximum height of Rocket 2 is at the vertex of the parabola.
Height (ft)
You can use the graph to find the approximate maximum height of the rocket.
800
y
700
(6, 580)
600
500
400
300
200
100
x
0
2 4 6 8 10 12 14 16
Time (s)
The maximum height of Rocket 2 is at about 580 feet.
Rocket 2 flew higher than Rocket 1.
Got It? 2. a. Which rocket stayed in the air longer?
b. What is the reasonable domain and range for each quadratic model?
c. Reasoning Describe what the domains tell you about each of the models
and why the domains for the models are different.
210
Chapter 4
0209_hsm11a2se_cc_0403.indd 210
Quadratic Functions and Equations
5/16/11 7:46:29 AM
When more than three data points suggest a quadratic function, you can use the
quadratic regression feature of a graphing calculator to find a quadratic model.
Problem 3 Using Quadratic Regression
Sacramento, CA
The table shows a meteorologist’s predicted
temperatures for an October day in Sacramento,
California.
How do you write
times using a 24-hour
clock?
Add 12 to the number
of hours past noon. So,
2 P.M. is 14:00 in the
24-hour clock.
A What is a quadratic model for this data?
Step 1 Enter the data.
Use the 24-hour
clock to represent
times after noon.
Step 2 Use QuadReg.
L1
8
10
12
14
16
18
-----L3 L2
L3
52
64
72
78
81
76
------
------
3
Time
Predicted
Temperature (F)
8 A.M.
52
10 A.M.
64
12 P.M.
72
2 P.M.
78
4 P.M.
81
6 P.M.
76
Step 3 Graph the data and the function.
QuadReg
y = ax 2 + bx + c
a = –.46875
b = 14.71607143
c = –36.12142857
R 2 = .9919573999
A quadratic model
is reasonable.
A quadratic model for temperature is y 5 20.469x2 1 14.716x 2 36.121.
B Use your model to predict the high temperature for the day. At what time does
the high temperature occur?
Use the Maximum feature or tables.
Maximum
X=15.697148 Y=79.378709
X
Y1
79.337
15.4
79.36
15.5
79.374
15.6
79.379
15.7
79.374
15.8
79.359
15.9
79.336
16
Y179.3787053571
Predict the high temperature for the day to be 79.48F at about
3:42 p.m.
Got It? 3. The table shows a meteorologist’s predicted
temperatures for a summer day in Denver, Colorado.
What is a quadratic model for this data? Predict the
high temperature for the day. At what time does
the high temperature occur?
16 represents 4 P.M.
The maximum occurs
at approximately 15.7,
or about 3:42 P.M.
Denver, CO
Time
Predicted
Temperature (F)
6 A.M.
63
9 A.M.
76
12 P.M.
86
3 P.M.
89
6 P.M.
85
9 P.M.
76
Lesson 4-3 Modeling With Quadratic Functions
0209_hsm11a2se_cc_0403.indd 211
211
3/28/11 8:44:07 PM
Lesson Check
Do you know HOW?
1. (1, 0), (2, 23), (3, 210)
3.
x
y
x
y
2
1
0
1
2
3.5
3.5
7.5
15.5
27.5
2
1
0
1
41.5
25.5
13.5
5.5
5. Reasoning Explain how you can determine which of
the quadratic functions in Exercise 1 and Exercise 2
attains the greatest values.
6. Error Analysis Your classmate says he can write the
equation of a quadratic function that passes through
the points (3, 4), (5, 22), and (3, 0). Explain his error.
Practice and Problem-Solving Exercises
A
Practice
MATHEMATICAL
PRACTICES
See Problem 1.
Find an equation in standard form of the parabola passing through the points.
7. (1, 22), (2, 22), (3, 24)
8. (1, 22), (2, 24), (3, 24)
9. (21, 6), (1, 4), (2, 9)
10. (1, 1), (21, 23), (23, 1)
11. (3, 26), (1, 22), (6, 3)
12. (22, 9), (24, 5), (1, 0)
13.
14.
15.
x
f(x)
1
x
f(x)
17
1
4
1
17
1
2
2
8
2
4
x
f(x)
1
1
1
3
2
8
16. A player throws a basketball toward a hoop. The basketball follows a
parabolic path that can be modeled by the equation
y 5 20.125x2 1 1.84x 1 6. The table models the parabolic path of
another basketball thrown from somewhere else on the court.
If the center of the hoop is located at (12, 10), will each ball pass
through the hoop?
STEM
212
PRACTICES
4. Compare and Contrast How do you know whether
to perform a linear regression or a quadratic
regression for a given set of data?
Find a quadratic function that includes each set of
values.
2.
MATHEMATICAL
Do you UNDERSTAND?
Chapter 4
0209_hsm11a2se_cc_0403.indd 212
17. Physics A man throws a ball off the top of a building and records
the height of the ball at different times, as shown in the table.
a. Find a quadratic model for the data.
b. Use the model to estimate the height of the ball at 2.5 seconds.
c. What is the ball’s maximum height?
See Problems 2 and 3.
x
y
2
10
4
12
10
12
Height of a Ball
Time (s)
Height (ft)
0
46
1
63
2
48
3
1
Quadratic Functions and Equations
5/17/11 6:49:46 AM
B
Apply
Determine whether a quadratic model exists for each set of values. If so, write the model.
18. f(22) 5 16, f(0) 5 0, f(1) 5 4
19. f(0) 5 5, f(2) 5 3, f(21) 5 0
20. f(21) 5 24, f(1) 5 22, f(2) 5 21
21. f(22) 5 7, f(0) 5 1, f(2) 5 0
22. a. Geometry Copy and complete the table. It shows the total number of segments
whose endpoints are chosen from x points, no three of which are collinear.
Number of points, x
2
3
O
O
Number of segments, y
1
3
O
O
Water Levels
b. Write a quadratic model for the data.
c. Predict the number of segments that can be drawn using 10 points.
Elapsed
Time (s)
Water
Level (mm)
0
120
20
83
40
50
23. Think About a Plan The table shows the height of a column of water as it
drains from its container. Use a quadratic model of this data to estimate
the water level at 30 seconds.
• What system of equations can you use to solve this problem?
• How can you determine if your answer is reasonable?
24. A parabola contains the points (21, 8), (0, 4), and (1, 2). Name another point also
on the parabola.
25. a. Postal Rates Find a quadratic model for the data. Use 1981 as year 0.
Price of First-Class Stamp
Year
Price (cents)
1981
1991
1995
1999
2001
2006
2007
2008
18
29
32
33
34
39
41
42
SOURCE: United States Postal Service
b. Describe a reasonable domain and range for your model. (Hint: This is a
discrete, real situation.)
c. Estimation Estimate when first-class postage was 37 cents.
d. Use your model to predict when first-class postage will be 50 cents. Explain why
your prediction may not be valid.
Speed (mi/h)
20
30
40
50
55
Stopping Distance on Dry Roadway (ft)
17
38
67
105
127
a. Compare the models. What are the reasonable domain and
range for each road condition?
b. Writing Explain what that means about stopping distances
under certain road conditions.
Stopping Distance
on Wet Roadway (ft)
26. Road Safety The table and graph below give the stopping distance for an
automobile under certain road conditions.
200
y
175
150
125
100
75
50
25
0
10
30
50
x
70
Speed (mi/h)
Lesson 4-3 Modeling With Quadratic Functions
0209_hsm11a2se_cc_0403.indd 213
213
5/10/11 2:25:42 AM
CC Challenge
27. a. A parabola contains the points (0, 24), (2, 4), and (4, 4). Find the vertex.
b. Reasoning What is the minimum number of data points you need to find a
single quadratic model for a data set? Explain.
28. A model for the height of an arrow shot into the air is h(t) 5 216t2 1 72t 1 5,
where t is time and h is height. Without graphing, answer the following questions.
a. What can you learn by finding the graph’s intercept with the h-axis?
b. What can you learn by finding the graph’s intercept(s) with the t-axis?
Standardized Test Prep
29. The graph of a quadratic function has vertex (23, 22). What is the axis of
symmetry?
SAT/ACT
x 5 23
x53
y 5 22
y52
30. Which function is NOT a quadratic function?
y 5 (x 2 1)(x 2 2)
y 5 3x 2 x2
y 5 x2 1 2x 2 3
y 5 2x2 1 x(x 2 3)
31. Which is the composition f (g(x)), if f (x) 5 2x 2 3 and g(x) 5 7 1 5x?
f (g(x)) 5 4x 1 4
f(g(x)) 5 25x 2 8
f (g(x)) 5 4x 2 10
f (g(x)) 5 25x 2 10
32. Mark has 42 coins consisting of dimes and quarters. The total value of his coins is
$6. How many of each type of coin does he have? Show all your work and explain
what method you used to solve the problem.
Extended
Response
Mixed Review
See Lesson 4-2.
Graph each function.
33. y 5 x2 2 6x 2 3
34. y 5 2x2 1 9x 2 4
35. y 5 3x2 2 4x 1 1
See Lesson 3-2.
Solve each system by elimination.
36. e
x 1 y 5 7
5x 2 y 5 5
37. e
2x 2 3y 5 214
3x 2 y 5
7
x 2 3y 5 2
x 2 2y 5 1
See Lesson 2-2.
For Exercises 39–40, y varies directly with x.
39. If y 5 2 when x 5 5, find y when x 5 2.
38. e
40. If y 5 22 when x 5 4, find y when x 5 7.
Get Ready! To prepare for Lesson 4-4, do Exercises 41–43.
See Lesson 1-3.
Simplify by combining like terms.
41. x2 1 x 1 4x 2 1
214
Chapter 4
0209_hsm11a2se_cc_0403.indd 214
42. 6x2 2 4(3)x 1 2x 2 3
43. 4x2 2 2(5 2 x) 2 3x
Quadratic Functions and Equations
3/28/11 8:44:18 PM