5-3
Modeling With Quadratic Functions
TEKS FOCUS
VOCABULARY
TEKS (4)(A) Write the quadratic function given three specified points in
the plane.
TEKS (1)(C) Select tools, including real objects, manipulatives, paper and
pencil, and technology as appropriate, and techniques, including mental
math, estimation, and number sense as appropriate, to solve problems.
ĚScatter plot – a graph that relates two
different sets of data by plotting the
data as ordered pairs
ĚNumber sense – the understanding of
what numbers mean and how they are
related
Additional TEKS (1)(A), (1)(F), (4)(B)
ESSENTIAL UNDERSTANDING
Three noncollinear points, no two of which are in line vertically, are on the graph of
exactly one quadratic function.
Problem 1
P
TEKS Process Standard (1)(E)
Writing an Equation of a Parabola
A parabola contains the points (0, 0), (−1, −2), and (1, 6). What is the
equation
of this parabola in standard form?
e
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.
Substitute
the (x, y) values into y = ax2 + bx + c to write a system of equations.
S
Use (0, 0).
y=
ax2
+ bx + c
Use ( -1, -2).
y=
ax2
Use (1, 6).
+ bx + c
y = ax2 + bx + c
0 = a(0)2 + b(0) + c
-2 = a( -1)2 + b( -1) + c
6 = a(1)2 + b(1) + c
0=c
-2 = a - b + c
6=a+b+c
Since c = 0, the resulting system has two variables. e
a - b = -2
a+b=6
Use elimination.
a ⫽ 2 and b ⫽ 4.
Substitute a = 2, b = 4, and c = 0 into standard form: y = 2x2 + 4x + 0.
y = 2x2 + 4x is the equation of the parabola that contains the given points.
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Problem 2
P
y
(4, 12)
Using a Quadratic Model
Basketball A player throws a basketball
toward the hoop. The basketball follows a
parabolic path through the points shown.
If the center of the hoop is at (12, 10), will
the ball pass through the hoop? (You can
think of the units as feet.)
(10, 12)
(2, 10)
Step 1 Find a quadratic model.
Substitute the x and y values into
the standard form of a quadratic
function. The result is a system
of three linear equations.
x
y = ax2 + bx + c
10 = a(2)2 + b(2) + c
12 =
a(10)2
+ b(10) + c
12 = a(4)2 + b(4) + c
Use (2, 10).
Use (10, 12).
Use (4, 12).
4a + 2b + c = 10
Use one of the methods from Topic 3. Solve. W 100a + 10b + c = 12
16a + 4b + c = 12
The solution is a = -0.125, b = 1.75, and c = 7.
Substitute the values into the standard form of a quadratic function. An equation
of
o the parabola is y = -0.125x2 + 1.75x + 7.
In terms of the
quadratic model,
what does it mean
for the ball to pass
through the hoop?
The point (12, 10) is on
the parabola. That is, the
point (12, 10) satisfies the
quadratic equation.
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Lesson 5-3
Step 2
S
Use the quadratic model to see if the player makes the basket.
y = -0.125x2 + 1.75x + 7
10 ≟ -0.125(12)2 + 1.75(12) + 7
Substitute (x, y) = (12, 10).
10 = 10 ✔
The point (12, 10) is on the parabola. The ball will pass through the hoop.
Modeling With Quadratic Functions
Problem 3
P
TEKS Process Standard (1)(C)
Sacramento, CA
Using Quadratic Regression
The table shows a meteorologist’s predicted
temperatures for an October day in Sacramento,
California.
A What is a quadratic model for these data?
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.
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.
Step 1
Step 2
Enter the data.
Use the 24-hour
clock to represent
times after noon.
Use QuadReg.
L1
8
10
12
14
16
18
-----L3 =
L2
52
64
72
78
81
76
------
Step 3
L3
------
Time
Predicted
Temperature (ⴗF)
8 A.M.
52
10 A.M.
64
12 P.M.
72
2 P.M.
78
4 P.M.
81
6 P.M.
76
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 = -0.469x2 + 14.716x - 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
Y1=79.3787053571
16 represents 4 P.M.
The maximum occurs
at approximately 15.7,
or about 3:42 P.M.
Predict the high temperature for the day to be 79.4°F at about 3:42 p.m.
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HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Find an equation in standard form of the parabola passing through the points.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. (1, -2), (2, -2), (3, -4)
2. (1, -2), (2, -4), (3, -4)
3. ( -1, 6), (1, 4), (2, 9)
4.
5.
6.
x
f(x)
−1
−1
1
3
2
8
x
f(x)
17
−1
−4
1
17
1
−2
2
8
2
−4
x
f(x)
−1
7. Apply Mathematics (1)(A) 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.
Height of a Ball
Time (s)
Height (ft)
a. Find a quadratic model for the data.
0
46
b. Use the model to estimate the height of the ball at 2.5 seconds.
1
63
c. What is the ball’s maximum height?
2
48
3
1
Determine whether a quadratic model exists for each set of values.
If so, write the model.
8. f ( -2) = 16, f (0) = 0, f (1) = 4
9. f (0) = 5, f (2) = 3, f ( -1) = 0
10. f ( -1) = -4, f (1) = -2, f (2) = -1
11. f ( -2) = 7, f (0) = 1, f (2) = 0
12. Apply Mathematics (1)(A) 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.
13. A parabola contains the points ( -1, 8), (0, 4), and (1, 2). Name
another point also on the parabola.
14. a. Analyze Mathematical Relationships (1)(F) 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
■
■
Number of segments, y
1
3
■
■
Water Levels
Elapsed
Time (s)
Water
Level (mm)
0
120
20
83
40
50
b. Write a quadratic model for the data.
c. Predict the number of segments that can be drawn using 10 points.
15. A model for the height of an arrow shot into the air is h(t) = -16t 2 + 72t + 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?
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Lesson 5-3
Modeling With Quadratic Functions
16. a. Apply Mathematics (1)(A) Find a quadratic model for the data. Use 1981 as year 0.
Price of First-Class Stamp
Year
1981
1991
1995
1999
2001
2006
2007
2008
18
29
32
33
34
39
41
42
Price (cents)
SOURCE: United States Postal Service
b. Describe a reasonable domain and range for your model. (Hint: This is a
discrete, real situation.)
c. Use a Problem-Solving Model (1)(B) 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.
17. Apply Mathematics (1)(A) The table and graph below give the stopping distances
of an automobile for dry and wet road conditions.
Speed (mi/h)
0
20
30
40
50
Stopping Distance on Dry Roadway (ft)
0
40
75
120
175
b. Use Representations to Communicate Mathematical
Ideas (1)(E) Use your models to compare the stopping
distance of an automobile traveling at 65 mph on dry and
wet road conditions.
400
Stopping Distance
on Wet Roadway (ft)
a. Find a quadratic model for the stopping distance of
an automobile for each type of road condition.
y
300
200
100
0
10
30
50
x
70
Speed (mi/h)
TEXAS Test Practice
T
18. The graph of a quadratic function has vertex (-3, -2). What is the axis of
symmetry?
A. x = -3
B. x = 3
C. y = -2
D. y = 2
19. Which function is NOT a quadratic function?
F. y = (x - 1)(x - 2)
G. y = x2 + 2x - 3
H. y = 3x - x2
J. y = -x2 + x(x - 3)
20. 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.
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