Modeling With
Quadratic Functions
4-3
Vocabulary
Review
1. Cross out the graphs that are NOT parabolas.
y
y
y
x
y
x
x
x
Vocabulary Builder
model (verb)
MAH
dul
Definition: A function or equation models an action or relationship by describing its
behavior or the data associated with that relationship.
Example: The equation a 5 3g models the relationship between the number of
apples, a, and the number of oranges, g, when the number of apples is triple the
number of oranges.
Use Your Vocabulary
Draw a line from each description in Column A to the equation that models it in
Column B.
Column A
Column B
2. The string section of the orchestra has twice
as many violins as cellos.
y 5 2x 1 1
3. There are two eggs per person with one
extra for good measure.
y 5 100 2 2x
4. There were 100 shin guards in the closet, and
each player took two.
y 5 2x
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Main Idea: Modeling is a way of using math to describe a real-world situation.
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Problem 1 Writing an Equation of a Parabola
Got It? What is the equation of a parabola containing the points (0, 0), (1, 22),
and (21, 24)?
5. Substitute the three points one at a time into y 5 ax 2 1 bx 1 c to write a system
of equations.
2
R1c
2
R1c
2
R1c
Use (0, 0).
5 aQ
R 1 bQ
Use (1, 22).
5 aQ
R 1 bQ
Use (21, 24).
5 aQ
R 1 bQ
6. Solve the system of equations.
7. The equation of the parabola is y 5
x2 1
x1
.
Problem 2 Using a Quadratic Model
Got It? The parabolic path of a thrown ball can be modeled by the table. The top
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
of a wall is at (5, 6). Will the ball go over the wall? If not, will it hit the wall on the way
up, or the way down?
8. Circle the system of equations you find by substituting the three given points that
are on the parabola.
1 5 9a 1 3b 1 c
2 5 25a 1 5b 1 c
3 5 36a 1 6b 1 c
35a1b1c
5 5 2a 1 2b 1 c
6 5 9a 2 3b 1 c
x
y
1
3
2
5
3
6
35a1b1c
5 5 4a 1 2b 1 c
6 5 9a 1 3b 1 c
9. Now, solve the system of equations.
10. The solution of the system is a 5
,b5
,c 5
11. The quadratic model for the ball’s path is
.
.
12. How can you determine whether the ball goes over the wall? Place a ✓ if the
statement is correct. Place an ✗ if it is not.
The value of the model at x 5 5 is at least 6.
The value of the model at x 5 6 is at least 5.
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13. Will the ball go over the wall? Explain.
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_______________________________________________________________________
14. The value of the model at x 5 6 is less than / greater than value of the model at
x 5 5, therefore the ball was on its way down / up as it approached the wall.
Problem 3
Using Quadratic Regression
Got It? The table shows a meteorologist’s predicted temperatures
for a summer day in Denver, Colorado. What is a quadratic model
for the data? Predict the high temperature for the day. At what time
does the high temperature occur?
15. Using the LIST feature on a graphing calculator, identify the data that
you will enter.
L1 5
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
L2 5
6 a.m.:
3 a.m.:
9 a.m.:
6 p.m.:
12 p.m.:
9 p.m.:
17. Circle the calculator screen that shows the correct data entry.
L1
6
9
12
3
6
9
L2
63
76
86
89
85
76
L2(6)76
L3
L1
6
9
12
15
18
21
L2
63
76
86
89
85
76
L3
L2(6)63
L2(6)76
18. Enter the data from the table into your calculator. Use
the QuadReg function. Your screen should look like the
one at the right.
Write the quadratic model for temperature.
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L1
6
9
12
15
18
21
QuadReg
y = ax 2bxc
a = –.329365
b = 9.797619
c = 15.571429
L2
76
86
89
85
76
63
L3
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
16. Using a 24-hour clock, write the values for the L1 column.
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19. Use your calculator to find the maximum value of the model. The vertex of the
parabola is Q
R.
,
20. The high temperature will be
°F.
21. At what time will the high temperature occur?
_______________________________________________________________________
Lesson Check • Do you UNDERSTAND?
y
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.
4
22. Graph the points (3, 4), (5, 22), and (3, 0).
2
23. Underline the correct words to complete the rule for finding
a quadratic model.
Two / Three noncollinear points, no two / three of which
are in line horizontally / vertically , are on the graph of exactly
2
O
x
2
4
6
2
one quadratic function.
24. What is your classmate’s error?
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_______________________________________________________________________
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Math Success
Check off the vocabulary words that you understand.
model
quadratic model
Rate how well you can use a quadratic model.
Need to
review
0
2
4
6
8
10
Now I
get it!
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Lesson 4-3
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