Exponential functions

Exponential Functions

More Mathematical Modeling

Internet Technology

• The Internet is growing faster than all other technologies that have preceded it.

• Radio existed for 38 years before it had 50 million listeners.

• Television took 13 years to reach that mark.

• The Internet crossed the line in just four years.

Internet Traffic

• In 1994, a mere 3 million people were connected to the Internet.

• By the end of 1997, more than 100 million were using it.

• Traffic on the Internet has doubled every 100 days.

• Source: The Emerging Digital Economy, April 1998 report of the United States Department of Commerce.

Exponential Functions

• A function is called an exponential function if it has a constant growth factor.

• This means that for a fixed change in x, y gets multiplied by a fixed amount.

• Example: Money accumulating in a bank at a fixed rate of interest increases exponentially.

Exponential Functions

• Consider the following example, is this exponential?

x y

5 0.5

10 1.5

15 4.5

20 13.5

Exponential Functions

• For a fixed change in x, y gets multiplied by a fixed amount. If the column is constant, then the relationship is exponential.

x y

5 0.5

10 1.5 1.5 / 0.5 3

15 4.5 4.5 / 1.5 3

20 13.5 13.5 / 4.5 3

Exponential Functions

• Consider another example, is this exponential?

x y

0 192

1 96

2 48

3 24

Exponential Functions

• For a fixed change in x, y gets multiplied by a fixed amount. If the column is constant, then the relationship is exponential.

x y

0 192

1 96 192 / 96 0.5

2 48 96 / 48 0.5

3 24 48 / 24 0.5

Other Examples of Exponential Functions

• Populations tend to growth exponentially not linearly. • When an object cools (e.g., a pot of soup on the

dinner table), the temperature decreases exponentially toward the ambient temperature.

• Radioactive substances decay exponentially.• Viruses and even rumors tend to spread exponentially

through a population (at first).

Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent.

xxf 2

Let’s look at the graph of this function by plotting some points. x 2x

3 8 2 4 1 2 0 1

-1 1/2 -2 1/4 -3 1/8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

2

121 1 f

Recall what a negative exponent means:

BASE

xxf 2

xxf 3

Compare the graphs 2x, 3x , and 4x

Characteristics about the Graph of an Exponential Function where a > 1 xaxf

What is the domain of an exponential function?

1. Domain is all real numbers

xxf 4

What is the range of an exponential function?

2. Range is positive real numbers

What is the x intercept of these exponential functions?

3. There are no x intercepts because there is no x value that you can put in the function to make it = 0

What is the y intercept of these exponential functions?

4. The y intercept is always (0,1) because a 0 = 1

5. The graph is always increasing

Are these exponential functions increasing or decreasing?

6. The x-axis (where y = 0) is a horizontal asymptote for x -

Can you see the horizontal asymptote for these functions?

All of the transformations that you learned apply to all functions, so what would the graph of look like?

xy 232 xy

up 3

xy 21up 1

Reflected over x axis 12 2 xy

down 1right 2

xy 2

Reflected about y-axis This equation could be rewritten in a different form: x

xxy

2

1

2

12

So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be decreasing. It will have the same domain, range, intercepts, and asymptote.

These two exponential functions have special names.

Exponential Factors

• If the factor b is greater than 1, then we call the relationship exponential growth.

• If the factor b is less than 1, we call the relationship exponential decay.

Exponential Growth

• Exponential growth occurs when some quantity regularly increases by a fixed percentage.

• The equation for an exponential relationship is given by

• y = Abx

• where A is the initial value of y when x = 0, and b is that growth factor.

• An example of the equation of the last relationship above is simply y = $100 (1.05)x.

Exponential Functions

• If a quantity grows by a fixed percentage change, it grows exponentially.

• Example: Bank Account – Suppose you deposit $100 into an account that earns

5% annual interest.

– Interest is paid once at the end of year.

– You do not make additional deposits or withdrawals.

– What is the amount in the bank account after eight years?

Bank Account

year Amount Interest Earned

ConstantGrowthFactor

0 $100.00 = $100.00 * 0.05 = $5.00  

1 $100.00 + $5.00 = $105.00 = $105.00 * 0.05 = $5.25 = $105.00 / $100.00 = 1.05

2 $105.00 + $5.25 = $110.25 = $110.25 * 0.05 = $5.51 = $110.25 / $105.00 = 1.05

3 $110.25 + $5.51 = $115.76 = $115.76 * 0.05 = $5.79 = $115.76 / $110.25 = 1.05

4 $115.76 + $5.79 = $121.55 = $121.55 * 0.05 = $6.08 = $121.55 / $115.76 = 1.05

5 $121.55 + $6.08 = $127.63 = $127.63 * 0.05 = $6.38 = $127.63 / $121.55 = 1.05

6 $127.63 + $6.38 = $134.01 = $134.01 * 0.05 = $6.70 = $134.01 / $127.63 = 1.05

7 $134.01 + $6.70 = $140.71 = $140.71 * 0.05 = $7.04 = $140.71 / $134.01 = 1.05

8 $140.71 + $7.04 = $147.75   = $147.75 / $140.71 = 1.05

Exponential Growth Graph

Exponential Decay

• Exponential Decay occurs whenever the size of a quantity is decreasing by the same percentage each unit of time.

• The best-known examples of exponential decay involves radioactive materials such as uranium or plutonium.

• Another example, if inflation is making prices rise by 3% per year, then the value of a $1 bill is falling, or exponentially decaying, by 3% per year.

trvalueinitialvaluenew 1__

Exponential Decay: Example

• China’s one-child policy was implemented in 1978 with a goal of reducing China’s population to 700 million by 2050. China’s 2000 population is about 1.2 billion. Suppose that China’s population declines at a rate of 0.5% per year. Will this rate be sufficient to meet the original goal?

Exponential Decay: Solution

The declining rate = 0.5%/100 = 0.005

Using year 2000 as t = 0, the initial value of the population is 1.2 billion.

We want to find the population in 2050, therefore, t = 50

New value = 1.2 billion × (1 – 0.005)50

New Value = 0.93 billion ≈ 930 million

Example of Radioactive Decay• Suppose that 100 pounds of plutonium (Pu) is

deposited at a nuclear waste site. How much of it will still be radioactive in 100,000 years?

• Solution: the half-life of plutonium is 24,000 years. The new value is the amount of Pu remaining after t = 100,000 years, and the initial value is the original 100 pounds deposited at the waste site:

• New value = 100 lb × (½)100,000 yr/24,000 yr

• New value = 100 lb × (½)4.17 = 5.6 lb• About 5.6 pounds of the original amount will still be

radioactive in 100,000 years.

Exponential Decay Graph

Exponential Equations and Inequalities

Ex: All of the properties of rational exponents apply to real exponents as well. Lucky you!

Simplify:

3232 555 Recall the product of powers property, am an = am+n

Ex: All of the properties of rational exponents apply to real exponents as well. Lucky you!

Simplify:

10

2525

6

6)6(

Recall the power of a power property, (am)n= amn

Suppose b is a positive number other

than 1. Then b x1 b x 2 if and only if

x1 x2 .

The Equality Property for Exponential

Functions

• This property gives us a technique to solve• equations involving exponential functions.• Let’s look at some examples.

Basically, this states that if the bases are the same, then we can simply set the exponents equal.

This property is quite useful when we are trying to solve equations

involving exponential functions.

Let’s try a few examples to see how it works.

Basically, this states that if the bases are the same, then we can simply set the exponents equal.

This property is quite useful when we are trying to solve equations

involving exponential functions.

Let’s try a few examples to see how it works.

This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal.

If au = av, then u = v

82 43 x The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something?

343 22 xNow we use the property above. The bases are both 2 so the exponents must be equal.

343 x We did not cancel the 2’s, We just used the property and equated the exponents.

You could solve this for x now.

Let’s try one more:8

14 x The left hand side is 4

to the something but the right hand side can’t be written as 4 to the something (using integer exponents)

We could however re-write both the left and right hand sides as 2 to the something.

32 22 x

32 22 xSo now that each side is written with the same base we know the exponents must be equal.

32 x

2

3x

Check:

8

14 2

3

8

1

4

1

2

3 8

1

4

12 3

Example 1:

32x 5 3x 3(Since the bases are the same wesimply set the exponents equal.)

2x 5 x 3x 5 3

x 8

Here is another example for you to try:

Example 1a:

23x 1 21

3x 5

The next problem is what to do when the bases are not the

same.

32x 3 27x 1

Does anyone have an idea how

we might approach this?

Our strategy here is to rewrite the bases so that they are both

the same.Here for example, we know

that 33 27

Example 2: (Let’s solve it now)

32x 3 27x 1

32x 3 33(x 1) (our bases are now the sameso simply set the exponents

equal)2x 3 3(x 1)

2x 3 3x 3

x 3 3

x 6

x 6

Let’s try another one of these.

Example 3

16x 1 1

32

24(x 1) 2 5

4(x 1) 54x 4 5

4x 9

x 9

4

Remember a negative exponent is simply another way of writing a fraction

The bases are now the sameso set the exponents equal.

By now you can see that the equality property is

actually quite useful in solving these problems.

Here are a few more examples for you to try.

Example 4: 32x 1 1

9

Example 5: 4 x 3 82x 1

The Base “e” (also called the natural base)

To model things in nature, we’ll need a base that turns out to be between 2 and 3. Your calculator knows this base. Ask your calculator to find e1. You do this by using the ex button (generally you’ll need to hit the 2nd or yellow button first to get it depending on the calculator). After hitting the ex, you then enter the exponent you want (in this case 1) and push = or enter. If you have a scientific calculator that doesn’t graph you may have to enter the 1 before hitting the ex. You should get 2.718281828

Example for TI-83

xxf 2

xxf 3

xexf