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MATH NATION SECTION 7
H.M.H. RESOURCES
SPECIAL NOTE:
These resources were assembled to assist in student readiness for their upcoming Algebra 1 EOC. Although these resources have been compiled for your convenience from the recently adopted textbook materials from Houghton Mifflin Harcourt, digital versions of these materials can also be accessed via the textbook link found in the employee portal. Please be reminded that these materials are copyrighted and should not be posted on school or private websites without prior written permission from the publisher.
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69
Understanding Geometric Sequences
Reteach
It is important to understand the difference between arithmetic and
geometric sequences.
Arithmetic sequences are based on adding a common difference, d.
Geometric sequences are based on multiplying a common ratio, r.
• If the first term of an arithmetic sequence, a1, is 2 and the common
difference is 3, the arithmetic sequence is: 2, 5, 8, 11, …
• If the first term of an arithmetic sequence, a1, is 72 and the common
difference is 3, the arithmetic sequence is: 72, 69, 66, 63, …
• If the first term of a geometric sequence, a1, is 2 and the common
ratio is 3, the geometric sequence is: 2, 6, 18, 54, …
• If the first term of a geometric sequence, a1, is 72 and the common
ratio is 1
3, the geometric sequence is: 72, 24, 8,
8
3, …
Complete each table.
1. An arithmetic sequence has a1 4 and d 3:
an a1 a2 a3 a4 a5
Value
2. A geometric sequence has a1 4 and r 3:
an a1 a2 a3 a4 a5
Value
3. An arithmetic sequence has a1 96 and d 4:
an a1 a2 a3 a4 a5
Value
4. A geometric sequence has a1 96 and r 1
:4
an a1 a2 a3 a4 a5
Value
LESSON
15-1
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70
Understanding Geometric Sequences
Practice and Problem Solving: Modified
Find the common ratio, r, for each geometric sequence and use r to
find the next three terms. The first one is done for you.
1. 2, 10, 50, 250, … r ______ 2. 4, 24, 144, 864, … r ______
Next three terms: ______________________ Next three terms: _______________________
Complete.
3. The 4th term in a geometric sequence is 24 and the common ratio is 2.
The 5th term is _________ and the 3rd term is ________.
4. 6 and 24 are successive terms in a geometric sequence. The
term following 24 is __________________________ .
Find the common difference, d, of the arithmetic sequence and write
the next three terms. The first one is started for you.
5. 6, 9, 12, 15, … d ______ 6. 5, 2, 1, 4, … d ______
Next three terms: ______________________ Next three terms: _______________________
Complete the tables. The first one is started for you.
7. 8.
Arithmetic
Term Number
Arithmetic
Term
Common
Difference
Geometric
Term Number
Geometric
Term
Common
Ratio
1 6 ____ 1 4 _____
2 11 _____ 2 24 _____
3 16 _____ 3 144 _____
4 21 _____ 4 864 _____
9. A population of animals declines in a manner that closely resembles a
geometric sequence.
Year Number of Animals Given this table of values, how large is the population:
1 36 In year 4? ________ animals
2 27 In year 5? ________ animals
3 20.25
LESSON
15-1
1250, 6250, 31,250
5
3
5
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71
Constructing Geometric Sequences
Reteach
In a geometric sequence, each term is multiplied by the same number to get to
the next term. This number is called the common ratio.
3 12 48 192
4 4 4
Determine if each sequence is a geometric sequence. Explain.
1. 2, 4, 6, 8, … ______________________________________________________________________
2. 4, 8, 16, 32, … __________________________________________________________________
3. 32, 16, 8, 4, … ___________________________________________________________________
You can write a geometric sequence using either a recursive rule or an explicit rule.
Recursive rule: Given f(1), ( ) ( 1) for 2f n f n r n
Explicit rule: Given f(1), f(n) f(1) r n–1
Examples
Write a recursive rule and an explicit rule for the geometric sequence 1, 4, 16, 64, … .
Step 1. Find the common ratio. r 4
Step 2. Write a recursive rule. (1) 1, ( ) ( 1) 4 for 2f f n f n n
Step 3. Write an explicit rule. f(1) 1, 1( ) 1 4nf n
Each rule represents a geometric sequence. If the given rule is recursive, write it as an explicit rule. If
the rule is explicit, write it as a recursive rule. Assume that f(1) is the first term of the sequence. Write
the first 4 terms of the sequence.
1
(1) ,4
f ( ) ( 1) 2 for 2f n f n n 1( ) 3 (2)nf n
Step 1. 11( ) 2
4
nf n Step 1. (1) 3, ( ) ( 1) 2 for 2f f n f n n
Step 2. 1 1
, , 1 , 2, ...4 2
Step 2. 3, 6, 12, 24, …
Each rule represents a geometric sequence. If the given rule is recursive,
write it as an explicit rule. If the rule is explicit, write it as a recursive rule.
Assume that f(1) is the first term of the sequence.
4. (1) 2, ( ) ( 1) 3 for 2f f n f n n 5. 1( ) 5 (2)nf n
____________________________________ ____________________________________
LESSON
15-2
The common ratio is 4.
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72
Constructing Geometric Sequences
Practice and Problem Solving: Modified
Complete: The first one is done for you.
1. Below are the first five terms of a geometric series. Fill in the bottom
row by writing each term as the product of the first term and a power of
the common ratio.
N 1 2 3 4 5 The general rule is f(n) __________.
f (n) 3 12 48 192 768
f (n) 3(4)0 3(4)1 3(4)2 3(4)3 3(4)4
2. Below are the first five terms of a geometric series. Fill in the bottom row by writing each
term as the product of the first term and a power of the common ratio.
N 1 2 3 4 5 The general rule is f(n) __________.
f (n) 6 12 24 48 96
f (n)
Evaluate each geometric sequence written as an explicit rule for n 4. The
first one is done for you.
3. f(n) 10(3)n1 4. f(n) 2(5)n1
_______________________________________ ________________________________________
Evaluate each geometric sequence written as a recursive rule for n 4. Assume
that f(1) is the first term of the sequence. The first one is done for you.
5. f(1) 7; f(n) f(n 1) 3 for n 2 6. f(1) 4; f(n) f(n 1) 2 for n 2
_______________________________________ ________________________________________
Write an explicit rule for each geometric sequence based on the given terms
from the sequence. Assume that the common ratio r is positive. The first one
is done for you.
7. a1 9 and a2 18 8. a1 2 and a2 20
_______________________________________ ________________________________________
The population of a town is 20,000. It is expected to grow at 4% per year. Use
this information for 9–10. The first one is started for you.
9. Write a recursive rule and an explicit rule to predict the population
p(n) n years from today.
________________________________________________________________________________________
10. Use a rule to predict the population in 5 years and in 10 years.
LESSON
15-2
f(4) 10(3)3 270
f(4) 7(3)3 189
f(n) 9(2)n1
p(1) 20,000; p(n) p(n 1) 1.04 for n 2
3(4)n1
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73
Constructing Exponential Functions
Reteach
In this lesson, you need to know how to write an exponential equation given two points.
Write an equation for the following:
1. An exponential function that includes points (2, 50) and (3, 250)
a. b __________________
b. a __________________
c. f(x) _________________
2. An exponential function that includes points (1, 3) and 9
2,2
a. b _________________
b. a _________________
c. f(x) _________________
3. A safari park’s lion population is experiencing an exponential growth.
In year 3 the park has a population of 32 lions. In year 4 the park has
128 lions. Write an exponential function that includes these two points.
a. Write the two points. _________________
b. Find b _________________. Find a _________________
c. f(x) ________________
LESSON
15-3
f(x) abx
through points
(2, 6) and (4, 24)
1
2 24
6
yb
y
4
6 a(4)2
6 3
16 8a
3(4)
8
xy
Pick one point and plug into
f(x) abx with the b value
and solve for a.
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74
Constructing Exponential Functions
Practice and Problem Solving: Modified
Find the value of each exponential expression. The first one is done
for you.
1. 22(3) 2. 05 3. 12 4. 24
_______________ _______________ _______________ ________________
5. 3100(0.6) 6.
31
82
7. 312(4 ) 8.
01
187
_______________ _______________ _______________ ________________
Use two points to write an equation for the function shown. The first
one is done for you.
9. 10.
_______________________________________ ________________________________________
Solve. The first problem is started for you.
11. Make a table of values and a graph for the function 1
( ) 6 .2
x
f x
12. A blood sample has 50,000 bacteria present. A drug fights the
bacteria such that every hour the number of bacteria remaining, r(n),
decreases by half. If r(n) is an exponential function of the number, n, of
hours since the drug was taken, find the bacteria present four hours
after administering the drug.
_______________________________________________________________________________________ .
LESSON
15-3
18
x 0 1 2 3
f(x) 1 5 25 125
x 0 1 2 3
f(x) 81 27 9 3
x 2 1 0 1 2
f(x) 24
f(x) 5x
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75
Graphing Exponential Functions
Reteach
An exponential function has the form f(x) abx.
The independent variable is in an exponent.
The graph is always a curve in two quadrants.
Graph y 3 (2)x.
Create a table of ordered pairs.
Plot the points.
Because a 0 and b 1,
this graph should look similar
to the second graph above.
a 0 and b 1 a 0 and b 1 a 0 and 0 b 1 a 0 and 0 b 1
Graph each exponential function.
1. y 4 (0.5)x 2. y 2 (5)x 3. y 1 (2)x
LESSON
15-4
a 0
b 0 and 1
x is any real number
x y 3 (2)x y
1 y 3 (2)1 1.5
0 y 3 (2)0 3
1 y 3 (2)1 6
2 y 3 (2)2 12
x y 4 (0.5)x y
2
1
0
1
x y 2 (5)x y
1
0
1
2
x y 1 (2)x y
1
0
1
2
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76
Graphing Exponential Functions
Practice and Problem Solving: Modified
Solve. The first problem is started for you.
1. Make a table of values and a graph for the function ( ) 3 2x
f x .
2. Make a table of values and a graph for the function 1
( ) 62
x
f x
.
Graph each exponential function. Identify a, b, the y-intercept, and the
end behavior of the graph.
3. f(x) 4(2)x 4. 1
( ) 33
xf x
a ____ b ____ y-intercept ____ a ____ b ____ y-intercept ____
end behavior: x ____, x ____ end behavior: x ____, x ____
LESSON
15-4
x 2 1 0 1 2
f(x) 3
4
x 2 1 0 1 2
f(x)
x 2 1 0 1 2
f(x)
x 2 1 0 1 2
f(x) 24
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77
Transforming Exponential Functions
Reteach
The parent function for f(x) a(2)x is f(x) 2x. When a 1,
the graph looks like this.
When a is greater than 1, the curve is steeper and has a
higher y-intercept. When a is between 0 and 1, the curve is less steep and
has a lower y-intercept.
1. Compare the graph of f(x) 2x and the graph of f(x) 3(2x ).
Give the y-intercept for each graph.
________________________________________________________________________________________
2. Compare the graph of f(x) 2x and the graph of f(x) 0.25(2x ).
Give the y-intercept for each graph.
________________________________________________________________________________________
This graph compares f(x) a(4x), when a 1 and when a 1. When a is less than 0, the curve is reflected across the x-axis,
so the curve is in Quadrants III and IV and has a negative
y-intercept.
3. Compare the graph of f(x) 3(2x) and the graph of f(x) 3(2x).
Give the y-intercept for each graph.
________________________________________________________________________________________
This graph compares f(x) 3x, f(x) 3x 5, and
f(x) 3x 5.
For the function f(x) 3x c,
the curve has the same shape as for f(x) 3x
and is translated up or down the y-axis by c units.
4. Compare the graph of f(x) 2x and the graph of f(x) 2x 5.
Give the y-intercept for each graph.
________________________________________________________________________________________
5. Compare the graph of f(x) 2x and the graph of f(x) 2x 3.
Give the y-intercept for each graph.
LESSON
15-5
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78
Transforming Exponential Functions
Practice and Problem Solving: Modified
The graphs of the parent function 1 0.4x
Y and the function
2 3 0.4x
Y are shown to the right.
Use the graphs for 1–4. The first one is done for you.
1. What is the value of a for1Y and
2 ?Y
___________________________
2. Explain how you can tell that 2Y is a vertical stretch of
1Y .
_______________________________________________________________
3. Write an equation for a function that is a vertical compression of 1Y .
________________________________________________________________________________________
4. Write an equation for a function that translates 1Y 5 units up.
________________________________________________________________________________________
Values for f(x), a parent function, and g(x), a function in the same
family, are shown below. Use the table for Problems 5–8. The first
one is done for you.
x 2 1 0 1 2
f(x) 1
4
1
2 1 2 4
g(x) 1 2 4 8 16
5. Write an equation for the parent function.
________________________________________________________________________________________
6. How does the value for g(x) compare with the value for f(x)
in each column?
________________________________________________________________________________________
7. Write an equation for g(x).
________________________________________________________________________________________
8. Is g(x) a vertical stretch or a vertical compression of f(x)? Explain how
you can tell.
________________________________________________________________________________________
LESSON
15-5
Y1 : 1, Y2 : 3
f(x) 2x
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79
Comparing Linear, Quadratic, and Exponential Models
Reteach
Graph to decide whether data is best modeled by a linear, quadratic, or exponential function.
Graph (2, 0), (1, 3), (0, 4), (1, 3), (2, 0). What kind of model best describes the data?
You can also look at patterns in data to determine the correct model.
x y x y x y
2 5 1 8 0 2
4
4 2 2 5
1 8 4
6 1 3 0 2 32
4
8 4 4 7 3 128
Graph each data set. Which kind of model best describes the data?
1. (2, 4), (1, 2), (0, 0), (1, 2), (2, 4) 2. (1, 4), (0, 2), (1, 1),
2, 1
2
,
3, 1
4
_______________________________________ ________________________________________
3. 4. 5.
LESSON
23-2
x y
0 10
1 18
2 28
3 40
x y
3 4
6 2
9 8
12 14
x y
0 6
1 12
2 24
3 48
Connect the points.
Linear functions have
constant 1st differences.
Quadratic functions have
constant 2nd differences.
Exponential functions
have a constant ratio.
3
3
3
3
5
7
2
2
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80
Comparing Linear, Quadratic, and Exponential Models
Practice and Problem Solving: Modified
Determine if each function is linear, quadratic, or exponential.
The first one is done for you.
1. f(x) 5x2 ______________ 2. f(x) x 3 ______________ 3. f(x) 4x ______________
Complete the following to determine if each function is linear,
quadratic, or exponential.
4. f(x) 3x 1 7. f(x) x2 2 10. f(x) 2x
x f(x)
1st d
iffe
ren
ce
2n
d d
iffe
ren
ce
rati
o
1
0
1
2
3
4
x f(x)
1st d
iffe
ren
ce
2n
d d
iffe
ren
ce
rati
o
1
0
1
2
3
4
x f(x)
1st d
iffe
ren
ce
2n
d d
iffe
ren
ce
rati
o
1
0
1
2
3
4
5. End behavior as x 8. End behavior as x 11. End behavior as x
increases: increases: increases:
f(x) ____________________ f(x) ____________________ f(x) ____________________
6. f(x) is: _________________ 9. f(x) is: _________________ 12. f(x) is: _________________
Use the following information for 13.
Flavia had $125 in an account and began adding money each month.
The table shows the amount in Flavia’s account in dollars after each of the
first four months.
Month 0 1 2 3 4
Amount 125 140 155 170 185
13. Does the data follow a linear, quadratic or
exponential model? __________________
LESSON
23-2
quadratic
increases without
bound