Exponential Equations - University of Georgiamath.coe.uga.edu/olive/EMAT3500f08/TI_NSPIRE/3... ·...

3Exponential Equations

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Exploring Algebra 1 with TI-Nspire™ 3: Exponential Equations 57© 2009 Key Curriculum Press

You have $600 that you want to save, and you know that it can earn money for you.

When you let someone else use your money, they must pay you rent. That rent is

called interest. In this activity, you will explore different types of interest.

INTRODUCTION

A relative agrees to pay you interest at a rate of 10%. This means that 10% of the

$600 will be added to your savings each year. This is called simple interest.

Q1 How much will your savings increase each year?

A bank advertises that it will pay you 6% interest. The bank’s interest is compounded.

In other words, each year the bank adds to your savings 6% of the growing amount

of savings, not just 6% of the original $600.

Q2 How much will your savings increase the fi rst year? The second year?

Your goal in this activity is to decide whether the bank’s offer can ever be better

than your relative’s offer.

INVESTIGATE

1. Open the TI-Nspire document

Interest.tns on your handheld. Go

to page 1.2. You will be making a

spreadsheet showing your earnings

over 50 years.

2. First enter a formula that will generate

the years. The formula for year looks

like this:

� seqn(u(n � 1) � 1, {0}, 50)

3. To calculate the simple interest, enter

the formula

� seqn(u(n � 1) � 60, {600}, 50)

The u(n � 1) in the formula tells TI-

Nspire to use the previous value of the

variable. The 600 tells TI-Nspire to use the

value 600 for the initial case (here, it is the

0th case).

Q3 What does the 60 in the formula do?

You will need• Interest.tnsYou will need• Interest.tns

Exponential Growth—Interest


58 3: Exponential Equations Exploring Algebra 1 with TI-Nspire™© 2009 Key Curriculum Press

4. For compound, set up a formula to

represent the bank’s offer. You can use

the formula

� seqn(u(n � 1) + .06 � u(n � 1),

{600}, 50)

Q4 Explain in your own words what this

formula does.

Q5 Will your savings ever earn more

money with the bank’s offer than with

your relative’s? Explain.

You would like formulas that give you the value of your savings for any year,

without having to fi nd the values for all previous years. You will use scatter plots to

fi nd these formulas.

5. At the top of page 1.3, make a scatter plot of the (year, simple) data.

6. The data points appear linear, so you’ll

be looking for an equation of the form

simple � a � b � year. Choose Plot Function from the Actions menu, and

enter the equation f(x) � a[1] � b[1] �

x. Use the sliders at the bottom of the

page to adjust a and b to fi t the data

points as well as possible.

Q6 What values of a and b give the best fi t?

How do these values relate to $600 of your

initial savings and to the 10% simple interest rate?

7. At the top of page 1.4, make a scatter plot of the (year, compound) data and plot

the line from page 1.3.

Q7 Can you adjust a and b to make the

graph fi t the compound interest data?

Why or why not?

Because compound interest cannot be

modeled by a linear equation, you will

look for another type of equation that can

model it.

Exponential Growth—Interestcontinued



Exponential Growth—Interestcontinued

8. Perhaps this curve is part of a parabola.

Go to page 1.5 and make another scatter

plot of the (year, compound) data. Plot

the function f(x) � c[1] � d[1] � x2.

(To get the exponent, press l or

q.) Adjust the c and d sliders to

match the data as well as possible.

Q8 How well does this graph model the

data? Do you think compound interest

growth can be modeled by a parabolic

graph?

9. Another curve is the graph of an

equation in which year is in the

exponent. Go to page 1.6 to explore an

equation of the form compound � m �

n year. You will need to plot the function

f(x) � m[1] � n[1] x. This is called an

exponential equation. To adjust n, you

may need to zoom in on values near 1.

Q9 What values of m and n give the best fi t

for the data? Do you think compound

interest can be modeled by exponential equations?

Q10 When compound interest in this situation is modeled by an equation of the

form A � mnx, how do you think m and n relate to the $600 initial investment

and to the rate of 6%?

EXPLORE MORE

Insert a new Lists & Spreadsheets application as page 1.7. What formulas give the

following sequences?

2, 4, 6, 8, . . . , 16

3, 6, 12, 24, . . . , 1536

1, 1, 2, 3, 5, . . . , 89

To zoom, use the Window/Zoom menu

or drag the ends of the axes.

To zoom, use the Window/Zoom menu

or drag the ends of the axes.



Adapted from Exploring Algebra 1 with Fathom by Eric Kamischke, Larry Copes, and Ross Isenegger.

Objectives: Students will learn that a linear equation

models simple interest and an exponential equation

models compound interest. They will relate the values of

a and b in the equations A � a � bx and A � ab x to the

principal and interest rate, respectively.

Activity Time: 35–50 minutes

Materials: Interest.tns

Mathematics Prerequisites: Students should be able to

calculate a percentage of an amount, write a percent as a

decimal, evaluate equations by substitution, and substitute

variable expressions into a formula.

TI-Nspire Prerequisites: Students should be able to

open and navigate a document, use formulas in the Lists

& Spreadsheet application, create scatter plots, plot an

equation on a scatter plot, adjust the scale of an axis, and

insert a new page. (See the Tip Sheets.)

TI-Nspire Skill: Students will insert formulas using the

seqn command.

Notes: As you facilitate student work, probe for

understanding of the quantity 1.06, especially in Q9. You

might have pairs who fi nish the main activity fi rst prepare

to present the Explore More questions and answers.

For a Presentation: As you lead a class discussion using a

presentation computer or projected handheld, emphasize

Q5–Q9.

INTRODUCTION

Q1 10% of $600 is $60.

Q2 Year 1: 6% of $600, or $36; Year 2: 6% of $636,

or $38.16

INVESTIGATE

Q3 The 60 adds 10% of $600, or $60.

Q4 The formula starts with 600. It takes that value,

multiplies it by 0.06, and adds that to the value to get

the next value. It does this 50 times.

Q5 The savings in the bank passes the savings with the

relative in year 18.

Q6 a � 600, the initial amount; b � 60, the amount of

interest being added each year. Introduce the terms

principal for the initial amount of $600 and interest

rate for the 10%.

Q7 No, the compound interest graph is not linear.

Q8 The compound interest graph is not quite parabolic.

It fi ts well for the fi rst 20 years but does not increase

fast enough after that.

Q9 The best value for m is the vertical intercept, 600. The

best value for n is 1.06. An exponential equation

fi ts well.

Q10 m is the principal, $600. n is 1 more than the

interest rate.

DISCUSSION QUESTION

• You entered several formulas at the beginning of this

activity. What does each part of the formula do?

Here is a quick look at the formula = seqn(u(n � 1) � 60,

{600}, 50):

� defi nes a formula

seqn creates a sequence

u(n � 1) the previous term in the list—u(n � 1) and u(n � 2) are the previous two terms.

� 60 what to do to previous term

{600} the initial term of the sequence. (The fi rst and second terms would be {a, b}.)

50 the number of terms to list

EXPLORE MORE

� seqn(u(n � 1) � 2, {2}, 8); � seqn(u(n � 1) � 2, {3},

10); � seqn(u(n � 1) � u(n � 2), {1, 1}, 11). Students

could also fi nd these formulas using cell operations and

fi ll down.


Exponential Growth—Interest Activity Notes



The amount of time a scuba diver can safely spend under water depends on the

amount of air in the diving tank. Your friend has asked for your help in determining

how much air is in a particular tank.

You can see that the volume of the tank is 0.340 cubic feet, but you also know

that the tank is pressurized to pack more air into a smaller space. As the air leaves

the tank, it expands. You can see that the tank is full and that the pressure gauge

reads 3500 pounds per square inch (psi). To determine how much air the tank will

supply while your friend is diving, you need to know what the volume of the air will

become as it is released from the tank and as the pressure becomes the normal air

pressure of 14.7 psi. Boyle’s law relates these two qualities.

INVESTIGATE

1. You have found some data on the

Internet and put it into a table to

help you look for patterns. Open the

document Boyles Law.tns and go to

page 1.2 to see the data.

2. Go to page 1.3 and create a scatter

plot of the (volume, pressure) data

at the right of the page. The data

are not linear, but perhaps they are

exponential.

Q1 As the volume is halved from 48 ft3 to

24 ft3, what happens to the pressure?

Q2 Does the pressure approximately

double when the volume goes from

30 ft3 to 15 ft3? Does the pressure

always double when the volume is cut

in half? Show the evidence for your

conjecture. You can use either the

scatter plot or the table to fi nd this

information.

You will need• Boyles Law.tnsYou will need• Boyles Law.tns

Click on the point representing

volume 48 to see its values. You can also

look in the table.

Click on the point representing

volume 48 to see its values. You can also

look in the table.

Inverse Variation—Boyle’s Law



Q3 If the equation were exponential, would the pressure double when the volume is

cut in half?

3. Because the data do not appear to

be exponential, you decide to try

modeling with a power equation of

the form y � k � x b. Unlike in an

exponential equation, the variable

pressure will be in the base rather than

in the exponent. Add the graph of the

power equation to your scatter plot.

Use the equation f1(x) � k[1] * x^b[1].

4. Adjust the k and b sliders to fi t the data as well as you can.

Q4 According to your model, what pressure is needed to reduce the volume

below 10 ft3?

You may have found that a good value for b is approximately –1. Using the

defi nition of a negative exponent, it can be simpler to write the equation as

pressure � k( 1_____volume ). The coeffi cient k of this equation is called the constant of

variation. Note that k � pressure � volume. Finding the value of this constant in

particular situations is important for answering pressure-volume questions like the

one in this activity.

5. To see this relationship in the data table, add a new variable constant with the

formula pressure � volume.

6. Add a new Data & Statistics page by pressing c and choosing Data & Statistics.

Q5 Create a new dot plot of constant and

plot the value mean(constant). How

does this mean relate to the values of

the sliders? Explain.

Q6 What is the constant of variation for

the scuba tank you are examining? Why

might it be different from the constant

of variation in the table?

Q7 What volume will the air have when it

is released from the tank and the pressure becomes 14.7 psi?

To plot a value, choose Plot Value from the

Actions menu.

To plot a value, choose Plot Value from the

Actions menu.

Inverse Variation—Boyle’s Lawcontinued



Inverse Variation—Boyle’s Lawcontinued

EXPLORE MORE

1. To explore more equations of the form

xy � k or the equivalent, go to the plot

of y � k ( 1_x ) on page 2.1. Explore the

graph for the values of slider k near 0.

Describe the graph when k � 0, k � 0,

and k � 0.

2. In Explore More 1, you studied a graph

in which the product of x and y was

the constant k. Now explore graphs in

which the sum of x and y is the constant k. You may need to add a new Data &

Statistics page after 2.1 to do this. Be as specifi c as you can about what is always

true about the graph and what changes as k changes.

3. What if the difference were always k? What if the quotient were k? Try this. Are

you surprised by the results?



Objectives: Students will model inversely proportional

quantities with equations of the form y � k ( 1 _ x ) , xy � k, and

y � kx�1. They will investigate the change in one variable

as the other doubles, relate y � k ( 1 _ x ) to a linear model, and

explore the graph of xy � k.


Materials: Boyles Law.tns

Mathematics Prerequisites: Students should be able to

solve literal equations and evaluate a formula.

TI-Nspire Prerequisites: Students should be able to create

a scatter plot, plot a function on a scatter plot, use

TI-Nspire to calculate a mean, and change the scale of

sliders and scatter plots.

TI-Nspire Skills: None

Notes: Boyle’s law states that if a gas is kept at constant

temperature, the pressure and volume are inversely

proportional, or have a constant product. Robert Boyle

published his fi ndings that pressure times volume is

constant in his 1662 article “A Defense of the Doctrine

Touching the Spring and Weight of the Air.”

As you facilitate student work, look for students who

have complete answers to Q2 and Q3. Have them share

their results with the class. A variety of answers to Q6

can also be shared; students taking physics will know that

temperature is an important factor.

INVESTIGATE

1. These data are Boyle’s original. The Internet source

is given on the last page of the document. Boyle’s

methodology and his published fi ndings are

intriguing reading.

Q1 The pressure roughly doubles from 29.125 psi to

58.8125 psi.

Q2 As the volume goes from 30 ft3 to 15 ft3, the pressure

roughly doubles from 47.0625 psi to 93.0625 psi.

Some students might say pressure is not quite

doubled as volume goes from 15 ft3 to 30 ft3. The table

demonstrates that the ratio of the pressures is very

close to 2 in every case:

Volume(ft3)

Pressure(psi)

Volume(ft3)

Pressure(psi)

Ratio of pressures

48 29.1250 24 58.8125 2.019313

46 30.5625 23 61.3125 2.006135

44 31.9375 22 64.0625 2.005871

42 33.5000 21 67.0625 2.001866

40 35.3125 20 70.6875 2.001770

38 37.0000 19 74.1250 2.003378

36 39.3125 18 77.8750 1.980922

34 41.6250 17 82.7500 1.987988

32 44.1875 16 87.8750 1.988685

30 47.0625 15 93.0625 1.977424

28 50.3125 14 100.4375 1.996273

26 54.3125 13 107.8125 1.985040

24 58.8125 12 117.5625 1.998937

Inverse Variation—Boyle’s Law Activity NotesAdapted from Exploring Algebra 1 with Fathom by Eric Kamischke, Larry Copes, and Ross Isenegger.



Q3 No. If the equation were exponential, the pressure

would double for constant changes in the volume, not

for proportional changes.

4. Finding the best values is tricky. It requires fi ne motor

control, the ability to adjust the scale of a slider, and

visual estimation skills. This model, with k � 1253

and b � �0.97, matches the graph remarkably well,

which might make you wonder whether Boyle’s

data were fudged. There is some evidence that

other historically important data can be statistically

shown to be too close to the predictions to have been

produced by experimentation.

Q4 Using pressure = 1253 · volum e �0.97 , a pressure of

160 psi will decrease the volume below 10 f t 3 .

Q5 If slider k is set to the mean value, 1408, the graph

goes through the data points. Doing a power

regression on the data yields the equation y � 1400.9 ____ x .

(Choose Regression from the Actions menu.)

Q6 pressure � volume � constant, so (0.34)(3500) � 1190.

Possible reasons the constant may differ from that of

the table include differences in temperature and the

nature of the gas being compressed.

Q7 About 81 f t 3 ( volume � constant _____ pressure � 1190

___ 14.7 )

EXPLORE MORE

1. When k < 0, the branches are in the second and

fourth quadrants. This graph is called a hyperbola. Its

branches approach, but never cross, its asymptotes,

which, in this case, are the x- and y-axes. When k � 0,

the graph is the x-axis with the point (0, 0) removed

(because 1 _ 0 is undefi ned). When k > 0, there will be

two branches of the graph, one in the fi rst quadrant

and one in the third.

2. The equation is y � k – x, so the graph is a straight

line with y-intercept k and slope –1. It crosses the

y-axis above the origin when k > 0, at the origin when

k = 0, and below the origin when k < 0.

3. If the constant difference equals x – y, then y � x – k;

if the constant difference equals y – x, then y � x + k.

The graph of each is a straight line with slope 1; k is

either the y-intercept or its opposite. If the constant

quotient equals x_y, then y � 1_k x ; if the constant

quotient equals x_y, then y � kx. In both cases, the

graph is a line through the origin with slope either k

or its reciprocal.

EXTENSIONS

1. Have students research the pressure experienced by

divers. How deep would a diver need to be in order

to be subject to twice the pressure experienced at

sea level? [A diver at depth 10.3 m under fresh water

experiences a pressure of about 2 atmospheres—

1 atm for the air and 1 atm for the water.]

Inverse Variation—Boyle’s Law Activity Notescontinued



2. Pose this problem: A data analyst uses the

TI-Nspire to fi t the exponential equation volume �

46.41(0.98032 ) pressure–29 to the data in the table. Make a

convincing argument for why either the exponential or

the power model is better. You may want to discuss the

vertical intercept and its role in the model, as well as

the halving time for volume. [The equation volume �

46.41(0.98032 ) pressure–29 goes through the point (29,

46.41), which is approximately pressure and volume in

the fi rst case. The volume is decreasing approximately

2% per increase of 1 unit of pressure at the beginning

of the experiment, but it decreases more slowly in the

later values. Not only does a graph of the form pv � k

fi t the points better, but it also ensures that the graph

has no vertical intercept (as pressure decreases to 0,

volume expands to infi nity). The exponential model

implies that when there is no pressure, the volume

will be about 82.6, which is physically incorrect. In

the activity, volume is halved as pressure is doubled

in Boyle’s model, as opposed to having a fi xed halving

period as in the exponential function model.]

Inverse Variation—Boyle’s Law Activity Notescontinued



The particles that make up the atoms of some elements, like uranium, are unstable.

Over a period of time specifi c to the element, the particles change and the atom

eventually becomes a different element. This process is called radioactive decay.

EXPERIMENT

In this experiment your counters represent atoms of a radioactive substance.

1. Take a paper plate and draw an angle from

the center of your plate, as illustrated.

2. Count the number of counters. Open the

TI-Nspire document Radioactive Decay.tnson your handheld and go to page 1.2. Enter the

number of counters as the number of “atoms”

after 0 years of decay. Pick up all of the counters.

3. Drop the counters randomly on the plate. Be

sure the counters are scattered evenly over all

parts of the plate—do not aim for the center. Counters that fall inside the

angle represent atoms that have decayed. Decide how you are going to handle

counters that fall on the lines of your angle and that miss the plate—they need

to be accounted for also.

4. Count and remove the counters that fall inside the angle—these atoms have

decayed. Subtract the decayed atoms from the previous value and enter the

number of counters remaining after 1 year of decay. Pick up the remaining

counters.

5. Repeat steps 3 and 4 until you have

fewer than ten atoms that have not

decayed. Each drop will represent

another year of decay. Enter the

number of atoms remaining each time

on your handheld.

You will need• Radioactive

Decay.tns• paper plate• protractor• supply of small

counters

You will need• Radioactive

Decay.tns• paper plate• protractor• supply of small

counters

Exponential Models—Radioactive Decay



INVESTIGATE

You’ll use your handheld to make a model for your experiment. Let x represent

elapsed time in years, and let y represent the number of atoms remaining.

6. On your handheld, make a scatter plot

of the data. To do this, go to page 1.3 and

add the appropriate variables to the plot.

Q1 What do you notice about the shape of

the graph?

7. Calculate the ratios of atoms remaining

between successive years. That is,

divide the number of atoms remaining

after 1 year by the number of atoms

remaining after 0 years, and so on. To do this, go to page 1.2. Arrow to the top of

column C and type the variable name, ratio. In cell C1, type the formula = b2/b1

and press ·. To fi ll in the rest of the values, highlight the fi rst cell and choose

Fill Down from the Data menu. You will see the cell surrounded by dashed lines.

Press ¤ to highlight all but the last row of data you entered, then press ·.

You should have a ratio for each row except the last.

Q2 How do the ratios compare?

8. Find a representative ratio by calculating either the mean or median. To use the

Calculator application, go to page 1.4. Type either mean(ratio) or median(ratio).

Q3 Which representative ratio did you choose and why?

Q4 At what rate did your atoms decay?

Q5 Write an exponential equation that models the relationship between the amount

of time elapsed and the number of atoms.

Exponential Models—Radioactive Decaycontinued



Exponential Models—Radioactive Decaycontinued

9. Graph the equation with the scatter plot: choose Plot Function from the

Actions menu and type the equation.

Q6 How well does the equation fi t the data? If the equation does not fi t well, adjust

the two values until you are satisfi ed. To do this, double-click the equation on

the screen and edit the values. Record your fi nal equation.

Q7 Measure the angle on your plate. Describe a connection between your angle and

the numbers in your equation.

Q8 Based on what you’ve learned and the procedures outlined in this activity, write

an equation that would model the decay of 400 counters using a central angle

of 60�.

Q9 What are some of the factors that might cause differences between actual data

and values predicted by your equation?

EXPLORE MORE

Perform an exponential regression with the handheld and compare this equation

with the others that you found.

Hint: Look in the Actions menu

on page 1.3.

Hint: Look in the Actions menu

on page 1.3.



Objectives: Students will write exponential equations that

model real-world decay data.


Materials: Radioactive Decay.tns, paper plates,

protractors, counters (Skittles, M&M bits, candy corn,

dried beans, etc.). Optional: Radioactive Decay Sample.tns

Mathematics Prerequisites: Students have had some

experience with exponential equations and function

notation.

TI-Nspire Prerequisites: Students should be able to open

and navigate a document, graph an equation, create a

scatter plot, and use the grab tool. (See the Tip Sheets.)

TI-Nspire Skills: Students will fi nd the mean or median

and type formulas into cells of the Lists & Spreadsheet

application.

Notes: There is a TI-Nspire document with sample data

for your use. All the screen shots below are with the sample

data. You may choose to have students work collaboratively

with a partner or as a whole class.

EXPERIMENT

1. Students should use protractors to make an angle

of less than 90°, but not too small. They can

approximate the center by using a ruler to fi nd the

midpoints of several diameters.

3. The objective is to have the counters spread evenly

and quickly. An acceptable plan would be that

counters on the line and counters that fall outside the

plate but lie within the extended rays of the angle are

accepted as being within the angle.

4. Remind students that they are interested in the

number of atoms that do not decay.

INVESTIGATE

6. Here is a graph of the sample data.

Q1 Students should notice a curved, exponential pattern.

7. You may have to assist students with this step. Some

may just type in the formulas for every cell, which is

not an effi cient approach.

Q2 The ratios should be approximately the same. For the

sample data, they range between 0.73 and 0.9.

Q3 Students could give reasons for selecting the mean,

the median, or another value. In these sample

data, the mean is about 0.802 and the median is

about 0.804.

Q4 For these sample data, using 0.802, the rate of decay is

19.8% per year. If you are not using the sample data,

you might use this opportunity to discuss various

student answers. You may need to encourage students

to look at their constant multiplier in the form of

(1 � r). That is, a ratio of 3 _ 4 would be (1 � 0.25).

Q5 For this sample, y � 201(1 � 0.198)x.

Q6 For the sample data, the equation does not appear

to fi t very well. Students may fi nd that the multiplier

A seems to adjust the curve’s position vertically and

r seems to change the steepness of the curve. The

equation y � 201(1 � 0.22)x fi ts the sample data better.

Exponential Models—Radioactive Decay Activity NotesAdapted from Discovering Algebra by Jerald Murdock, Ellen Kamischke, and Eric Kamischke.



Q7 To help students with this question, ask “What is

the ratio of your angle measure to the whole plate?”

The ratio of the angle measure to 360� should be

approximately the same as r. In the sample data, 68 ___ 360 �

0.19, which is close to the r-value used in Q5.

Q8 y � 400 ( 1 � 60 ___ 360 )

x

Q9 Factors given might include how evenly the counters

are distributed on the plate, what you do when

a counter is on an angle line, and how you treat

counters that fall outside the plate.

Exponential Models—Radioactive Decay Activity Notescontinued

EXPLORE MORE

To fi nd and graph the exponential regression from the

Data & Statistics application, choose Regression from

the Actions menu, then choose Exponential Regression.

For the sample data, the regression equation is y �

177.22(0.80)x. This equation is close to the predicted

equation, with a smaller A-value and a very similar r-value.

Comparisons to students’ adjusted equations will vary.




In this activity you will do an experiment, write an equation that models the

decreasing exponential pattern, and fi nd the half-life—the amount of time needed

for a substance or an activity to decrease to one-half its starting value.

EXPERIMENT

1. Make a pendulum with a soda can half-fi lled

with water tied to at least 1 m of string—use the

pull tab on the can to connect it to the string.

Place the meterstick underneath or behind the

pendulum so you can take readings.

2. Pull the can back about 1 m from its resting

position, then release it. Measure how far the

can swings from the resting position for as

many swings as you can.

3. Open the TI-Nspire document

Pendulums.tns on your handheld and

go to page 1.2. Enter your data in the

Lists & Spreadsheet application.

Q1 Go to page 1.3 and make a scatter plot

of your data. What type of pattern does

the graph seem to show?

INVESTIGATE

4. Go to page 1.4. Here you will see a

scatter plot of data on 19 pendulum

swings collected using a CBR2. These

data show the full swing of the

pendulum.

Q2 Describe this graph as fully as you can.

How does this graph compare with the

graph of your data? Explain.

You will need• Pendulums.tns• soda can or bottle

of water• string• meterstick

You will need• Pendulums.tns• soda can or bottle

of water• string• meterstick

Resting positionResting position

You may have to collect data for every fourth or fi fth swing

to get an accurate reading.

You may have to collect data for every fourth or fi fth swing

to get an accurate reading.

Half-Life—Pendulums



5. The maximum swings from the sample

data are on page 1.5. Go to page 1.6

and make a scatter plot of these data.

Q3 Does this graph look similar to the

graph of your data? Explain.

Q4 Find an equation in the form

y � A(1 – r)x that models the sample

data. Graph this equation with the

scatter plot. Adjust the values until you

have as good a fi t as possible. What is your function?

Q5 Find the half-life of your data. To fi nd the half-life, approximate the value of x

that makes y equal to 1_2 A.

Q6 What does the half-life mean for the situation in your experiment?

Q7 Find the maximum distance after one half-life, two half-life cycles, and three

half-life cycles. How do these values compare?

Q8 Write a summary of your results. Include descriptions of how you found your

exponential model, what the rate r means in your equation, and how you found

the half-life.

EXPLORE MORE

Perform an exponential regression with your handheld: go to page 1.7, which shows

a Data & Statistics scatter plot, and choose Regression from the Actions menu.

Compare this equation with the one you found.

Choose Function from the Graph Type menu to graph. Double-click the equation to edit it.

Choose Function from the Graph Type menu to graph. Double-click the equation to edit it.

Half-Life—Pendulumscontinued



Objectives: Students will write exponential equations that

model real-world decay data and fi nd the half-life for the

equation.


Materials: Pendulums.tns, soda cans or bottles of water,

string, metersticks

Mathematics Prerequisites: Students have had some

experience with exponential equations and function

notation.

TI-Nspire Prerequisites: Students should be able to open

and navigate a document, graph an equation and modify

it, create a scatter plot, and use the grab tool. (See the Tip

Sheets.)

TI-Nspire Skills: None

Notes: You have four options for this experiment,

depending on your time constraints and classroom setup.

Option 1: Have students collect data, enter the data into the

handheld, analyze the data, and compare them with the

sample data. Option 2: Do a demonstration for the whole

class and collect data using the meterstick, enter the data

into the handheld, analyze the data, and compare them

with the sample data. Option 3: Demonstrate the basic

experiment so students understand the situation, then go

right to the sample data. Option 4: Discuss the experiment,

then go right to the sample data. You could, theoretically,

also collect data similar to the sample data using a CBR2

or other data collection device. However, the process of

manually capturing the maximum swing points requires

patience and precision, and may not be worth class time.

You may choose to have students work collaboratively with

a partner or as a whole class.

EXPERIMENT

Q1 Students will probably recognize an exponential

pattern.

Q2 The location of the pendulum bob is harmonic, but

its maximum distance from the resting position is

roughly exponential. Students should notice the

sinusoidal pattern (in their own words) and that the

graph is decaying, or getting smaller. They should

notice that the top points of the graph look like

their graph.

5. Discuss with students any patterns they see with the

data table.

Q3 If students collected data accurately, this graph should

resemble their data. They may note that this graph

shows every swing, whereas theirs shows every fi ve

swings or so.

Q4 Sample equation:

Q5 Using the sample equation, the half-life is about

82 swings. One way to fi nd this is to graph y2 � 1 _ 2 A

and fi nd the intersection with y1 � A(1 � r)x. To fi nd

the intersection, choose Intersection Point(s) from

the Points & Lines menu and select the two functions.

Q6 Sample answer: The half-life of the pendulum swing

is approximately 82 swings. This means that on the

82nd swing the pendulum’s maximum distance is half

of its original maximum distance.

Q7 Using the sample equation: approximately 0.393 m,

0.197 m, and 0.098 m. With each consecutive half-life,

the value of y will be half the previous value of y.

Q9 Student summaries will vary.

Half-Life—Pendulums Activity NotesAdapted from Discovering Algebra by Jerald Murdock, Ellen Kamischke, and Eric Kamischke.



DISCUSSION QUESTIONS

• How does the shape of the graph in Q2 correspond to

the situation with the pendulum?

• You might want to introduce the equation

y � A ( 1 _ 2 ) x /t , where t is the half-life. Students can see

that the graph of this equation is similar to that of

their equation in the form y � A(1 � r ) x . Ask them to

think about why the graphs are the same.

EXPLORE MORE

The regression equation y � 0.78 (0.99) x is probably similar

to the equation students found. Neither equation seems

to fi t the data remarkably well over this small set of data.

However, if students saw a greater domain and range for

the graph using the method given above for Q5, they may

think the fi t is quite good.

Half-Life—Pendulums Activity Notescontinued


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