Session 7 Nonlinear Functions

Patterns, Functions, and Algebra 169 Session 7

Session 7

Nonlinear Functions

Key Terms for This SessionPreviously Introduced

• closed-form description [Session 2] • recursive description [Session 2]

• function [Session 3] • independent variable [Session 5]

• dependent variable [Session 5]

New in This Session• exponential function • base • exponent

• exponential growth function • exponential decay function • quadratic function

• parabola • common differences • figurate number

Introduction and ReviewIn the previous two sessions, we looked at linear relationships and developed strategies for solving linear equa-tions. Linear functions are important in mathematics and are usually studied first because they are the easiestequations to solve. In this session, we’ll expand our exploration of functions and relationships to include two typesof nonlinear functions: exponential functions and quadratic functions. [SEE NOTE 1]

Learning ObjectivesThis session explores nonlinear functions. We will:

• Become familiar with exponential and quadratic functions

• Understand that exponential functions are expressed in constant ratios between successive outputs

• Understand that quadratic functions have constant second differences

• Work with graphs of exponential and quadratic functions

• Explore exponential and quadratic functions in real-life situations

NOTE 1. In this session, we’ll explore nonlinear functions and situations in which these functions arise. People tend to think offunctions mostly in terms of linear functions, but exponential, quadratic, and other nonlinear functions are also common in theworld, and they’re also important to understand.

NOTE 1 cont’d. next page

Session 7 170 Patterns, Functions, and Algebra

NOTE 1, CONT’D.

Another important idea in this section is that families of functions can be described by certain characteristics. For example,exponential functions are characterized by a constant ratio between successive outputs. Emphasizing these common charac-teristics helps us to think about functions as objects of study in and of themselves, and not just as rules that transform inputsinto outputs.

Materials Needed: Computers with spreadsheet program, graph paper, toothpicks. Groups: If it’s easy to move back and forthto computers, you may want to use the spreadsheet program throughout the session. If not, you may use it just for Part A, andwork with a calculator and graph paper after that time.

ReviewGroups: Discuss any questions that came up on the homework. Share solutions to the mobile problems. Talk about the think-ing used in solving the problems. Did you think “algebraically”? Did you use any symbols to solve the problems?

Move into today’s session by reviewing what you know or remember about linear functions. Some key points are:

• There is a constant difference between successive outputs of a linear function

• Linear equations look like y = ax + b, where a and b are numbers and x is the input

• The graphs of linear functions are lines

• The slope of a linear function measures how much the output changes for each change of 1 in the input; in otherwords, it measures rise over run

• Slope is constant everywhere on a line

• There is only one linear equation that fits any two given points

• Rate problems are related to linear functions

• Direct variation is a kind of linear function that takes the form y = ax

Linear functions are important in mathematics, but there are a host of other mathematical functions used to model bothabstract and real-world phenomena.This session will introduce just a few of those functions. Groups: Brainstorm any functionsthat are not linear. You may want to record your comments on an overhead, flip chart, or blackboard. You can leave the listposted in the room and add to it throughout the session.


Changing + to *In Session 5, we learned that linear functions have constant first differences. That is, every time x increases by 1,y increases by a constant amount. In this section, we will come up with a description for another type of function:exponential functions. [SEE NOTE 2]

Here’s a table generated by a spreadsheet. The rule in the Output column is recursive: Start at 3 and add 10 eachtime. Here are the values in the table:

If you need a refresher on how to use a spreadsheet, see the tutorial inSession 5, page135.

To generate this table in a spreadsheet using formulas, type the number “3” in cell B2, then type “=B2+10” in cellB3. Use the “Fill Down” menu command to continue the rule to cell B7.

Remember that the value in a cell is what is displayed, but the for-mula is used to generate the value. Click on cell B4 (the cell con-taining “23”) to see the rule that generates the table: “=B3+10”.You also could have used a formula to generate the values in column A.

Because the successive outputs have a constant difference of 10,these points should all lie on a line.

Part A: Exploring Exponential Functions (40 MINUTES)

NOTE 2. Groups: Work in pairs at the computer.

Start your spreadsheet program and create a worksheet. Remember the following tips:

• Click on a cell to see the function it contains displayed in the edit line above the worksheet.

• To edit the entry in a cell, click on the cell, highlight any characters that were already there, and type over them.

• To “fill down” (when you have a function used throughout a column of cells), highlight the cell you want to use as astarting point, then drag down to the cell you want to use as your ending point. Choose the Fill Down command tofill down.

• To create the graph of a function, highlight the input and output cells and click on the “chart” button. Go through themenus, using the first column as the inputs.

For the functions in this session, you may choose to connect the points with line segments or with smooth curves. Considerthe benefits of each type of graph. A smooth curve might not make sense if the function is only defined by integer values, butit might help to see the graph more clearly.


Problem A1. Use the spreadsheet to graph this table, then verify that all the points lie on the same line. [SEE NOTE

3] [SEE TIP A1, PAGE 190]

Click on cell B3 again, and change the + to * in the output rule.The rule should now read:“=B2*10”, and you shouldsee the value “30” displayed in the cell after you enter the new formula. Now use the “Fill Down” command to copythe rule into the rest of the Output column.

If you have done this correctly, the final value in cell B7 should be 300,000.

Part A, cont’d.

NOTE 3. Remember to use the Fill Down command after changing the output rules. The graphs and tables will change auto-matically with any changes you make to the columns. Groups: After completing Problems A1-A7, discuss your responses toProblem A7.

The tendency may be to answer Problem A7 with particular numbers. For example, the first table will never contain the output2, because the table starts at 3 and then increases. Think also about what types of numbers will never appear. For example,unless you use negative inputs, the first table will never contain negative numbers, numbers less than 3, fractions, or numbersthat aren’t multiples of 10 (except 3). Justify your claims. This will push you to think more deeply about exponential functions.

Groups: Before moving on, take a few minutes to read through the chart or put up an overhead of the chart. Take a momentto work a bit with the notation. For example, if you have a rule that is y = 10x, what would you get for x = 2? For x = 4? You alsomay want to discuss 0 as an exponent. Note that this is completely optional; knowledge of 0 as an exponent is not assumedanywhere during the session.There are a couple of ways to explain the fact that any number (except 0) to the 0 power is 1. First,discuss what 20 would mean and why. Then think about the following two explanations:

Multiplying by 1 doesn’t change anything, so you can think of powers of 2 (for example) as 1 times some number of 2s.

23 = 1 × 2 × 2 × 2 (1 times three 2s)

22 = 1 × 2 × 2 (1 times two 2s)

21 = 1 × 2 (1 times one 2)

20 = 1 (1 times zero 2s)

Alternately, look at decreasing powers of 2:

23 = 8 22 = 4 21 = 2

Each power is half the previous one, so if the pattern is to continue, it must be the case that 20 = (1/2) × 2 = 1. Extending thispattern can help you find the meaning of negative exponents, as well. It should be the case that 2-1 = (1/2) × 1 = 1/2, and in factthis is what negative exponents mean.

Think about why a closed-form rule like y = 3x would give rise to a table with a constant ratio between successive outputs. Apossible explanation is that if the input increases by 1, then the product is multiplied by another 3, so each output will be 3times the previous one.


Problem A2. Describe the pattern of outputs in your spreadsheet. [SEE TIP A2, PAGE 190]

Problem A3. Graph this table. Describe the graph.

Take It Further Problem A4. Predict how much higher the input would have to be for the output to be more than 1 billion. Thenuse the spreadsheet to find the answer.

Problem A5. Let’s make another small change to the formulas on our spreadsheet and see how this changeaffects our graph. Click on cell B3 again, and change the * to / in the output rule. The rule should now read:“=B2/10”, and you should see the value “0.3” displayed in the cell after you’ve entered the new formula. Now, usethe “Fill Down” command to copy the rule into the rest of the Output column.

Describe the pattern of outputs in your spreadsheet.

Problem A6. Graph this table. Describe the graph.

Write and ReflectProblem A7. Think about these two new tables. For each table, list three numbers that will never appear in theOutput column, and then explain why they will never appear there. [SEE TIP A7, PAGE 190]

Part A, cont’d.


Introduction to Exponential FunctionsIt’s amazing how different the three tables and graphs presented in Problems A1-A7 are, and all you did waschange a “+” to a “*”, then a “*” to a “/”!

The changes we made to the spreadsheet may have seemed small, but each change made an enormous impactin the tables and graphs. When we changed the “+” to a “*”, we changed each output from a constant difference of10 to a constant ratio of 10.This created a new type of rule called an exponential function, any function where eachoutput is a constant multiple of the previous output.

Exponential functions often come up in real-world situations.The interest earned on an investment and the decayof nuclear waste are two good examples.

Before we move on, let’s take a few moments to think about exponential notation. Just as multiplication showsrepeated addition,exponents show repeated multiplication.Here are a few examples so that you can see the parallels.

You may have noticed that one example in the table above shows repeated multiplication of a fraction. Since divi-sion by a constant whole number is equivalent to multiplying by a fraction, dividing by a constant multiple alsocreates exponential functions.

A few terms are handy to know when you’re talking about exponential functions. In the equation y = bx, b is calledthe base and x is called the exponent.

So far, the exponential functions we’ve created have used recursive rules: Each output is a multiple of the last out-put. As is true with linear functions, it’s often more useful to write an exponential function using a closed-form rule.To do this, we’ll need to use exponents.

Part A, cont’d.


The spreadsheet uses a ^ symbol to make exponents. Use the table below as a guide to set up the first spread-sheet of Problem A8. After typing in the rule, you should use the “Fill Down” command to copy it to the rest of theOutput column. When you’re finished, the formulas in the table should look like this:

Problem A8. Use a spreadsheet to investigate the graphs of exponential functions. For each output rule below,create an input/output table for the rule and then graph the function. Describe the graphs and how they vary fordifferent bases, making sure to include parentheses around a fractional base when you enter these rules into aspreadsheet. [SEE NOTE 4]

a. y = 2x

b. y = (2/3)x

c. y = (3/2)x

d. y = (7/10)x

e. y = 8x

f. Make up one of your own to try.

Part A, cont’d.

NOTE 4. Create a new worksheet and then work on Problem A8. As you make your own functions, experiment with interestingor strange cases:What if you use 1x? What about using a number close to 1? Or using very large or very small numbers? In somecases, the numbers get so big that the graphs are distorted between the input points. For example, this graph was created withthe rule 100x. It does not look like a smooth, constantly increasing function. Surely it shouldn’t dip below the x-axis! Groups:Discuss why this is a mistake, and why the software might do that.

Before moving on, think about exponential functions and describe twodifferent kinds. Some exponential functions—those with a base that isgreater than 1—are increasing (bigger inputs always produce biggeroutputs, and the graph never slopes down), and others—those with abase less than 1—are decreasing (bigger inputs always produce smalleroutputs). The exception is 1, which produces a constant function thatgraphs as a line. Using numbers close to 1 as bases produce graphs thatlook like lines, but actually are not. Extending the graphs to more inputsbetter shows the behavior of the functions.

NOTE 4 cont’d. next page


Problem A9. Which of the graphs in Problem A8 were increasing? Which were decreasing? Explain how you coulddecide whether or not y = (4/5)x is increasing or decreasing without graphing it. [SEE TIP A9, PAGE 190]

Can you explain why the graph of y = (2/3)x is decreasing?

Take It FurtherProblem A10. Is there any exponential function that increases for a while, then decreases? Is there an exponen-tial function that never increases or decreases?

Part A, cont’d.

NOTE 4 CONT’D.

All exponential functions (except with a base of 1) have graphs like these:

VIDEO SEGMENT (approximate times: 4:45-6:49): You can find this segment onthe session video approximately 4 minutes and 45 seconds after the Annenberg/CPB logo. Zero the counter on your VCR clock when you see theAnnenberg/CPB logo.

In this video segment, the class discusses the graphs of y = 2x and y = (2/3)x

from Problem A8.They find rules to determine whether an exponential func-tion will be increasing or decreasing. Watch this segment to review yourwork in Part A and to get a clearer understanding of why exponential func-tions are increasing or decreasing.


A Salary SituationSuppose you are given two different options for salary at a temporary job. [SEE NOTE 5]

• Plan A: You can earn $2,000 each week, or

• Plan B: You can earn 1 penny the first week, 2 pennies the next week, 4 pennies the next week, and so on,doubling your salary each week.

The job lasts 25 weeks, and your goal is to earn the largest total salary for the 25 weeks.

Problem B1. Which salary plan would you choose and why? Write down your initial reaction.

Problem B2. Use a spreadsheet or some other method to figure out which salary plan is better.[SEE TIP B2, PAGE 190]

Problem B3. Are you surprised that the second salary grew so rapidly? Use the spreadsheet to determine howmuch money you would make from plan B in 40 weeks. [SEE TIP B3, PAGE 190]

Describe the difference in the ways salary plans A and B grow. Which one is a proportional change, and which isan absolute change? For more on this, refer back to Session 4, Part A, page 92.

Part B: Exponential Growth (40 MINUTES)

NOTE 5. In this section, we’ll explore situations that give rise to increasing exponential functions. Groups: Begin by putting upan overhead of the description of two different salary options for Problems B1 and B2. Discuss your gut reactions: Which is thebetter choice? No need to justify your answers; go instead with your first instincts.

Work with a spreadsheet or calculator to figure out the final value of each salary scenario. Remember that the most importantfactor in this comparison is the total amount earned over the 25 weeks.

A calculator can be used to create the same table as in a spreadsheet. Even if you are filling in the table by hand, there are short-cuts to using the calculator. For example, type in “.01” then “Enter.”Then type “x 2” and “Enter.” Repeatedly hitting the Enter keywill produce the list of outputs for the second salary option. Keep track of each output, and then sum the outputs at the end.For the first salary option, because the salary never changes, simply multiply the salary ($2,000) by the number of weeks (25)to get the total. Groups: When you have finished the comparison, share your results and reactions. Many will probably be sur-prised that starting with such a small amount—a penny compared with $2,000—you could end up with more than 10 timesas much total money, and in only half a year.

VIDEO SEGMENT (approximate times: 7:51-9:18): You can find this segment onthe session video approximately 7 minutes and 51 seconds after the Annen-berg/CPB logo. Zero the counter on your VCR clock when you see theAnnenberg/CPB logo.

In this video segment, onscreen participants discuss the differencesbetween salary plans A and B in Problems B1 and B2 above. In particular, theyexplore the differences between linear and exponential functions.Watch thissegment after you have worked on Problems B1 and B2. If you get stuck, youcan watch the video segment to help you.


Population GrowthMany populations—human, plant, and bacteria—grow exponentially, at least at first. In time, these populationsstart to lose their resources (space, food, and so on). Here’s an example: [SEE NOTE 6]

“Whale Numbers up 12% a Year” was a headline in a 1993 Australian newspaper. A 13-year study had found thatthe humpback whale population off the coast of Australia was increasing significantly. The actual data suggestedthe increase was closer to 14 percent!

When the study began in 1981, the humpback whale population was 350. Suppose the population has beenincreasing by about 14 percent each year since then. To find an increase of 14 percent, you could do either of thefollowing (P stands for population):

Pthis year = Plast year + (0.14) * Plast year

Pthis year = (1.14) * Plast year

Each of these is a recursive rule. The first rule says that to know this year’s population, start with last year’s popu-lation (which is Plast year), then add the population growth. Since the growth is 14 percent of last year’s population,add (0.14) * Plast year.

The second rule is used more frequently because it’s easier to calculate—it incorporates the adding in one calcu-lation. This equation shows that the second computation is equivalent to the first:

x + (0.14) * x = (1.14) * x

The second computation fits the format of an exponential function, because successive outputs have the sameratio (in this situation, the ratio is 1.14).

Part B, cont’d.

NOTE 6. The salary activity segues nicely into the population model. If a population reproduced by doubling, it would quicklyrun out of resources, even starting with a small number.Though exponential models are used for some populations, the bases(the constant multiple between outputs) is usually much closer to 1. Groups: Read through the whale problem. Spend amoment discussing why increasing by 14 percent is mathematically equivalent to multiplying by 1.14. Percents are not a focusof this session, but it’s worth spending a little time on this idea in order to understand why it is true. You can also relate it toother things the students probably know:

• If you have a 10 percent decrease, you can calculate 0.9 × (original number). This is the same as (1 - 0.1) × (originalnumber), or (original number) - 0.1 × (original number).

• To compute the final price of an item when you have to pay 10 percent sales tax, you can use 1.1 × price, which is thesame as (1 + 0.1) × price, or (price) + 0.1 × (price).

• 3 × (number) + 5 × (number) = (3 + 5) × number

• or in symbols: 3n + 5n = (3 + 5)n = 8n

• or in words: If you have 3 of something and add 5 of that thing, you end up with 8 of the thing.

These are specific cases of an important algebraic idea: the distributive property of multiplication over addition.

Here, we have: 1 × (population) + 0.14 × (population) = (1 + 0.14) × (population) = 1.14 × (population)


Problem B4. Make a table that shows the estimated whale population for the 5 years after 1981. [SEE NOTE 7]

Years Estimated After 1981 Population

0 350

1

2

3

4

5

Part B, cont’d.

NOTE 7. If working on a computer, open a new worksheet to model the situation. Groups: Work in pairs on Problems B4-B6.

Think about a strategy for calculating how long the population takes to double. The Fill Down command on the spreadsheetcan be used until a population of 700 is reached. To answer the question of whether it depends on the initial population,change that starting number in the spreadsheet and see if it doubles in the same place. The doubling time does not dependon the starting value; thus an exponent n can be found so that 1.14n = 2.

Here’s one way to see that the time to double doesn’t depend on the starting value, and it also highlights some importantalgebraic thinking.

Year P o p u l a t i o n Growth

1 1.14

2 1.30

3 1.48

4 1.69

5 1.93

6 2.19

7 2.50

8 2.85

9 3.25

10 3.71

You’re looking for a year in which 1.14 × 1.14 × 1.14x ... × 1.14 × n = 2 × n. There is an n multiplied on each side, so the onlything that could possibly make the multiple of 2 is all those 1.14s multiplied together. You just have to find the right numberof them, and the number of 1.14s only depends on the year.

This also tells you that, for example, if you get a 5 percent raise at your job every year, the number of years it takes you to dou-ble your salary is fixed, and it doesn’t depend on how much you start out earning.


Problem B5. If the whale population continues to grow by 14 percent per year, predict how many whales therewould be in 2001 (20 years after 1981). [SEE TIP B5, PAGE 190]

Take It FurtherProblem B6. How many years would it take the whale population to double if it grows at this rate? Does youranswer depend on the starting value of the whale population?

Do you think the whale population discussed in Problems B3-B6 could increase at 14 percent per year forever?Why or why not?

Part B, cont’d.

VIDEO SEGMENT (approximate times: 22:40-25:10): You can find this segmenton the session video approximately 22 minutes and 40 seconds after theAnnenberg/CPB logo. Zero the counter on your VCR clock when you see theAnnenberg/CPB logo.

In this video segment, taken from the “real world” example at the end of theSession 7 video, Mary Bachman of the Harvard School of Public Health dis-cusses the exponential model of population growth, the factors that affectthe model, and the uses of population modeling.

The population growth problem is taken from IMPACT Mathematics Course 2, developed by Education Development Center, Inc.(New York: Glencoe/McGraw-Hill, 2000), p.186.


Take It FurtherProblem B7. Look at the following toothpick pattern.The number of toothpicks needed to build each stage of thepattern is a linear function. [SEE NOTE 8]

Create a toothpick pattern in which thenumber of toothpicks you need for eachstage is an exponential function.[SEE TIP B7, PAGE 190]

Part B, cont’d.

NOTE 8. Groups: Problem B6 relates back to earlier work with toothpick patterns, but requires a bit of creativity.You may wantto have actual toothpicks available. If you have difficulty coming up with patterns that work, think of a particular case. Forexample, try to come up with a pattern that uses twice as many toothpicks at each successive stage.To maintain a pattern visu-ally, it helps to think about making copies of the shape at any stage, and arranging them in some regular way. Here are a cou-ple of possible solutions:

Groups: To wrap up this part, talk about how “growing exponentially” is used as slang in the press to mean “growing very fast,”but in fact “exponentially” has a specific mathematical definition. Think of possible definitions for “growing exponentially” inyour own words, and add exponential functions to the list of nonlinear functions started at the beginning of the session.


Square NumbersNow that we have a basic understanding of exponential functions, let’s move on to another kind of relationship.As you work through the following problems, try to get a sense of how these relationships are different from bothlinear and exponential functions.

“Square numbers” describe the numberof dots needed to make squares like theones at right. The first square number is1, the second is 4, and so on. [SEE NOTE 9]

Problem C1. Draw the next two squares in this pattern. [SEE NOTE 10]

Part C: Figurate Numbers (45 MINUTES)

NOTE 9. This section focuses on figurate numbers as a visual context for introducing and exploring quadratic functions. Theidea is to get a sense for another kind of function without worrying too much about the details. In working through these prob-lems, some people might come up with closed-form rules to describe the figurate numbers, and others might come up withrecursive or even doubly-recursive rules (using the fact that the second differences are constant). All of these approaches makeit possible to work through the problems and see how these functions are different from both the linear and exponential func-tions studied previously.

NOTE 10. Graph paper can be used to work through Problems C1-C9. If using the spreadsheet program to create the graphs,only use the data for the first six figures (or fill in the in-between entries in the table). Large jumps in inputs cause the spread-sheet graphs to distort and look less like parabolas. Since there are only a few data points for each, it might be just as easy tograph them by hand.

If you have trouble coming up with a rule for the triangularnumbers, remember that the triangles you are building look alot like the staircases built in a previous session. (In fact, the pat-tern is exactly the same.)

Groups: To wrap up this part, discuss how tables were filled in,particularly how missing inputs were found when given anoutput. Also compare rules. Some people will likely have comeup with closed-form rules and others with rules like “add theprevious output and the current input” (for triangular num-bers). They might even have extended patterns like these:


Problem C2. Fill in the table below:

Number Total Number of Dots of Dotson Side (Square

of Square Number)

1 1

2 4

3

4

5

6

100

169

Problem C3. Graph the data in your table using graph paper or a spreadsheet, then describe your graph. How isit different from the linear and exponential graphs you’ve seen? [SEE TIP C3, PAGE 190]

Problem C4. Describe a rule relating the number of dots on the side of a square (the independent variable) andthe total number of dots (the dependent variable).

Triangular Numbers“Triangular numbers” describe the number of dots needed to make triangles like the ones below. The first trian-gular number is 1, the second is 3, and so on.

Part C, cont’d.


Problem C5. On a sheet of paper, draw the next two triangles in this pattern.

Problem C6. Fill in the table on the right:

Problem C7. Graph the data in your table using graph paper or a spread-sheet, then describe your graph. How is it different from the linear andexponential graphs you’ve seen?

Problem C8. Describe a rule relating the number of dots on the side of atriangle (the independent variable) and the total number of dots (thedependent variable).

Problem C9. Describe any similarities and differences between your rules and graphs for the square and triangu-lar numbers. In a way, they both have the same kind of rule. How would you describe it? [SEE NOTE 11][SEE TIP C9, PAGE 190]

You can create “figurate numbers” for any polygon shape. Below are pictures of the first few pentagonal andhexagonal numbers.

The growth of figurate numbers is an example of a quadratic function. A quadratic function’s formula will alwaysinvolve squaring the input number: y = 3x2 + 5 is a quadratic function, and y = 3x + 5 is not. In Part D, we will explorethe formulas and properties of quadratic functions in more detail.

Part C, cont’d.

NOTE 11. Groups: Share thoughts on how the two graphs in this part were similar to each other and different from others theyhave seen. If possible, look at a picture of a parabola that includes negative inputs in order to clarify the difference in shapebetween a parabola and the exponential functions they saw earlier.

Number Total Number of Dots of Dots on Side (Triangular

of Triangle Number)

1 1

2 3

3

4

5

6

45

190


Quadratic Functions and DifferencesYou know that for linear functions, the difference between successive outputs is a constant. For exponential func-tions, the ratio between successive outputs is a constant. Is there some similar pattern for quadratic functions? Thenext few problems will help you decide. [SEE NOTE 12]

Problem D1. As described at the end of Part C, quadratic functionsinvolve squaring an input. The simplest quadratic function is simplyoutput = (input)2. Here’s the start of a table for this function, with threecolumns: input, output, and the common difference between successive outputs:

Fill in both the missing outputs and missing differences. Describe apattern in the differences. Are the differences constant? [SEE NOTE 13][SEE TIP D1, PAGE 190]

Problem D2. Add a new column to your table. In this column,put the “differences between differences,” called the second differences. What do you notice?

Part D: Quadratic Functions (25 MINUTES)

NOTE 12. In this last part of the session, we’ll look at differences between successive outputs of quadratic functions. Successiveoutputs do not produce constant differences (as with linear functions) or constant ratios (as with exponential functions), butthe differences do have a pattern nonetheless. It turns out that for quadratic functions the differences are linear, and the sec-ond differences—the differences of differences—are constant.

NOTE 13. Groups: Work on Problems D1-D4 with a partner. Share results and describe the functions created for Problem D4.Out of this work, a conjecture should emerge that the second differences of quadratic functions are constant. If there’s time,finish the session by looking for what the second difference tells you about the function. For a linear function, the differencebetween outputs is the same as the coefficient of x in the linear equation, and the same as the slope of the line. Look at sev-eral different quadratic functions and the corresponding second differences. Notice that half of the second difference is thecoefficient of the x2 term in the equation.

Looking back at the figurate numbers, for example, the square numbers had a formula y = x2. The coefficient of x2 is 1, and thesecond differences were constant 2s. The triangular numbers had a formula y = x2/2 + x/2. The coefficient of x2 is 1/2, and thesecond differences were constant 1s.

Look for the constant second differences for the pentagonal and hexagonal numbers, and use those to help find the more diffi-cult formulas for these numbers. Groups: Sharing results will allow you to collect information about quadratic functions. End thesession by adding quadratic and exponential functions to the list of nonlinear functions started at the beginning of the session.


Problem D3. In Part C, you built a table for triangular num-bers. One way to write the rule for triangular numbers is:

output = (n2 + n) / 2.

Fill in the table for this rule for triangular numbers, andlook for patterns in the first and second differences.

A quadratic function is any function that can be written as:

output = A(input)2 + B(input) + C

A, B, and C can be any number. The only exception is that A cannot equal 0—if A were 0, there would be no needfor the input to be squared!

Problem D4. Create your own quadratic function.Tabulateit and look at the differences and second differences. Whatseems to be true about quadratic functions?

Quadratic functions are studied in detail in physics and calculus, because the path of a quadratic functiondescribes anything falling under the force of gravity.

Part D, cont’d.

VIDEO SEGMENT (approximate times: 15:42-16:49): You can find this segmenton the session video approximately 15 minutes and 42 seconds after theAnnenberg/CPB logo. Zero the counter on your VCR clock when you see theAnnenberg/CPB logo.

This video segment shows how to create a table of common differences andsecond differences in the equation y = x2. Watch this segment after you’vecompleted Problem D2. If you get stuck on the problem, you can watch thevideo segment to help you.


SummaryIn this session, we’ve seen some of the differences between linear functions, quadratic functions, and exponentialfunctions and learned about their practical uses. On a piece of graph paper, graph each of the following threefunctions, which have similar equations but very different graphs:

• y = 2x

• y = x2

• y = 2x

Problem D5. Which of these is an exponential function? Which is aquadratic function? Which is a linear function? For each, explainhow you know.

Part D, cont’d.

Try It Online!

This problem can be explored online asan Interactive Activity. Go to the Patterns,Functions, and Algebra Web site atwww.learner.org/learningmath and findSession 7, Part D, Summary.


You’ve already seen examples of expo-nential growth.When you have an expo-nential function for which the constantmultiple between outputs is less than 1,you have a decreasing function. Insteadof exponential growth, you have expo-nential decay. Here’s an example:

The brightness of a light can bedescribed with a unit called a lumen. Acertain type of mirror reflects 3/5 of thelight that hits it. Suppose a light of 2,000lumens is shined on a series of severalmirrors.

Problem H1. Complete the table to indicate how much light would be reflected by each of the first 3 mirrors.(Mirror 0 represents the original light.)

Mirror ReflectedNumber Light (Lumens)

0 2,000

1

2

3

Problem H2. One mirror in the series reflects about 12 lumens of light. Which mirror number is it? How did youfind your answer?

Problem H3. Which mirror number reflects about 1/10 the original amount of light? Does it depend on the start-ing amount of light?

Homework


Problem H4. The graph to the right shows the amount of lightreflected by another series of mirrors. The amount of lightreflected by this type of mirror is different from the amountreflected by the other mirrors you investigated. The intensity ofthe light being shone on the first mirror is also different.

a. What is the intensity of the light being shone on thefirst mirror?

b. How much light was reflected by the first mirror?

c. What fraction of light does the first mirror reflect?

d. Do the other mirrors reflect the same fraction of light?That is, does this graph show exponential decay?

Problem H5. Suppose you have the function y = (1/5)x.

a. As the value of the inputs increases, what happens to the outputs?

b. Is this an exponential growth function, or an exponential decay function?

c. Will you ever get 0 as an output? Explain your answer.

d. Will you ever get negative numbers as outputs? Explain your answer.

e. How does your answer to (c) relate to Problems H1-H4?

Homework, cont’d.

Homework problems are taken from IMPACT Mathematics Course 3, developed by Education Development Center, Inc. (New York:Glencoe/McGraw-Hill, 2000), pp. 175-176.

Session 7: Tips 190 Patterns, Functions, and Algebra

Part A: Exploring Exponential FunctionsTIP A1. If you’re having trouble making a graph, see the section in the spreadsheet tutorial about graphs, page 138.

TIP A2. Use the recursive rule to describe the table.

TIP A7. Can you find any types of numbers that will never appear in the Output column?

TIP A9. The recursive rule for y is very helpful here. What changes when x grows by 1?

Part B: Exponential GrowthTIP B2. When using a spreadsheet, create three columns, one for the week number, one for plan A, and one for planB. When creating the rule for plan B, remember that the salary is doubling each week, and that the first week’ssalary is one penny, not one dollar.

TIP B3. To make a column wider, click and hold down on the right edge of the column heading letter, then dragthe mouse to the right.

TIP B5. See if you can do this problem without extending your table from Problem B4. You would need to use aclosed-form rule to “jump” directly to the 2001 answer.

TIP B7. Think of a pattern that would require you to double the number of toothpicks you use at each step. Thatwould form an exponential function, because successive outputs have a common ratio (2).

Part C: Figurate NumbersTIP C3. Try to be specific about how this graph is different from an exponential graph. What is the key property ofan exponential graph?

TIP C9. Think about how you would go from one output to the next. What changes? Can you describe thesechanges with a rule?

Part D: Quadratic FunctionsTIP D1. Remember,“constant” means the number remains the same: 5, 5, 5, ... . A pattern may or may not be a con-stant pattern.

Tips

Patterns, Functions, and Algebra 191 Session 7: Solutions

Part A: Exploring Exponential Functions

Problem A1. It is the line y = 10x + 3.

Problem A2. The outputs are: 3, 30, 300, 3,000, 30,000, and 300,000.In each, the input number is the number of 0s in the output. Thereis a constant ratio between each term. A recursive rule is to multi-ply by 10 each time to get the next output number.

Problem A3. The graph is increasingly steep as the input valuegrows, and successive values of the output dwarf all the others.Thefirst output value, 3, is virtually impossible to see if we scale thegraph to include the last output value, 300,000.

Problem A4. The input only needs to grow by 4 more to reach 1billion. The input of 9 gives an output of 3 billion.

Problem A5. The outputs are: 3, 0.3, 0.03, 0.003, 0.0003, and0.00003. In each, the input number is the number of digits to theright of the decimal point. Again, there is a constant ratio. A recur-sive rule is to divide by 10 each time to get the next output number.

Solutions

Session 7: Solutions 192 Patterns, Functions, and Algebra

Problem A6. The graph is similar to the graph in Problem A3,but in reverse: the graph is increasingly flat, and successive out-puts are increasingly close to zero.The first output, 3, dwarfs thelast output, which is 100,000 times smaller.

Problem A7. The output table can produce neither negative numbers nor 0 unless the initial value is negative or0. This happens because we are multiplying and dividing, step by step, by a positive number, and the only way toyield a negative or 0 would be to start with a negative or 0. There are many other positive numbers that do notappear in the tables of Problems A2 and A5, including any number not starting with 3.

Problem A8. The graphs for y = 2x, y = (3/2)x, and y = 8x are increasing. The graph for y = 8x is the steepest graph,while y = 2x is steeper than y = (3/2)x. The graphs for y = (2/3)x and y = (7/10)x are decreasing. The graph fory = (2/3)x decreases a little faster than y = (7/10)x, but the graphs are much closer together than the increasinggraphs. All five graphs pass through the point (0, 1).

Solutions, cont’d.

a.

b.


Problem A9. The functions y = 2x, y = (3/2)x, and y = 8x were increasing.The functions y = (2/3)x and y = (7/10)x weredecreasing. When the base is larger than 1, successive outputs will be larger, because we are multiplying to createa larger number. Additionally, the larger the base (greater than 1), the steeper the graph will be. When the base isbetween 0 and 1, successive outputs will be smaller, since we are multiplying by a number less than 1. Addition-ally, the smaller the base (closer to 0), the more quickly the graph will advance toward 0.

Problem A10. No exponential function can reverse direction, because the ratio between successive outputsalways remains constant. There is an exponential function that never increases or decreases, y = 1x. The functiony = 0x is constant when x is positive, but is not defined if x is negative or 0.


d.

e.

c.


Part B: Exponential GrowthProblem B1. Most initial reactions are to choose plan A.

Problem B2. The total for 25 weeks of plan A, at $2,000 per week, is $50,000. Plan B, which starts at 1 penny anddoubles each week, totals $335,544.31, including $167,772.16 in the final week.

Problem B3. After 40 weeks, the grand total at this job would be $10,995,116,277.75, including just under $5.5 billion in the last week. Before the end of the year, you would exhaust the total money supply of the United States.

Problem B4. Here is the completed table.


Years Estimated After 1981 Population

0 350

1 399

2 455

3 519

4 591

5 674


Problem B5. You can extend the spreadsheet, but an easier way to obtain an answer would be to calculate 350 * (1.14)20 = 4,810 whales. One way to estimate this is to observe that the population seems to be nearly dou-bling in 5 years (doubling would be 700 whales). This means that we could expect the population to nearly dou-ble 3 more times in a total of 20 years. This doubling would be 350 >> 700 >> 1,400 >> 2,800 >> 5,600, so anestimate is roughly 5,000 whales.

Problem B6. It takes a little more than 5 years. And no, it doesn’t matter what the initial population value was,because the ratio is created only by multiplying by 1.14 five times.

Problem B7. One possible pattern is to build, at each stage, a triangle oftoothpicks with sides twice as long as the previous triangle.

Part C: Figurate NumbersProblem C1.

Problem C2. The completed table is on the right.

Problem C3. It is neither a linear nor an exponential graph. Itssuccessive outputs do not have the same ratio; therefore, it can-not be an exponential graph. It is certainly not a straight line,because successive outputs do not have the same difference, soit cannot be a linear graph.

Problem C4. The rule is O = n2, where O is the output, the totalnumber of dots, and n is the number of dots on the side of asquare. A recursive rule is Dn = Dlast + (2n - 1).


Number Total Number of Dots of Dots on Side (Square

of Square Number)

1 1

2 4

3 9

4 16

5 25

6 36

10 100

13 169


Problem C5.

Problem C6. The completed table is on the right.

Problem C7. As with Problem C3, the graph does not demonstrateexponential behavior, because successive terms do not have the sameratio. It’s not a linear graph, either, because it is certainly not a straightline. Actually, the graphs of the table of square numbers and the table oftriangular numbers look pretty similar.

Problem C8. There is more than one answer, but one is O = (n)(n + 1) / 2.The recursive form is easier to find: Dn = Dlast + n, because n new dots areadded in the nth triangle.

Problem C9. Both closed-form rules involve multiplying n by itself atsome point, and both recursive rules involve adding something linear ton. Compare this to linear functions, which have only a single use of thevariable in their closed-form rules, and a constant in the recursive rule.

Part D: Quadratic FunctionsProblem D1. Here is the completed table:

The differences are not constant, but they increase by 2 foreach successive difference.


Number Total Number of Dots of Dots on Side (Triangular

of Triangle Number)

1 1

2 3

3 6

4 10

5 15

6 21

9 45

19 190


Problem D2. Here is the completed table:

The difference between outputs forms a linear func-tion, and the second differences are constant (in thiscase, they’re always equal to 2).

Problem D3. Here is the completed table:

As before, the first differences form a linear function,and the second differences are constant.

Problem D4. You should find that with any quadratic function, the first differences will always form a linear function, and the second differences will always be constant.

HomeworkProblem H1. Here is the completed table. One formula is L = 2,000(0.6)m.

Mirror Reflected Number Light (Lumens)

0 2,000

1 1,200

2 720

3 432

Problem H2. The best way to do this, without using more advanced math such as logarithms, is to continue following the table. This is particularly easy with a spreadsheet. The 10th mirror will reflect about 12 lumens.

Problem H3. We’d be looking for the mirror that reflects closest to 200 lumens, which is the 5th mirror (156lumens).The 4th mirror reflects 259 lumens; this does not depend on the starting amount of light.This means thatthe 6th mirror reflects about 1/10 the light of the 1st mirror, the 7th mirror reflects 1/10 the light of the 2nd mirror, and so on.



Problem H4.

a. The initial intensity is 1,000 lumens.

b. The 1st mirror reflects 700 lumens.

c. This fraction is 700 / 1000 = 7/10, or 70 percent.

d. Yes, the 2nd mirror appears to reflect about 7/10 of 700 lumens, which is 490 lumens. The 3rd mirrorreflects 7/10 of 490 lumens, which is 343 lumens. This pattern appears to continue indefinitely at a con-stant ratio, so it is an exponential decay situation.

Problem H5.

a. The outputs keep getting smaller, but remain positive, because at each stage a positive number isdivided by 5.

b. This is an exponential decay function, because successive outputs are getting smaller, and the base isbetween 0 and 1.

c. Zero will never be an output, even though the outputs will become increasingly close to zero. This hap-pens because the numerator remains 1, no matter what the value of x is, while the denominator becomesincreasingly larger as x increases.

d. Because two positive numbers are used in the division, a negative number can never result.

e. In Problems H1-H4, this implies that even after 100 or more mirrors, some light will still be reflected,although the amount of light reflected will become increasingly small.


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Session 7 Nonlinear Functions

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