problem solving with exponential …pkeenan.weebly.com/uploads/4/5/6/7/45670275/unit_7_topic...326!...

Topic 21: Problem solving with exponential functions 323

Copyright © 2014 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc.

PROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS Lesson 21.1 Finding function rules from graphs

21.1 OPENER 1. Plot the points from the table onto the graph below.

x y -‐2 -‐4 -‐1 -‐2 0 0 1 2 2 4 3 6 4 8

2. Draw slope triangles between the points you plotted.

324 Unit 7 – Exponential relationships


21.1 CORE ACTIVITY 1. Look back at your slope triangles from the Opener.

a. What do you notice about the heights of the slope triangles as x increases?

b. Write an algebraic rule for the relationship between x and y in question 1 of the Opener.

2. Given the following table and graph

a. Plot the points from the table onto the curve and draw slope triangles between the points you plotted.

b. How does the height of each slope triangle compare with the one before it (moving from left to right)? By what factor does it change for each new triangle?

c. What does this factor, or constant multiplier, tell us about the algebraic rule for this relationship?

d. What is the y-‐intercept (the value of y when x = 0)?

e. What does the y-‐intercept tell us about the algebraic rule for this relationship?

f. Write an algebraic rule for the relationship between x and y.

x y -‐2 0.25 -‐1 0.5 0 1 1 2 2 4 3 8 4 16



3. Given the following table and graph

a. Plot the points from the table onto the curve and draw slope triangles between the points you plotted.

b. How does the height of each slope triangle compare with the one before it (moving from left to right)? By what factor does it change for each new triangle?

c. What does this factor, or constant multiplier, tell us about the algebraic rule for this relationship?

d. What is the y-‐intercept (the value of y when x = 0)?

e. What does the y-‐intercept tell us about the algebraic rule for this relationship?

f. Write an algebraic rule for the relationship between x and y.

x y -‐1 120 0 60 1 30 2 15 3 7.5



21.1 CONSOLIDATION ACTIVITY

In this activity, you will work with your partner to match different representations of functions to descriptions of how the functions grow.

Objective: Create sets of “matching” cards. “Matching” is defined as representing the same relationship. Each set will have a table card, an equation card, a graph card, and a “growth” card that describes the growth of the relationship. On the growth card, you may be asked to write some additional information about the relationship to complete the set.

Materials: Your teacher will give you and your partner pages with cards on them to cut out. There are six graph cards (labeled A-‐F), six table cards (labeled G-‐L), and six growth cards (labeled M-‐R).

Instructions: Work with your partner to find a set of matching cards. When you both agree on a set of matching cards, tape the cards that form that set together. Justify the growth card you selected by filling in the information that is asked for on the card. So that you can more easily check your answers, tape each set with the graph card on the left, the table card in the middle, and the growth card on the right, as shown here.

Graph card

Table card

Growth card



HOMEWORK 21.1

Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. Plot the points from the table onto the curve and draw slope triangles between the points you plotted.

2. How does the height of each slope triangle compare with the one before it? By what factor does it increase for each unit

increase of x?

3. What is the value of y when x = 0?

4. Write an algebraic rule for the relationship between x and y.

x y

-‐1 3.33

0 10

1 30

2 90

3 270



STAYING SHARP 21.1 Practic

ing algebra skills & con

cepts

1. What is the product of x4 and x5?

2. The product of two exponential expressions is x5y2z. One of the expressions is x2y. What is other expression?

Prep

aring for u

pcom

ing lesson

s

3. Describe the pattern of the y-‐values in this table.

x y

0 1

1 !!

�

12

2 !!

�

14

3 18

4. Graph the values in Question 3. Is the graph linear? Explain.

Focus skill: R

easoning

with

qua

ntities

5. Write each number in scientific notation.

a. 1,320,000

b. 0.0006

c. 75,000,000,000,000,000,000

d. 0.00000000204310

6. Write each number in standard notation.

e. 8.00 x 10–3

f. 3.206 E 12

g. 9.9 x 105

h. 3.206 E –12

! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !

!



Lesson 21.2 Geometric sequences and exponential functions

21.2 OPENER Martina and Karina each wrote sequence puzzles on a strip of paper for each other to figure out.

Martina’s Sequence

2, 6, 10, 14, ___, ___, ...

Karina’s Sequence

2, 6, 18, 54, ___, ___, ...

1. Find the next two terms for Martina’s sequence. Describe the pattern and explain how you found it.

2. Find the next two terms for Karina’s sequence. Describe the pattern and explain how you found it.

21.2 CORE ACTIVITY

Compare the process you used to find the next two terms for Martina’s pattern and Karina’s pattern in the Opener. 1. How did each term compare to the one before it for Martina’s sequence? 2. How did each term compare to the one before it for Karina’s sequence? 3. For which sequence were the terms related by constant differences? What was the constant difference? 4. For which sequence were the terms related by constant ratios? What was the constant ratio?



5. You already saw that each term in Martina’s sequence is 4 units more than the previous term. In other words, the common difference is 4. Fill in the table to express this relationship using function notation.

Term number, n Process Term, f(n)

1 2

2 f(2) = 2 + 4 = f(1) + 4 6

3 f(3) = 6 + 4 = f(2) + 4 10

4 f(4) = 10 + 4 = f(3) + 4 14

5 f(5) = 14 + 4 = f(4) + 4 18

6 f(6) = 18 + 4 = _________ + 4 22

7 f(7) = _________ + 4 = _________ + 4 _________

8 f(8) = _________ + 4 = _________ + 4 _________

9 f(9) = _________ + 4 = _________ + 4 _________

6. Write a rule to represent the sequence.

7. Complete the table to write both types of rules for the sequence in symbolic and verbal form.

Recursive rule

Explicit rule

Verbal rule

Function rule



Today is Amanda’s fifteenth birthday. On the day Amanda was born, her grandmother invested $100 for her in a special account. Today she wants to know how much it is worth. Suppose the account earns 6% interest on Amanda’s birthday each year. Since the interest is always added to the amount in the account, each year’s interest is based on a larger amount than the year before.

8. Fill in the table to figure out how much money is in the account on Amanda’s fifteenth birthday.

9. Once you found the amount in the account after Amanda’s eighth birthday (n = 8), how did you figure out the amount on her ninth birthday (n = 9)?

10. Describe how you used the amount in the previous year to calculate the amount in Amanda’s account for each year.

11. Write two different rules for finding the amount in Amanda’s account. Your recursive rule should tell how to find the amount based on the amount from the previous year. Your explicit rule should tell how to find the amount based on the number of years, n.

Recursive rule Explicit rule

Verbal rule

Function rule

Number of years, n

Amount, A ($)

0 100

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15



A Round-‐A-‐Bound is a toy ball that bounces unusually high. The diagram shows the path of a Round-‐A-‐Bound ball that is dropped from a height of 50 feet. On its first bounce, it reaches a height of 40 feet. On its second bounce, it reaches a height of 32 feet. Each successive bounce height decreases by a constant ratio.

12. Find a pattern to complete the rest of the table for the path of the Round-‐A-‐Bound ball.

Bounce, b Height, h (ft) 0 50

1 40

2 32

3

4

5

6

13. What is the constant ratio for this function? Explain how you found it.

Drop height: 50 ft

bounce 1

bounce 2

bounce 3

bounce 4 bounce 5

bounce 6



14. Write two different rules for finding the bounce height of the Round-‐A-‐Bound. Your recursive rule should tell how to find the bounce height using the height from the height of the previous bounce. Your explicit rule should tell how to find the bounce height based on the number of bounces, b.

Recursive rule Explicit rule


In this activity, you will work with your partner to find different representation of the same function. Each of the functions can be represented as a recursive function, as an explicit function, and as an input-‐output table. One representation of the function is given in each row of the table. Complete the table by filling in the missing two representations for each function.

Recursive function Explicit function Input/output table

x y 0 3

1 7

2 11

3 15

4 19

The first term is 80. To get each next term, divide the previous term by 2

(or multiply it by ½).

x y 0

1

2

3

4

x y 0 80

1 79.5

2 79

3 78.5

4 78




The first term is 10. Then double each term to get the next term.

x y 0

1

2

3

4

x y

0 1

1 5

2 25

3 125

4 625

y = 1.1( )x

x y

0

1

2

3

4

Now create a geometric sequence (i.e., exponential function) of your own, and represent it using only one of the boxes below. Then have your partner fill in the other two representations. Check each other’s work when you are both done.


x y

0

1

2

3

4



HOMEWORK 21.2

Notes or additional instructions based on whole-‐class discussion of homework assignment:

Marcos invested $1,000 in a savings account. His money will grow at a rate of 15% a year.

1. Create a table and graph showing the amount of money in his account for the first 5 years of his investment.

x (Number of years)

y (Amount of money in $)

0 1,000

1

2

3

4

5

2. Write a recursive function rule to model the situation.

(How can you find each value of y from the value that comes before it?)

What is the constant multiplier in this relationship?

3. Write an explicit function rule to model the situation. (How can you find each value of y from each value of x?)

How does each number in this function rule relate to the data in the table and the graph?

4. Use the table, graph, or function rule you wrote to answer the following questions. Explain how you found the answer.

a. How much money will be in the account after 5 years?

How I figured out the answer:

b. After about how many years will the amount of money in the account be $1,500?


c. How much money will be in the account in 9 years?




STAYING SHARP 21.2

Practic


cepts

1. Solve the following equation: 2(x + 1) = 3x – 1. Justify each step you take to solve the problem.

2. Use first differences to determine whether or not the table represents a linear relationship:

x y

-‐2 8

-‐1 2

0 0

1 2

2 8

Prep

aring for u

pcom

ing lesson

s

3. Find second differences of the y-‐values in the table from problem 2. What do you notice?

4. Graph the points from problem 2. What do you notice?

Focus skill: R

easoning

with

qua

ntities

5. Justify each step in the multiplication of (8.7 x 104) • (6.1 x 109).

Step Justification

8.7 x (104 x 6.1) x 109

8.7 x (6.1 x 104) x 109

(8.7 x 6.1) x (104 x 109)

6. Write the following numbers in order from least to greatest.

9.999999 x 10–3

0.00000007668

100,000,000

126 million

3.9 E -‐9

5,900,000,000

2.75 E 9

1.1 X 101

8.0006 x 10–10

1.0001 x 1018



Lesson 21.3 Transforming exponential functions

21.3 OPENER 1. Determine whether each table represents a linear relationship, an exponential relationship, or neither. Then, explain how

you know.

a. x y

1 -‐2

2 3

3 8

4 13

5 18

6 23

Circle one:

Linear Exponential Neither

Explain how you know:

b. x y

-‐3 1

-‐2 2

-‐1 4

0 8

1 16

2 32

Circle one:



c. x y

1 8

2 12

3 18

4 27

5 40.5

6 60.75

Circle one:



d. x y

-‐3 10

-‐2 5

-‐1 2

0 1

1 2

2 5

Circle one:



2. What did you calculate to determine whether the tables above were linear or exponential?

21.3 CORE ACTIVITY 1. Recall Barry and Red’s experiment with insects from the

topic Comparing Linear and Exponential Growth. Red’s data from raising fire ants is shown in the table. Use the process column to show how the number of fire ants, y, can be calculated mathematically from the number of weeks, x, in each row of the table.

2. Write an algebraic rule for the number of fire ants, y, in terms the number of weeks, x.

Weeks x

Process Fire ants y

0 20

1 40

2 80

3 160

4 320



3. Red and Barry are discussing Red’s data. Work with your partner to evaluate each of the statements they make in their

discussion. Tell whether you agree or disagree with the statement. Give reasons why you agree or disagree.

Statement Agree or disagree (with explanation of reasoning)?

Barry: The growth is linear with a slope of 2.

Red: No, the growth is exponential with a constant ratio of 2.

Barry: If there is a constant ratio of 2, that means the base of the exponential expression in the algebraic rule is 2.

Red: Since I started with 20 ants, the base must be 20. Maybe the function rule should be:

y = 20x

Barry: Starting with 20 ants means you have to multiply by a constant of 20. That’s where the 20 comes from in

y = 20 • 2x.

Red: I see. That means the y-‐intercept of the graph should be at y = 20.

Barry: It also means that every value of our function is 20 times larger than it would be for y = 2x. That’s because you started out with 20 ants instead of just one.

4. Sketch the graph of y = 20 � 2x on the same axes as the function y = 2x. Label the graph of the new function.

5. How does the graph of y = 20 � 2x compare to the graph of y = 2x?

y = 2x



6. Suppose Red tries a second experiment. This time he starts with 15 fire ants instead of 20. The population of fire ants grows

the same way as it did in the first experiment.

a. Fill in the data table to show how Red’s data would change. Show how you calculated the number of fire ants for each

week in the process column.

Weeks x Process

Fire ants y

0 15

1

2

3

4

b. What is the new function rule for this second experiment?

c. Sketch and label the graph of the new function on the same axes as your graph from question 4.

d. How does the graph of this new function compare to the graph of y = 20 � 2x and the graph of y = 2x?

e. How does this function compare to the experiment that started with 20 ants? How are the two functions similar? How are they different?

f. For the new experiment, predict how many weeks it will take for the number of fire ants to reach 1000.




1. Make a table comparing the y-‐values of Red’s new function rule, y = 40 ⋅2x−1, to those of the function rule you found earlier.

Weeks x

Number of fire ants

y = 20 ⋅2x

Number of fire ants

y = 40 ⋅2x−1

0

1

2

3

4

2. Verify the rules y = 20 ⋅2x and y = 40 ⋅2x−1

are equivalent using graphs.

3. Think about how you can apply the laws of exponents you learned in a previous topic to rewrite the expression 40 ⋅2x−1 .

Use these laws to verify the rules y = 40 ⋅2x−1 and y = 20 ⋅2x

are equivalent by algebraic manipulation.

40 ⋅2x −1 = 40 ⋅2x ⋅2−1

= 40 ⋅2x ⋅ 12

= 40 ⋅ 12⋅2x

= 20 ⋅2x



HOMEWORK 21.3 Notes or additional instructions based on whole-‐class discussion of homework assignment:

A team of biologists is researching the population of white-‐tailed deer that live in a certain area of the country. They have found that the deer population in the area is growing at a rate of about 25% per year. There are currently 32 deer living in the area.

1. Create a table and graph showing a prediction of number of deer in the area for each of the next 10 years.

x (Number of years)

y (Number of deer)

0 32

1

2

3

4

5

6

7

8

9

10

2. Find the following for this situation:

a. The constant multiplier:

b. The multiplication constant, or “stretch factor”:

c. A function rule that fits this population model:

3. Use the table and graph you created to predict the following. Explain how you made each prediction.

a. The amount of time it will take for the deer population to reach 400 deer

b. The number of deer that will be in the area in 15 years

4. Suppose there were currently 100 deer living in the area instead of 32.

a. What new function rule would fit this situation?

b. Make a sketch of the graph of this function on the same axes as the function you graphed in question 1. How does the graph compare to the graph from question 1?





cepts

1. A 9th grade math class has 27 students. There are twice as many girls in this class as there are boys. Write a system of equations that could be used to model this situation.

2. Solve the system of equations from problem 1 using any method. How many girls are in the class?

Prep

aring for u

pcom

ing lesson

s

3. Describe the patterns you see in this sequence of ordered pairs:

x y

-‐4 -‐14

-‐3 -‐7

-‐2 -‐2

-‐1 1

0 2

1 1

2 -‐2

3 -‐7

4 -‐14

4. Graph the ordered pairs from Problem 3. (Choose your scale carefully.) What do you notice?

! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !

!

Focus skill: R

easoning

with

qua

ntities

5. A large tank is 120 meters long, 65 meters wide, and 48 meters high. Express the volume of the tank in cubic meters using scientific notation.

6. One liter is equal to 0.001 cubic meters. Find the volume of the tank from question 5 in liters. Express your answer using scientific notation.



Lesson 21.4 Exploring parameters

21.4 OPENER

1. Given the function y = 80 � 5x

a. What is the value of y when x = 0?

b. What is the common multiplier?

c. Complete this table of values.

x y

-‐2

-‐1

0

1

2

3

2. Given the function y = 80 � 12

⎛

⎝⎜

⎞

⎠⎟

x

a. What is the value of y when x = 0?

b. What is the common multiplier?

c. Complete this table of values.

x y

-‐2

-‐1

0

1

2

3

3. Describe two different methods you could use to find the values in the table in question 2.

4. How are the function rules in questions 1 and 2 different? How are they the same?



21.4 CORE ACTIVITY In this activity, you will use your graphing calculator to investigate the effect of changing parameters of exponential functions. Many exponential functions can be written in the form:

y = a � bx

Two of the parameters of an exponential function are the values of a, the multiplication factor, and b, the base (or constant multiplier). As you have already seen, changing the values of a function’s parameters changes the behavior of the function. You will predict how each parameter affects the function and then test your predictions using your graphing calculator.

1. Investigate the effect of the base, b, by following the steps below.

a. Enter the function y = 2x into Y1. (This is the original function that you will compare your transformations with.) b. Select a new value for the base, b. Pick values that are greater than 2, between 1 and 2, between 0 and 1. c. Record your new function. Enter it into your calculator as Y2. d. Predict what the new graph will look like. Sketch it with a dotted line. Then graph the new function on your calculator.

Sketch it with a solid line. How does your prediction compare to the actual graph? e. Describe how the graph of the new function compares to the original function y = 2x.

(a) Original function (in Y1)

Multipli-‐cation factor,

a

(b) New base, b

(c) New

function y = bx (in Y2)

(d) Graph

Prediction: Dotted line Actual: Solid line

Compare

(e) Describe how the graph of the new

function compares to the original function

y = 2x

1

Pick a value greater than 2 and less than 10, 2 < b < 10

b = ______

1


b = ______

1


b = ______

Explain how the value of the base affects the graph of an exponential function.



2. Investigate the effect of the multiplication factor, a, by following the steps below.

a. Enter the original function y = 2x into Y1. b. Select a new value for the multiplication factor, a. Pick values that are between 1 and 10 and between 0 and 1. c. Record your new function. Enter it into your calculator as Y2. d. Predict what the new graph will look like. Sketch it with a dotted line. Then graph the new function on your calculator.



(b) New multipli-‐cation factor,

a

Base, b

(c) New

function y = a � 2x (in Y2)

(d) Graph


Compare

(e) Describe how the graph of the new function

compares to the original function

y = 2x

Pick a value greater than 1

and less than 10, 1 < a < 10

a = ______

2

Pick a value greater than 0 and less than 1,

0 < a < 1

a = ______

2

Explain how the value of the multiplication factor affects the graph of an exponential function.



One additional parameter, c, can also be included in an exponential function. This parameter is called a constant term. It can be added to the function as shown below. The result is called the general form of an exponential function.

y = a � bx + c

3. Investigate the effect of adding a constant term by following the steps below.

a. Enter the original function y = 2x into Y1. b. Select a new value for the constant term, c. Use a variety of values: Ones that are between 0 and 10, between -‐2 and 0. c. Record your new function. Enter it into your calculator as Y2. d. Predict what the new graph will look like. Sketch it with a dotted line. Then graph the new function on your calculator.



Multipli-‐cation factor,

a

Base, b

(b) New

constant term, c

(c) New

function y = 2x + c (in Y2)

(d) Graph


Compare

(e) Describe how the graph of the new

function compares to the original function

y = 2x

1 2

Pick a value greater than 1 and less than 10, 1 < c < 10

c =

______

1 2

Pick a value greater than 0 and less than 1, 0 < c < 1

c =

______

Explain how the constant term affects the graph of an exponential function.

21.4 ONLINE ASSESSMENT




1. Without graphing the functions on a graphing calculator, describe the similarities and differences between the graphs of the functions y = 100 � 0.2x and y = 100 � 0.95x. Sketch a graph of what you think these two functions will look like.

2. Without graphing the functions on a graphing calculator, describe the similarities and differences between the graphs of the functions y = 2 � 3x and y = 10 � 3x. Sketch a graph of what you think these two functions will look like.

3. Match the curves on the graph with the function rule that best represents the curve.

Graph A

Graph B

Graph C

Graph D

Graph E

Graph F

y = 1 � 5x

y = 1 � 2x

y = 1 � 0.2x

y = 1 � 0.5x

y = 1 � 2.7x

y = 1 � 0.35x

4. Describe how the parameters a and b affected the graph of y = abx. Don’t forget to mention specific values that these parameters cannot have.

How the parameter a affects the graph of y = abx How the parameter b affects the graph of y = abx



STAYING SHARP 21.4

Practic


cepts

1. While taking a road trip, Jose decides to keep track of his mileage. After 2 hours he has traveled a total of 120 miles and after 3 hours he has traveled a total of 180 miles. What is the rate of change between the two points?

2. What does the rate of change from Problem 1 represent?

Prep

aring for u

pcom

ing lesson

s

3. Plot the following points on the coordinate plane provided: (-‐1,1), (0,3), (1,1), (2,-‐5).

4. Does the graph from problem 3 represent a linear function, an exponential function, or neither? Explain how you know.

Focus skill: R

easoning

with

qua

ntities

5. What was the estimated world population in 1940? Express your answer in scientific notation.

6. Use the graph in question 5 to find the year when the population of the world was approximately 5.25 x 109.

! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !

!



Lesson 21.5 Behavior of exponential functions

21.5 OPENER

Without calculating exact values, predict whether each of the following values will be very small or very large. Explain the reasoning for each of your predictions. Then check your predictions using a calculator.

Expression Prediction (circle one) Reasoning Calculated value

(1/3)10 very small very large

3.0510 very small very large

5000 • (0.005)10 very small very large

5000−10 very small very large

(1.005)10 very small very large

21.5 CORE ACTIVITY The behavior of a function has to do with how its value changes at different locations. Work with a partner to answer questions 1 and 2 by analyzing the behavior of the functions shown in the graph below.

| P

| Q

y = (1.3) � x

y = (1.3)x

y = –0.8 � x + 10

y = 10 � (0.8)x



1. Compare the four functions in the graph at the locations described in the table below. Then write the function from the

graph that best fits into each box in the table below.

Location along x-axis

Very far to the

left (beyond what you can see on

the graph)

At x = 0

A little farther to the right on the

graph where x = P

A little farther to the right on the

graph where x = Q

Very far to the right (beyond

what you can see on the graph)

a. Which function has the greatest value?

b. Which function has the least value?

c. Which function shows the fastest growth?

d. Which function shows the fastest decay?

2. Recall that the domain of a function is the set of possible x-‐values. The range of a function is the set of possible y-‐values. Find the domain and range of each of the functions in the graph.

a. y = (1.3) � x b. y = (1.3)x

Domain: Range: Domain: Range:

c. y = –0.8 � x + 10 d. y = 10 � (0.8)x

Domain: Range: Domain: Range:



An insurance company estimates that the value of a particular car depreciates by 15% each year. The company uses an exponential function to predict the value of a car, v, as a function of time, t, in years. The function rule and graph are shown here.

v = 22,000 � (0.85)t

3. Write a paragraph to describe the

behavior of this function model. In your description, discuss the following: • Does the function represent

exponential growth or decay? Explain how you know in as many ways as you can.

• What does the 22,000 represent in the function equation?

• What does the 0.85 represent in the equation? How is it related to the 15% depreciation?

• What is the domain and range of the function? Explain your reasoning.

• When is the value of the car greatest? The least? Why?

4. Write an exponential function to predict the value of a car, v, as a function of time, t, in years, for a car that is worth $28,000

when new and depreciates by 20% every year.

Value of the car, v ($)

Time, t (years)

Car Depreciation



21.5 REVIEW ONLINE ASSESSMENT You will work with your class to review the online assessment questions.

Problems we did well on:

Skills and/or concepts that are addressed in these problems:

Attributions for our successes:

Problems we did not do well on:

Skills and/or concepts that are addressed in these problems:

Attributions for our difficulties:

Addressing areas of incomplete understanding

Use this page and notebook paper to take notes and re-‐work particular online assessment problems that your class identifies.

Problem #_____ Work for problem:






Next class period, you will take an end-‐of-‐unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts by reviewing related course materials.

Important skills and ideas you have learned in the unit Exponential and quadratic functions:

1. Rewriting expressions using the laws of exponents

2. Converting numbers between scientific and standard notation and computing in scientific notation

3. Connecting common differences and common multipliers to linear and exponential functions

4. Examining the effects of a and b on the behavior of exponential functions

5. Examining the effects of a and c on the behavior of quadratic functions

6. Comparing linear, exponential, and quadratic functions

Homework Assignment

Part I: Study for the end-‐of-‐unit assessment by reviewing the key ideas in the topic as listed above.

Part II: Take the More practice from the topic Problem solving with exponential functions through the online services. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study.

Part III: Complete Staying Sharp 21.5





cepts

1. Write an equation or inequality that could be used to mathematically represent the following statement: “Five more than a number is less than twice the number minus one”.

2. Solve the equation or inequality from Question 1.

Prep

aring for u

pcom

ing lesson

s

3. The following table relates the area of a square given a certain side length. Complete the table and sketch a graph of the data.

Side length Area

1 1

2

3

4

4. What function rule could be used to represent the data from Question 3? What is an appropriate domain for this function? Explain.

Focus skill: R

easoning

with

qua

ntities

5. Write the following distances in order from least to greatest. 25,000 cm; 3.2 x 10–4 km; 6.08 E 3 km; 5 million meters

6. Without using a calculator, rewrite the following product using scientific notation.

(7 x 103) • (2 x 104) • (3 x 108)



Lesson 21.6 Checking for understanding

21.6 OPENER

Three situations are described below. One is represented with a graph, one with a verbal description, and one with a table. For each situation, write a function rule to model the relationship. Then explain how you found your function rule.

Relationship Function rule Explanation

a. The number of trees growing in an orchard each year is plotted on a graph.

b. A sand hill is 50 feet high. The wind and rain cause its height to decrease by 20% each year.

c. A runner keeps track of how many miles she runs each week.

Weeks Number of miles run

0 10.00

1 11.00

2 12.10

3 13.31

4 14.64

21.6 END-OF-UNIT ASSESSMENT Today you will take the end of unit assessment.




1. There are some important similarities between linear functions and exponential functions. Explore these similarities by completing the table below for Function A and Function B.

Function A Function B

y = 3 + 2x y = 3 � 2x

Type of function: (linear or exponential):

Type of function: (linear or exponential):

Table:

x y

-‐2

-‐1

0

1

2

3

Table:

x y

-‐2

-‐1

0

1

2

3

Sketch of graph:

Sketch of graph:

The constant difference between terms is: The constant multiplier between terms is:

The coefficient multiplied to the variable, x, is:

The base of the exponent, x, is:

The y-‐intercept is:

The y-‐intercept is:

To find the next term for this function, I would…

To find the next term for this function, I would…



2. Answer the following questions to reflect on your performance and effort this unit.

a. Summarize your thoughts on your performance and effort in math class over the course of this unit of study. Which areas were strong? Which areas need improvement? What are the reasons that you did well or did not do as well as you would have liked?

b. Set a new goal for the next unit of instruction. Make your goal SMART.

·∙ Description of goal:

·∙ Description of enabling goals that will help you achieve your goal:




1. Evaluate the following expressions. (Don’t forget to use the correct order of operations!)

a. 4 – 3 + 22 + -‐6 ÷ 3 b. (2 + 3 ·∙ 4) – 4 + (-‐23 +1)2

2. Complete the following table. The first row has been completed for you.

x 3x2 2x2 3x2 + 2x2 5x2

-‐4 3 ·∙ (-‐4)2 = 48 2 ·∙ (-‐4)2 = 32 3 ·∙ (-‐4)2 + 2 ·∙ (-‐4)2 = 80 5 ·∙ (-‐4)2 = 80

-‐2

0

3

5

3. What relationships do you notice between the expressions 3x2 + 2x2 and 5x2? Explain.

4. Complete the following table. The first row has been completed for you.

x x2 + x 2x – 9 (x2 + x) + (2x – 9) x2 + 3x – 9

-‐4 (-‐4)2 + -‐4 = 12 2(-‐4) – 9 = -‐17 ((-‐4)2 + -‐4) + (2(-‐4) – 9) = 12 + -‐17 = -‐5 (-‐4)2 + 3(-‐4) – 9 = -‐5

-‐2

0

3

5

5. What relationships do you notice:

a. between the values in the table in Question 4 for the (x2 + x) + (2x – 9) and x2 + 3x – 9 columns?

b. between the expressions (x2 + x) + (2x – 9) and x2 + 3x – 9? Explain.



STAYING SHARP 21.6

Practic


cepts

1. At Pizzamania, the cost of a large pizza is $12 plus $1.75 for each topping. What function rule could you use to find the cost c of a pizza with x toppings?

2. Using the function rule from Problem 1, determine the number of toppings a large pizza has if it costs $20.75.

Prep

aring for u

pcom

ing lesson

s

3. The following diagram models the expression

(x2 + x) + (2x2 + 3x + 1)

Write a simpler expression for this sum.

4. Find the perimeter of a rectangle with length 2x and width 8. Provide a sketch to support your work.

Focus skill: R

easoning

with

qua

ntities

5. Calculate the value of E in the equation below. Express your answer in scientific notation.

E = (7.6 x 10–30) • (3.0 x 108)2

6. Complete the table by writing the amounts in scientific notation.

Year National debt ($)

National debt in scientific notation ($)

1791 75 million

1916 1 billion

1946 280 billion

2010 13.5 trillion

!"# !#

!"!!

!"! !"! !!

!! !! $#!

!"# !#

!"!!

!"! !"! !!

!! !! $#!

!"# !#

!"!!

!"! !"! !!

!! !! $#!

problem solving with exponential …pkeenan.weebly.com/uploads/4/5/6/7/45670275/unit_7_topic...326!...

Documents