Section 6 Introduction to Linear and Exponential Functions Having been introduced to the ideas in unit analysis, linear and exponential change, and functions, we are now ready to begin our exploration of two of the most important classes of functions: linear functions and exponential functions. We will begin with linear functions, but first, we need to extend the idea of unit analysis one step further. Part 1 Applying Unit Analysis to Sums and Differences The underlying principle behind unit analysis can be illustrated by the following senseless statements:

6 horses = 6 turtles 18 bathtubs – 10 erasers 5 shirts + 7 computers

What is wrong with each of these statements? What important lessons about units do you learn from these statements? From the first statement, we learn that we can only equate two quantities if they have the same units. From the second and third statements, we learn that we can only add and subtract quantities that have the same units. Now, try out the lessons that you’ve just learned. Determine which of the following statements is insensible. Make corrections which will fix the unit problem(s).

(a)

For the units of every term in this equation to be the same, the last term has to have units of people. The problem is that the weeks and days don’t cancel. So, the last term in the equation does not have units of people. To fix the problem, we could insert a conversion factor of 7 days per week into the last term. This would give the following equation

Now, the weeks and days cancel and the last term has units of people.

(b) where C has units of dollars, M has units of miles, 39.93 has units of dollars and 0.80 has units of dollars per mile If we rewrite this equation with all of the units, the equation becomes

Now, we can see that all terms in this equation have units of dollars. So, the equation appears to be set up correctly.

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Part 2 Linear Functions Example 1: Suppose the present amount of some substance is 56 milligrams, and the amount

2 hours later is 44 grams. Let’s assume that the amount is decreasing at a constant rate. Then, according to last week’s material, the amount is decreasing linearly. Our goal in this example is to create a function that describes this linear decay.

Let x represent the number of weeks since the present week (independent variable), and let y represent the amount of substance present (dependent variable). First of all, what is the rate of change of this substance? The total change over the 2 hours is -12 grams. So, the average rate of change is -12/2 = 6 grams per hour. Use the rate of change to complete the following table:

x (weeks) 0 2 3 4 x y (grams) 56 44 56 - 6(3) = 38 56 - 6(4) = 32 56 – 6(x)

So, what is the equation of the function that describes this linear decay? y = 56 – 6x Let’s use unit analysis to check to make sure that there aren’t any obvious problems with this equation. If we insert all of the units into the equation y = 56 – 6x, we arrive at the following:

Since all of the terms in this equation have units of grams, there are no immediately obvious problems with this function.

Use the function to determine how much substance you have after 9 weeks. We set x = 9 weeks. Then y = 56 - 6(9) = 2. So, there will be 2 mg after 9 weeks. Make a sketch of the graph of the function y = 56 – 6x using the point (0, 56) as a starting point and knowledge that the constant rate of change is -6 to draw other points. (Don’t look at the table above to draw the graph!)

From the graph above, you see that each 1-unit increase in the input variable x causes a 6 unit decrease in the output variable y. In addition, you see that the y-intercept is y = 56.

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The last example of a linearly decreasing quantity motivates the following formulation of a linear function: A linear function is a function of the form where x represents the independent variable (input), y represents the dependent variable (output), m represents the total change in the value of y for each 1-unit increase in x and is called the slope of the line, and b represents the y-intercept (vertical intercept). If x represents time and y represents a quantity, then m represents the constant rate of change of the quantity over time and b represents the quantity at time x = 0. We call x = 0 the initial time and b the initial quantity. The graphs of linear functions are (straight) lines:

m > 0 m < 0 In the last example, where the linear function was y = 56 – 6x , what were the values of m and b? m = -6 and b = 56 Example 2: Suppose a population is growing by 2 people per day and the current population is 8 people.

(a) Is the population growing linearly or exponentially? Since the population is growing at a constant rate of change of 2 people per day, it is growing linearly.

(b) So, a function of the form should describe this growth over time. Explain

what the variables x and y would represent, and determine the values of m and b. x represents: the number of days since the present day y represents: the number of people m = 2 since the rate of change is 2 people per day b = 8 since the current population is 8 people

(c) What is the equation of the resulting function?

Example 3: Interpret each of the values of m given below.

(a) m = 7.9 For each 1-unit increase in the value of x (input), the value of y (output) increases by 7.9 units.

(b) m = -34.6 For each 1-unit increase in the value of x (input), the value of y (output) decreases by 34.6 units.

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Example 4: The graph of a linear function is shown below. Use it to answer the following questions. The letters T and P represent the independent and dependent variables.

(a) What is the value of b?

b = vertical intercept = 6 (b) What is the value of m?

Each 1-unit increase in the input T causes the output P to decrease by 1 unit. So, m = -1. (c) What is the equation of this function?

P = mT + b = (-1)T + 6 or P = 6 – T

Part 3 Exponential Functions Example 5: Suppose a population is growing by 2.6% every 4 years. Currently, the size of the

population is 6000 individuals. Since the population is growing by a constant percentage over equal periods of time, it is growing exponentially. Our goal is to determine a function that expresses this exponential growth.

First, note that the 4-year growth factor for this population is 1.026. So, the 8-year factor is

, the 20-year factor is , and the 32-year factor is . The population sizes in these years are then determined as follows:

Population after 4 years: Population after 8 years: Population after 20 years: Population after 32 years:

In this problem, the two variables are the time and the population size. Time represents the independent variable (input) and population size represents the dependent variable (output). Let’s introduce letter names for these variables: Let x represent the number of years since the current time. Let y represent the population size. What is the size of the population after x years? Based on our work above, the x-year growth factor ought to be . Consequently,

the population size after x years ought to be .

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Use the function to predict the size of the population after 64 years.

When x = 64 years, . So, the population is predicted to reach 9047 in 64 years. What is the y-intercept of this function? To find the y-intercept, we set the input x equal to 0: This is strange! What does mean? We know that the present population is equal to 6000. So, y ought to equal 6000 when x equals 0. Have we made a mistake? It turns out that we have made no mistake. In order for to equal 6000, has to equal 1. In a few weeks, you will see that must equal 1 for any positive number c. So, in summary,

and, therefore, the y-intercept is y = 6000.

This example of an exponentially growing quantity leads us to the following formulation of an exponential function: An exponential function is a function of the form where x represents the independent variable (input), y represents the dependent variable (output), b represents the y-intercept, and n represents a particular period of time over which the output grows/decays by a factor of c. From a practical perspective, b represents the initial quantity. Because exponential functions describe exponentially growing or decaying quantities, the graphs of these functions display the same trends as we saw in Week 5:

c < 1 c > 1 Example 6: The data below are exponentially decreasing over time. Determine a function which will describe/model this decay.

Time (in minutes) 0 30 60 90 Amount (in grams) 50 45 40.5 36.45

We represent exponential growth and decay by using an exponential function . Let’s take the independent variable x to be the time (in minutes) and the dependent variable y to be the amount. Then, b represents the y-intercept or the initial amount which equals 50. Since the amount is decreasing by a factor of 45/50 = 0.90 every 30 minutes, c = 0.90 and n = 30. So, the function that we need is .

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Use the function to determine the amount of substance remaining after 2 hours.

The variable x represents time in minutes. So, 2 hours is equivalent to x = 120 minutes. Substituting this into the function equation gives . So, 32.805 grams of substance will remain after 2 hours.

How much is remaining after 5 hours?

Five hours is equivalent to x = 300 minutes. Substituting this into the function equation gives . So, 17.434 grams of substance will remain after 5 hours.

Part 4 Practice Working with Linear and Exponential Functions Example 7: Suppose that each of the following functions describes the amount A of some

quantity after t units of time. In each case, give the initial amount of the quantity and discuss the growth or decay of the quantity over time. Time and amount units are given in each case. Then, make a rough sketch of what the graph would look like.

(a) where t is measured in hours and A is measured in grams This linear function describes a linearly decreasing quantity whose initial amount is 10.3

grams and is decreasing by 4.96 grams each hour.

(b) where t is measured in years and A is measured in millions of people This exponential function describes an exponentially growing population whose initial

size is 3.6 million people and is increasing by a factor of 1.009 (or by 0.9%) every 2 years.

(c) where t is measured in weeks and A is measured in kilograms This exponential function describes an exponentially decreasing quantity whose initial

amount is 15 kilograms and is decreasing by a factor of 0.45 (or by 55%) each week.

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(d) where t is measured in minutes and A is measured in cents This linear function describes a linearly growing amount of money. The initial amount of

money is 6 cents and the amount increases by 2 cents each minute.

Example 8: Shown below are the graphs of a linear and an exponential function. In each case, the variable x represents the independent variable and y represents the dependent variable. (You can think of x as representing time and y as representing a quantity.) For the linear function, determine the slope m and the y-intercept b. For the exponential function, determine the factor c over some n-unit period of time and the y-intercept b. Then, determine the equations of these functions.

(a) (b)

(a) This is the graph of a linear function. Notice that the y-intercept is 10. So, b = 10. Also, note that each 2-unit increase in x causes a 10-unit increase in y. So, the rate of change is 10/2 = 5. In other words, if the amount increases by 10 units over every 2-unit period of time, then the amount increases at the rate of 5 units over each 1-unit period of time. So, m = 5. Consequently, the equation of this function is y = 5x + 10.

(b) This is the graph of an exponential function. Notice that the y-intercept is 100. So, b = 100. If we make an input/output table for the function, we’ll be able to see what the factor equals:

x 0 10 20 y 100 50 25

Notice that the output (amount) decreases by a constant factor of 0.5 over 10-unit periods (since 50/100 = 25/50 = 0.50). So, n = 10, c = 0.50, and the resulting function is given by .

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Example 9: According to your telephone plan, you pay a fixed monthly service charge of $15.95 and you are charged an additional 10 cents per minute anytime you make a call.

(a) As you use more and more minutes each month, is the total amount you will have to pay

growing linearly or exponentially? Since the cost grows by a constant rate of change of $0.10 per minute, it is growing

linearly. (b) Determine the algebraic representation (i.e. the equation) for the function relating the

number of minutes used x (independent variable) and total monthly charge y (dependent variable). Since monthly cost grows linearly with the number of minutes used, we use the linear function form . Recall that b represents the value of y when the input x is equal to 0. In the context of this problem, b represents the total monthly cost when 0 minutes have been used. Thus, b = 15.95 dollars.

Recall that m represents the rate of change in the total cost. If we are measuring time in minutes and cost in dollars, then the rate of change would measure the change in the total cost over each minute. Since the cost increases by 0.10 dollars per minute, m = 0.10. So, the resulting function is .

(d) Use unit analysis to confirm your equation in part (c). Writing our equation with all of the units displayed gives the

following:

Since all of the terms in the equation have units of dollars, there are no obvious problems with this equation.

(e) Use your function to complete the following cost table:

Minutes Used 50 minutes 100 minutes 200 minutes Total Monthly Cost 15.95 + 0.1* 50

= $20.95 15.95 + 0.1*100

= $25.95 15.95 + 0.1*200

= $35.95 Example 10: The table below shows per capita meat consumption in the U.S. in the years 1990

and 2002. Year 1990 2002

Consumption (kg per person) 112.8 124.8 (a) Suppose consumption is increasing linearly. Determine an equation describing/modeling

the growth. If the growth is linear over time, then the correct representation to begin with is

where x represents time (in years) and y represents the per capita consumption (in kg per

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person). Since the earliest time at which we are given information is 1990, let’s measure time relative to 1990. In other words, we’ll let x represent the time in years since 1990. So, we have the following input/output table:

Year 1990 2002

x 0 12 y 112.8 124.8

We need to determine values for m and b.

• Since b represents the value of y corresponding to x = 0, it follows that b = 112.8. • Since m represents the yearly rate of change of per capita consumption,

In other words, the per capita meat consumption is increasing at a rate of 1 kg per person each year.

So, the resulting algebraic representation for this function is .

(b) Suppose now that consumption is increasing exponentially. Determine an equation describing/modeling the growth. If the growth is now exponential, then the correct representation to begin with is .

We need to determine the values of b, c and n:

• Since b represents the value of y corresponding to x = 0, it follows that b = 112.8. • Since c represents the constant factor over n-unit time periods, it follows that

and n = 12 years.

So, the resulting algebraic representation is .

(c) Use the linear and exponential functions from parts (a) and (b) to make projections for per

capita meat consumption in the years 2010 and 2020:

Year 2010 2020 x 20 30

y (linear function) 112.8 + 20 =132.8

112.8 + 30 = 142.8

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y (exponential function)

In other words, if you assume that per capita meat consumption grows at a constant rate of 1 kg per person per year, then the projected levels of consumption in the years 2010 and 2020 are 132.8 kg and 142.8 kg respectively. If, on the other hand, you assume that per capita meat consumption grows by 10.6% every 12 years, then the projected levels of consumption in the years 2010 and 2020 are 133.4 kg and 145.1 kg respectively.

(d) Use your graphing calculator to sketch graphs of the two functions in parts (a) and (b) over

a 40-year time period beginning with 1990. When you enter the equation of the exponential function, don’t forget to put parentheses around the exponent! What is a sensible viewing window to choose? Look at the table above and on the previous page to help you. xmin = 0 xmax = 40 ymin = 100 ymax = 160 With these chosen values, the graphs are displayed as follows:

(e) Locate the points on the graphs above that correspond to the per capita meat consumption projections in the years 2010 and 2020. Based on the graphs above, what levels of consumption are projected for the year 2030? The year 2030 corresponds to x = 40 on the graphs. The linear function will project consumption at approximately 152 kg per person, while the exponential function will project consumption at approximately 158 kg per person.

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Section 6 Homework Assignment

1. The algebraic representations of some linear and exponential functions are given below. Determine which functions are linear and which are exponential. For the linear functions, determine the values of m and b. For the exponential functions, determine the values of b, c and n.

(a) (b)

(c) (d)

(e) (f)

2. Suppose that each of the following functions describes the amount A of some quantity

after t units of time. In each case, give the initial amount of the quantity and discuss the growth or decay of the quantity over time. Write in complete sentences. Time and amount units are given in each case. Then, make a rough sketch of what the graph would look like. Don’t forget to label your axes!

(a) where t is measured in minutes and A is measured in cents

(b) where t is measured in years and A is measured in people (c) where t is measured in weeks and A is measured in grams (d) where t is measured in seconds and A is measured in milliliters

3. Determine equations for the functions described in the following scenarios. The input

and output variables are specified for you. (a) A population is growing 2% annually. The current size of the population is 98,000.

t = input = number of years since the current year p = output = population size

(b) St. Paul currently has about 6 inches of snow on the ground. It is expected to snow heavily over the next day averaging about 2 inches per hour. t = input = number of hours since the current hour I = output = total number inches of accumulated snow

(c) A radioactive substance is decaying about 18% every 4 days. Currently, there are 120 milligrams of this substance. t = input = number of days since the current day r = output = amount of radioactive substance remaining

(d) A money market account (imagine a U.S. Savings Bond) is currently worth $500 and carries an annual interest rate of 5%. t = input = number of years since the current year V = output = total value of the account

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4. Use your functions from exercise 3 to answer the following questions. Write your answer in a complete sentence. (a) What is the size of the population in 20 years? (b) What is the accumulated snowfall after 5 hours? (c) How many milligrams of this substance remain after 10 days? (d) How much is the money market account worth after 40 years?

5. The slope m and the y-intercept b for 3 different linear functions are given below. Use

this information to draw a graph of each function and determine 3 points on the graph. (a) m = 4 and b = 1 (b) m = -1 and b = 7 (c) m = 3 and b = -2

6. Consider a linear function y = m x + b. Determine the slope m in each of the following

scenarios. (a) Whenever the value of x increases by 1 unit, the value of y increases by 4 units. (b) Whenever the value of x increases by 1 unit, the value of y decreases by 3 units. (c) Whenever the value of x increases by 4 units, the value of y decreases by 8 units. (d) Whenever the value of x increases by 2 units, the value of y increases by 5 units.

7. The world’s population grows by about 225,000 people per day. Currently, there are 6.8

billion individuals. (Source: The New Atlas of Planet Management) Let x represent the number of years since the current year, and let y represent the population of the world in billions of people. For example, 6.8 billion people would correspond to y = 6.8, and the year 2008 would correspond to x = 0. (a) Based on the information above, is the world’s population growing linearly or

exponentially? (b) Suppose we want to find the equation for the function describing world population

growth over time. Use unit analysis to see why the equation y = 225,000 x + 6.8

can not be correct. (c) Determine the yearly rate of change of the population, and give the rate in billions

of people per year. (For example: If the rate of change were found to be 456,700,000 people per year, you’d write it as 0.4567 billion people per year.)

(d) Determine a correct equation for the function describing this growth over time. (e) If the population continues to grow at a rate of 225,000 people per day, what will be

the projected population in 2025? Use your function from part (d).

8. In 1980, the annual per capita consumption of a particular beverage was 51 liters. By 1994, this amount grew to 62 liters. (a) Let x represent the variable time in years and let y represent per capita beverage

consumption in liters. What year is a sensible choice for time x = 0? (b) Determine the average rate of increase in consumption during this period.

Remember to give the units. Interpret your answer in one sentence. (c) Determine the factor and percentage by which consumption increased during this

period.

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(d) Let’s assume that consumption is growing linearly over this period. Determine an equation for the function describing/modeling this growth.

(e) If we assume that consumption of this beverage is growing exponentially, determine an equation for the function describing/modeling this growth.

(f) What do each of your functions in (d) and (e) predict for consumption in 2008?

9. In 1991, the number of internet servers in the United States was approximately 1 million. By 2004, the number of servers had reached 280 million. (Source: The New Atlas of Planet Management) (a) Let x represent the variable time in years and let y represent the number of internet

servers in millions of servers. What year is a sensible choice for time x = 0? (b) Determine the average rate of increase in servers during this period. Remember to

give the units. Interpret your answer in one sentence. (c) Determine the factor and percentage by which consumption increased during this

period. (This is the largest percentage that you have probably ever seen!) (d) Let’s assume that the number of servers is growing linearly over this period.

Determine an equation for the function describing/modeling this growth. (e) Use unit analysis to confirm your equation in part (d). (f) If we assume that the number of servers is growing exponentially, determine an

equation for the function describing/modeling this growth. (g) What is the projected number of internet servers in the year 2015? Answer this

question using both the linear function and the exponential function. 10. Shown below are the graphs of 4 different functions. Determine equations for each

function. The letter names of the independent and dependent variables that you are to use are shown on each of the axes. Each graph has points that can be easily determined without having to guess approximate values. Use these points to help you! You can think of the input values as times and the output values as amounts, if that helps you.

(a) (b)

(c) (d)

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11. Determine the rate of change, and the horizontal and vertical intercepts of the following

linear functions. Use this information to sketch graphs of these functions by hand. (a) (b) (c) (d)

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Section 6 Answers to Selected Homework Exercises

1. (a) linear; m = 54.2; b = -6 (b) exponential; b = 304; c = 2.03; n = 1

2. (a) This function is linear with m = 12.6 and b = 150. So, the initial cost is 150 cents and the cost is increasing by 12.6 cents each minute. The graph is shown below:

(b) This function is exponential with b = 340, c = 1.065 and n = 9. So, the initial

population size is 340 people and it is increasing by a factor of 1.065 (or by 6.5%) every 9 years.

3. (a)

(b)

4. (a) After 20 years, the population will be 145,623 individuals. (You type ‘98000(1.02)^20’ on your calculator.)

(b) The accumulated snowfall after 5 hours is 16 inches. 5. (a) The graph is shown below. Three points on the graph are (0, 1), (1, 5), (2, 9).

6. (a) m = 4 (c) The total change in y when x is increased by 4 units is -8. So, m = -8/4 = -2.

7. (a) Because the population is growing at a constant rate of 225,000 people per day, it is

growing linearly. (b) Choose x = 0 to be the first time at which you have information. So, in this

situation, x = 0 should represent the year 2008.

Each term in this equation should have units of billion people. The first term on the right side of the equation does not have these units. There are two problems. First, there is no conversion factor to cancel the days and years. The second problem is that 225,000 is measured in units of people, rather than billion people.

(c) The yearly rate of change is found as follows:

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In other words, the world’s population is growing by 82,125,000 people per year. This is equivalent to approximately 0.082 billion people per year.

(d) The correct equation is . (e) The projected population of the world in 2025 is billion

people.

9. In 1991, the number of internet servers in the United States was approximately 1 million. By 2004, the number of servers had reached 280 million. (Source: The New Atlas of Planet Management) (a) Let 1991 be the year corresponding to x = 0. (b) The number of internet servers increased by 279 million during this 13-year period.

In other words, the number of servers increased at an average rate of 21.46 million servers each year.

(c) The number of internet servers increased by a factor of 280/1 = 280 during this period. This corresponds to a 27,900% increase!! (Can you believe that?)

(d) (e) Inserting units into the function in part (d) yields the following:

Note that both of the terms on the right side have units of “million servers” which, in turn, match the units on the left side.

(f) (g) What is the projected number of internet servers in the year 2015? Answer this

question using both the linear function and the exponential function. The year 2015 is represented by x = 24. Substituting x = 24 into the linear and exponential functions gives projections for the number of servers in the year 2008. For the linear function, y = 516.04 million servers. For the exponential function, y = 32,948.1 million servers (i.e. 32.948 billion servers).

10. (a)

(b)

11. (a) The rate of change equals 3. The vertical intercept is f(0) = -9. The horizontal

intercept is t = 3. The graph is shown below:

(b) The rate of change equals -2. The vertical intercept is b(0) = 16. The horizontal

intercept is u = 8. The graph is shown below:

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