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? 13.5 20 4.5 15 1.5 10 0.5 5 y x

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. 13.5 4.5 1.5 0.5 y 13.5 / 4.5 4.5 / 1.5 1.5 / 0.5 3 20 3 15 3 10 5 x

Exponential Functions Consider another example, is this exponential? 24 3 48 2 96 1 192 0 y x

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. 24 48 96 192 y 48 / 24 96 / 48 192 / 96 0.5 3 0.5 2 0.5 1 0 x

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. Let’s look at the graph of this function by plotting some points. x 2 x 3 8 2 4 1 2 0 1 -1 1/2 -2 1/4 -3 1/8 Recall what a negative exponent means: BASE 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7

Compare the graphs 2 x , 3 x , and 4 x Characteristics about the Graph of an Exponential Function where a > 1 What is the domain of an exponential function? 1. Domain is all real numbers 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? up 3 up 1 Reflected over x axis down 1 right 2

Reflected about y -axis This equation could be rewritten in a different form: 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 = Ab x 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 = $147.75 / $140.71 = 1.05 $140.71 + $7.04 = $147.75 8 = $140.71 / $134.01 = 1.05 = $140.71 * 0.05 = $7.04 $134.01 + $6.70 = $140.71 7 = $134.01 / $127.63 = 1.05 = $134.01 * 0.05 = $6.70 $127.63 + $6.38 = $134.01 6 = $127.63 / $121.55 = 1.05 = $127.63 * 0.05 = $6.38 $121.55 + $6.08 = $127.63 5 = $121.55 / $115.76 = 1.05 = $121.55 * 0.05 = $6.08 $115.76 + $5.79 = $121.55 4 = $115.76 / $110.25 = 1.05 = $115.76 * 0.05 = $5.79 $110.25 + $5.51 = $115.76 3 = $110.25 / $105.00 = 1.05 = $110.25 * 0.05 = $5.51 $105.00 + $5.25 = $110.25 2 = $105.00 / $100.00 = 1.05 = $105.00 * 0.05 = $5.25 $100.00 + $5.00 = $105.00 1 = $100.00 * 0.05 = $5.00 $100.00 0 Constant Growth Factor Interest Earned Amount year

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.

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 Equations and Inequalities

Ex: All of the properties of rational exponents apply to real exponents as well. Lucky you! Simplify: Recall the product of powers property, a m  a n = a m+n

Ex: All of the properties of rational exponents apply to real exponents as well. Lucky you! Simplify: Recall the power of a power property, (a m ) n = a mn

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.

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 a u = a v , then u = v The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something? Now we use the property above. The bases are both 2 so the exponents must be equal. 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: 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. So now that each side is written with the same base we know the exponents must be equal. Check:

Example 1: (Since the bases are the same we simply set the exponents equal.) Here is another example for you to try: Example 1a:

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

Example 2: (Let’s solve it now) (our bases are now the same so simply set the exponents equal) Let’s try another one of these.

Example 3 Remember a negative exponent is simply another way of writing a fraction The bases are now the same so 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.

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 e 1 . You do this by using the e x button (generally you’ll need to hit the 2nd or yellow button first to get it depending on the calculator). After hitting the e x, 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 e x . You should get 2.718281828 Example for TI-83

Exponential functions

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Exponential functions