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Section 10 Exponential Functions We now turn our attention back to exponential functions. In order to work with these functions effectively, it is important that we know how exponents can be manipulated. This, in turn, requires an understanding of roots. So, this is where we begin. Part 1 Roots Example 1: Complete the following table without using a calculator. (You don’t need to complete the gray cells.)

4 8 16 32 64

4 -8 16 -32 64

9 27 81 9 -27 81

16 64 16 -64

Example 2: Determine the missing numbers in each of the following equations.

(a) The missing number is 3.

(b) There are two possible missing numbers, 3 and -3.

(c) There are two possible missing numbers, 8 and -8.

(d) The missing number is 2.

(e) There is no real number which, when squared, equals -16. (A real number is a number that you would find on a number line. When you square a real number, the result is either positive or zero.)

The missing numbers you were looking for in the last example are called ‘roots’. For example, 3 is a ‘cube root’ of 27, 8 and -8 are ‘square roots’ of 64, and -2 is a ‘fifth root’ of -32. In general, an nth root of a number a is the number which, when raised to the nth power, equals a. In other words,

.

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Example 3: Find indicated roots of the following numbers. (a) square roots of 36 Since , 6 and -6 are both square roots of 36. (b) cube roots of 8 Since , 2 is a cube root of 8. (c) 4th roots of 81 Since , 3 and -3 are both 4th roots of 81. (d) cube roots of -64 Since , -4 is a cube root of -64.

Notation for Roots: We denote the nth root of a number a by . The symbol is called a

radical symbol. • For even-numbered roots (i.e. square roots, fourth roots, etc.),

refers only to the positive root. • For square roots, we write rather than .

Example 4: Find the indicated roots:

(a) since (b)

Both 5 and -5 are square roots of 25. However, refers to the positive square root. So, .

(c)

Both 2 and -2 are fourth roots of 16. However, refers to the positive fourth root. So, .

(d)

does not exist (as a real number) since it is not possible to square a real number and get -81.

(e)

since

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Part 2 Properties of Exponents We begin with the five most basic properties of exponents:

Motivating Example Property

Example 5: Simplify the following expressions using the properties above.

(a)

(b)

(c)

(d)

(e)

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The next three properties might appear less intuitive than the properties on the previous page, but they are the logical consequences of these properties.

Motivating Example Property

On the one hand, . On the other hand, .

So, .

On the one hand, . On the other hand,

. So, .

. So, .

Example 6: Simplify each of the following expressions using the above properties.

(a)

(b)

(c)

(d)

(e)

Example 7: Use the exponent properties that you’ve learned today to simply the following expressions.

(a)

(b)

5

(c)

(e)

(f)

Example 8: (a) Write using exponents.

(b) is not equal to an integer (i.e. like -3, -2, -1, 0, 1, 2, 3, …) . Determine a

decimal value (approximation) using your graphing calculator.

Type “9^(1/4)” into your calculator. (The parentheses are necessary! Why do you think this is?) The numerical value returned is 1.73205.

Part 3 Return to Exponential Growth and Decay Example 9: Suppose a population grows 6% every 3 years. Determine the annual percentage growth. The most obvious thing to try is to take . Let’s check to see if this is correct.

If the population is actually growing by 2% annually, then it must be growing by a factor of 1.02 each year. Use this factor to complete the table. Don’t round your values!

Year 0 1 2 3

Population 500 510 520.2 530.604 Now, did the population increase by 6% over these 3 years? (i.e. Did it grow by a factor

of 1.06?) 530.604/500 = 1.061208 So, the population actually increased by 6.1208%!! Clearly, dividing 6% by 3 years was not the correct way to determine the annual

percentage growth. So, what is the correct way?

Let R represent the annual growth factor. It follows that . Why is this? Well, to obtain a 3-year factor, you would need to multiply by the 1-year factor three times. So, then, what must

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R equal? From our earlier work, . If we evaluate this using a calculator, we find that R = 1.0196128… So, according to this calculation, the population would be growing by approximately 1.96% each year, which is a little less than 2%. Let’s recalculate the table on the previous page to make sure that this factor is correct.

Year 0 1 2 3 Population 500 509.806… 519.805… 530

Now, if we divide 530 by 500, we see that the population grew by a factor of 1.06 (6%)

over the three years. The results from this example are summarized below. If a quantity is growing or decaying exponentially by a factor of c over n-unit time periods, then the quantity is growing by a factor of over 1-unit time periods. Example 10: Assume the quantities below grow or decay exponentially. Determine the growth/decay factors and the corresponding percentage growth/decay over 1-unit time periods.

(a) Quantity grows by 45% every 10 years. The 10-year growth factor is 1.45. So, the annual growth factor = . In other words, the quantity grows 3.8% annually.

(b) Quantity decays by 21% every 3 weeks. The 3-week decay factor is 0.79. So, the weekly decay factor = which corresponds to a decay of 8% each week.

(c) Quantity grows by 26% every 10 days. The 10-day growth factor is 1.26. So, the daily growth factor = which corresponds to a growth of 2.3% each day.

Example 11: Shown below are some exponential functions. Use exponent properties to write the functions in the form .

(a)

Note: This shows that growing by a factor of 8 during 3-unit periods of time is equivalent to growing by a factor of 2 over 1-unit periods of time.

(b)

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This shows that growing by a factor of 81 during 4-unit periods of time is equivalent to growing by a factor of 3 over 1-unit periods of time.

(c)

This shows that growing by a factor of 1.57 during 7-unit periods of time is equivalent to growing by a factor of 1.06656 over 1-unit periods of time.

Example 12: In 1999, global consumption of bottled water was approximately 26 billion gallons.

By 2004, consumption reached 41 billion gallons. (Bottled Water: Pouring Resources Down the Drain. Earth Policy Institute (2006), http://www.earth-policy.org)

(a) By what percentage did consumption of bottled water increase during this period?

The 5-year growth factor is 41/26 = 1.577. Thus, consumption grew 57.7% during this period.

(b) What was the average annual percentage growth during this period?

Since the 5-year growth factor is 1.577, the average annual growth factor is , which corresponds to 9.54% average annual growth during this period.

(Note: We never assumed that consumption of bottled water was growing exponentially. The real annual percentage growth probably varies from year to year. The 9.54% is therefore only an average measure of growth during this period.)

(c) If we assume that bottled water consumption is growing exponentially, determine two

different function equations describing this growth. Let t represent time in years since 1999 and let y represent consumption (in billions of gallons). We begin with the basic equation for an exponential function . Recall that b represents the initial amount. Since t = 0 in the year 1999, b = 26 billion gallons. In determining values for c and m, we now have a choice: Either choose c = 1.577 and m = 5, or choose c = 1.0954 and n = 1. So, the two equivalent equations that describe the growth are

or .

(d) Complete the following input/output table for the two functions above.

t 0 1 2 3

26 28.48 31.197 34.172

26 28.48 31.197 34.174 Notice that the outputs of the two functions are effectively the same.

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(e) If bottled water consumption continues to grow exponentially in the future, what is the projected global consumption in the year 2010? In the year 2010, t = 11 years. When t = 11, . So, if bottled water consumption is growing exponentially, then the projected global bottled water consumption is approximately71 billion gallons in the year 2010.

Example 13: Determine the equations of each of the following exponential functions described

below.

(a) Outputs decay 10% over 4-unit periods and graph has a vertical intercept equal to 7. We begin with the algebraic form . The 10% decay over 4-unit periods tells us that c = 0.90 and n = 4. Since b represents the vertical intercept, b = 7. Thus, the equation is

.

(b) Graph passes through the points (0,1) and (3,4). The two points listed represent input/output pairs for this function: From the table, we determine that the growth factor c equals 4/1 = 4 over 3-unit periods, i.e. n = 3. Since b represents the value of the output variable corresponding to an input value of 0, b = 1. Thus, the equation is

. (c) Graph passes through the points (1,10) and (3,4).

The two points listed represent input/output pairs for this function: From the table, we determine that the growth factor c equals 4/10 = 0.4 and n = 2. Thus, the equation is

. To find b, we will use one of the input/output pairs. Since y = 10 when x = 1, it follows that

.

We solve this equation for b. Dividing both sides of this equation by gives

which is approximately 15.81. Thus, the resulting equation is

.

x 0 3 y 1 4

x 1 3 y 10 4

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(d) Graph passes through the points (5, 2) and (2, 1). The two points listed represent input/output pairs for this function:

From the table, we determine the factor for this function. If you think of the values of x as times, then x = 2 is earlier than x = 5. So, the factor c will equal 2/1 = 2 for n = 3. Consequently, the equation is

To find b, we again use one of the input/output pairs. Let’s choose (5, 2). Substituting this pair into our equation yields . Dividing both sides of this equation by

gives . Consequently, the equation is

.

x 5 2 y 2 1

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Section 10 Homework Assignment

1. Simplify each of the following expressions using only your knowledge of roots and the exponent properties from class. Don’t use a calculator!

(a) (b) (c) (d) (e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

2. Use your graphing calculator to determine a decimal value (approximation) for each of

the following expressions. (a) (b)

3. Shown below are some exponential functions. Use exponent properties to write the functions in the form . What do your results say about the growth/decay factors of each of the functions? (a) (b)

(c)

(d)

4. Consider the exponential function given by the equation .

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(a) Complete the following input/output table by hand (i.e. not using a calculator).

t 0 1 2 3 4 y

(b) Write this function as .

5. Determine equations for the exponential functions described below.

(a) Outputs grow by 20% over 6-unit periods and the graph passes through the point (0,10).

(b) Outputs decay by 30% over 10-unit periods and the graph has a vertical intercept equal to 18.

(c) The graph passes through the points (0,20) and (4, 15). (d) The graph passes through the points (2,4) and (5, 6). (e) The graph passes through the points (6,10) and (9,9).

6. Determine equations describing the exponentially growing and decaying quantities given

below. (a) The initial amount is 42 and quantity decays 15% every 5 years. (b) The initial amount is 124 and quantity grows by a factor of 1.12 every 4 months. (c) The initial amount is 200 and the quantity doubles every 6 weeks.

7. For each of the situations in exercise 6, determine the percentage growth/decay per unit

time (e.g. per year, per month, etc.) by using exponent properties. 8. This exercise again refers back to exercise 6.

(a) Use the equation for the function in part (a) to determine the amount after 20 years. (b) Use the equation for the function in part (b) to determine the amount after 1 year. (c) Use the equation for the function in part (c) to determine the amount after 26 weeks.

9. In this exercise, we look at per capita bottled water consumption in Italy (highest per

capita consumer in the world) and the U.S. (highest total consumption in the world). The data is shown in the table below:

Bottled water consumption (per

capita) in liters 1999 2004

Italy 154.8 183.6 U.S. 63.6 90.5

(Source: Bottled Water 2004: U.S. and International Statistics and Developments, Beverage Marketing Corporation (2005)) (a) Determine the percentage growth in per capita bottled water consumption for the Italy

and the U.S. during this period. (b) If we assume that consumption in both countries grew by constant annual percentages

during this period, determine these percentages. (c) Determine the average rate of change in per capita consumption for both countries

during this period. Be sure you give units for your answer.

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(d) Estimate per capita consumption in both countries for each year during this period, assuming that consumption grows by a constant annual percentage.

Year 1999 2000 2001 2002 2003 2004

Italy 154.8 183.6 Bottled water consumption (per capita) in liters U.S. 63.6 90.5

(e) Estimate per capita consumption in both countries for each year during this period,

assuming that consumption grows at a constant rate of change. Year 1999 2000 2001 2002 2003 2004

Italy 154.8 183.6 Bottled water consumption (per capita) in liters U.S. 63.6 90.5

10. Input/output tables for one linear function and one exponential function are given below.

For the exponential function, determine the constant growth/decay factor over 1-unit periods. For the linear function, determine the constant rate of change. Then, determine equations for these two functions. (a)

x 1 3 6 y 2.4 0.864 0.18662

(b)

x 1 3 6 y 7.25 9.75 13.5

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Section10 Answers to Selected Homework Exercises

1. (a) (b) (c)

(f)

(g)

(h)

(i)

2. (a)

(b)

3. (a)

Because , growing by a factor of 16 during 2-unit periods of time is equivalent to growing by a factor of 4 over 1-unit periods of time.

(b)

€

y = 80 27( )t3 = 80 3( )t

Because , growing by a factor of 27 during 3-unit periods of time is equivalent to growing by a factor of 3 over 1-unit periods of time.

5. (a)

(c)

(d)

6. (b)

(c)

7. (b) 2.9% monthly (c) 12% weekly

8. (b) 174.211

(c) 4031.75

9. (a) Italy 18.6%; U.S. 42.3% (b) Italy 3.5%; U.S. 7.3%

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(c) Italy 5.76 liters/year; U.S. 5.38 liters/year (d)

Year 1999 2000 2001 2002 2003 2004 Italy 154.8 160.1 165.7 171.5 177.4 183.6 Bottled water consumption

(per capita) in liters U.S. 63.6 68.2 73.2 78.6 84.3 90.5 (e)

Year 1999 2000 2001 2002 2003 2004 Italy 154.8 160.6 166.32 172.1 177.8 183.6 Bottled water consumption

(per capita) in liters U.S. 63.6 69.0 74.4 79.7 85.1 90.5