Algebra IIA
Unit VI: Properties and Attributes of Functions
• Foundational Material
o Prior study of various functions, graphs and equations
o Perform algebraic operations on various expressions
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o Perform transformations of various functions
o Model real world applications with functions
Goal
 Model real-world data with functions
Perform operations on functions
Why?
To further build a foundation for higher level mathematics
To solve problems in other classes such as chemistry, physics, and biology
Make predictions by modeling data involving time, money, speed, sports, travel, etc.
• Key Vocabulary
o Composition of functions
Lesson 2: Modeling Real World Data
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•
•
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Apply functions to problem situations.
Use mathematical models to make predictions.
Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and
interpret solutions as viable or nonviable options in a modeling context. (CC.9-12.A.CED.3)
Create equations in two or more variable to represent relationships between quantities, graph equations on
coordinate axes with labels and scales. (CC.9-12.A.CED.2)
Warm-Up:
For each graph that follows, determine the following:
Domain? _____
Range? _____
Function? _____
Domain? _____
Range? _____
Function? _____
Domain? _____
Range? _____
Function? _____
Domain? _____
Range? _____
Function? _____
Domain? _____
Range? _____
Function? _____
Domain? _____
Range? _____
Function? _____
Graphs of Functions You Should Know
Linear Function
Quadratic Function
Parent Function: ________
Parent Function: ________
Parent Function: ________
Domain? ______________
Domain? ___________________
Domain? ___________________
Range? _______________
Range? ____________________
Range? ____________________
End –Behavior? _________
End –Behavior? ______________
End –Behavior? ______________
Asymptotes? ____________
Asymptotes? _________________
Asymptotes? ________________
Square Root Function
Absolute Value Function
Cubic Function
Rational Function
Parent Function: ________
Parent Function: ________
Parent Function: ________
Domain? ______________
Domain? ___________________
Domain? ___________________
Range? _______________
Range? ____________________
Range? ____________________
End –Behavior? _________
End –Behavior? ______________
End –Behavior? ______________
Asymptotes? ____________
Asymptotes? _________________
Asymptotes? ________________
Logarithmic Function
Exponential Function
Parent Function: ________
Parent Function: ________
Parent Function: ________
Domain? ______________
Domain? ___________________
Domain? ___________________
Range? _______________
Range? ____________________
Range? ____________________
End –Behavior? _________
End –Behavior? ______________
End –Behavior? ______________
Asymptotes? ____________
Asymptotes? _________________
Asymptotes? ________________
Functions and Regression
Linear Function
Quadratic Function
Square Root Function
values.
Square root:
Constant second
differences between
x-values for evenly
spaced y-values.
Linear:
Constant first
difference between
the y-values for
evenly spaced xvalues.
Quadratic:
Constant second
difference between
the y-values for
evenly spaced x-
Exponential Function
Exponential:
Constant rations
between y-values for
evenly spaced xvalues.
IF:
x-values are evenly spaced and first differences of y-values are constant, a linear model fits the data.
x
1
2
3
4
5
y
12
27
42
57
72
First differences:
15
15
15
Linear model: first
differences are constant.
15
x-values are evenly spaced and second differences of y-values are constant, a quadratic model is used.
x
4
5
6
7
8
y
9
15
23
33
45
First differences:
6
8
Second differences:
2
10
12
2
If first differences are not
constant, try second differences.
2
x-values are evenly spaced and ratios of y-values are constant, an exponential model is used.
x
10
11
12
13
y
40
100
250
625
First differences:
60
Second differences:
150
90
225
100
 2.5
40
Ratios:
If first and second differences
are not constant, try ratios of
y-values.
375
250
 2.5
100
625
 2.5
250
y-values are evenly spaced & second differences of x-values are constant, a square root model is used.
x
42
45
52
63
78
y
3
4
5
6
7
First differences:
Second differences:
3
7
4
11
4
15
4
For evenly spaced y-values,
try first differences of x-values.
Modeling Real-World Data
Determine which parent function would best model the given data set.
Choose among linear, quadratic, exponential, and square root.
1.
x
y
5
1
8
2
13
3
20
4
29
5
40
6
a. Look at the table at right. Are the
data for one variable evenly spaced?
____________________________________
b. Look at the data for the other variable.
Which differences, if any, are constant?
____________________________________
c. Which parent function best models the data?
____________________________________
2.
3.
4.
x
y
26
1
2
16
2
2
4
52
24
22
3
8
8
24
32
46
4
16
10
12
40
70
5
32
12
56
6
64
x
y
84
8
4
72
6
x
y
2
________________________
________________________
________________________
Write a function that models the given data.
5. Use a graphing calculator to make a
scatter plot. Then use the regression
feature to find the function that best
represents the data.
x
2
0
2
4
6
y
8
10
8
2
8
_______________________________________
Solve.
6. The table shows the number of sport utility vehicles sold in the United States
from 1997 to 2003. Write a function that models the data.
Years after 1996
1
2
3
4
5
6
7
SUVs (millions)
2.3
2.8
3.1
3.2
3.8
4.0
4.3
________________________________________________________________________________________
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Holt McDougal Algebra 2
Modeling Real-World Data
Use constant differences or ratios to determine which parent function
would best model the given data set.
1.
x
12
16
20
24
28
y
0.8
3.6
16.2
72.9
328.05
2.
________________________________________
x
13
19
25
31
37
43
y
–1
17
35
53
71
89
________________________________________
3.
4.
x
2
7
12
17
22
x
0.10
0.37
0.82
1.45
2.26
y
100
55
40
185
380
y
0.3
0.6
0.9
1.2
1.5
________________________________________
________________________________________
Write a function that models the data set.
5.
x
2.2
2.6
3.0
3.4
3.8
y
0.68
4.52
9.0
14.12
19.88
6.
________________________________________
5
0
5
10
15
20
y
8
6
4
2
0
2
________________________________________
7.
8.
x
0.3
0.7
1.1
1.5
1.9
x
0.06
0.375
0.96
1.815
2.94
y
2.5
3
3.6
4.32
5.184
y
0.2
0.5
0.8
1.1
1.4
________________________________________
9.
x
x
6
1
8
15
22
y
15
1
30.12
102.36
217.72
________________________________________
10.
________________________________________
x
0.32
2.07
4.8
8.51
13.2
y
0.9
1.6
2.3
3.0
3.7
________________________________________
Solve.
11. The table shows the population growth of a small town.
Years after 1974
Population
1
6
11
16
21
26
31
662
740
825
908
1003
1095
1200
a. Write a function that models the data.
____________________________
b. Use your model to predict the population in 2020.
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Holt McDougal Algebra 2
You try:
1)
A printing company prints advertising flyers and tracks its profits. Write a function that models the
given data.
2)
Flyers
Printed
100
200
300
400
500
600
Profit ($)
10
70
175
312
500
720
Write a function that models the given data.
x
12
14
16
18
20
22
24
y
110
141
176
215
258
305
356
3)
The data shows the population of a small town since 1990. Using 1990 as a reference year, write a
function that models the data.
4)
Year
1990
1993
1997
2000
2002
2005
2006
Population
400
490
642
787
901
1104
1181
Write a function that models the data.
b)
Fertilizer/Acre (lb)
11
14
25
31
40
50
Yield/Acre (bushels)
245
302
480
557
645
705
What will the yield be if the amount of fertilizer/acre is 20 lbs?
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Algebra 2
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Algebra 2