MATH NATION SECTION 7 H.M.H. RESOURCES

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.

THIS PAGE

INTENTIONALLY

LEFT BLANK

Name _______________________________________ Date __________________ Class __________________

Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.

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

Name _______________________________________ Date __________________ Class __________________


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

Name _______________________________________ Date __________________ Class __________________


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.

Name _______________________________________ Date __________________ Class __________________


72

Constructing Geometric Sequences


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

Name _______________________________________ Date __________________ Class __________________


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.

Name _______________________________________ Date __________________ Class __________________


74

Constructing Exponential Functions


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

Name _______________________________________ Date __________________ Class __________________


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

Name _______________________________________ Date __________________ Class __________________


76

Graphing Exponential Functions


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

Name _______________________________________ Date __________________ Class __________________


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 ).


________________________________________________________________________________________

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).


________________________________________________________________________________________

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.


________________________________________________________________________________________

5. Compare the graph of f(x) 2x and the graph of f(x) 2x 3.


LESSON

15-5

Name _______________________________________ Date __________________ Class __________________


78

Transforming Exponential Functions


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

Name _______________________________________ Date __________________ Class __________________


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

Name _______________________________________ Date __________________ Class __________________


80

Comparing Linear, Quadratic, and Exponential Models


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

Documents

MATH NATION SECTION 7 H.M.H. RESOURCES