Boise State, 4 Eponential functions: week 3 Elementary Education As we have seen, eponential functions describe events that grow (or decline) at a constant percent rate, such as placing capitol in a savings account. Eponential functions will have a role to play in many areas outside of math: In business, eponential functions play a role in borrowing and lending; investing; discounting; and growth of companies. In science, eponential functions play a role in eamining population changes, growth of plants and modeling earthquakes. Many of these equations need additional mathematics topics such as series and sequences learned later in mathematics. As this is the first eperience for many into eponential functions we will spend today focusing on developing a deeper understanding of eponential functions, increasing the learning opportunities when eponential functions reappear. Let s first consider an eample of investing $. at % a year at simple interest. How much money will be invested at the end of each year? Complete the t-chart, where t represents time and A represents the amount. Then graph the resulting points. t A 3 4 5 6 7 8 9 $,. $,. $,. $,33. $,464. $,6.5 3 4 5 6 7 8 9 We have discussed rate of change over the semester. How does the rate of change differ between years and and between years 9 and? How does the rate of change transform as time changes? Do we have any graphs that behave in this manner that we have already studied?
Boise State, 4 $,6. $,4. $,. $,. $,8. $,6. $,4. $,. $,. 3 4 5 6 7 8 9 Here are the data points graphed. It is generally very difficult to freehand graph and develop a good picture. Does looking at our graph change any of your answers above? In the graph you were asked to complete, we left the y ais for you to fill in. In our graph, we started the y ais at $. How is this different from your graph? Typically in math classes the and y ais point of intersection is the point (,), but outside of math classes the intersection point is based on what best fits the problem, as we have demonstrated above. How does changing the ais affect the graph?
Boise State, 4 Now consider how we obtained the data points. t A Finding A from Previous Finding A from $. (verify that this is correct $,. = $, $,. = $,.. = $, $,. = $,.. = $, 3 $,33. = $,.. = $, 4 $,464. = $,33.. = $, 5 $,6.5 = $,464.. = $, 6 $,77.56 = $,6.5. = $, 7 $,948.7 = $,77.56. = $, 8 $,43.59 = $,948.7. = $, 9 $,357.95 = $,43.59. = $, $,593.75 = $,357.95. = $, Verify that the last column is correct by actually calculating the results. Use the last column to write an equation that compares the amount of money invested (A) to the amount of time the money has been invested (t). Use your equation to find the amount of money invested in years 5, 4 and 5. The equation you have created has the variable in eponent. For this reason, the equation is called an eponential function. As you work with eponential functions, you will recognize that they are significantly different from any equation we have already worked with. These equations are important and used any time we are discussing events that change at a constant percent rate of change.
Boise State, 4 The simplest form of an eponential function is ( ) where or. Note: the letter is used in an eponential function as it represents the base of the eponential function. Let s start with, let Graph the three functions below. - - - - - - Describe the characteristics that all three graphs have in common. Did you catch the relationship of the value of ( ) One relationship that is generally not seen by most students when they begin to study eponential functions is that as the answer, ( ) gets closer and closer to the -ais but never reaches it. This type of behavior is called asymptotic behavior. And it is said that these three functions are asymptotic to the line. The notation is read as goes to negative infinity. The arrow means goes to. The sentence as goes to negative infinity implies that we are considering what happens to the graph as moves further and further to the left on the -ais. So we have an idea as to what happens if. Let s know consider when. Graph the following functions. ( ) ( ) ( ) - - - - - -
Boise State, 4 Describe the characteristics of these three functions; include a discussion on when and on asymptotic behavior. A critical question that is very important for net week: Are eponential functions one-to-one? (One-to one means that for each input value there is eactly one output value and for each output value there is eactly one input value.) Recall that the eponent only acts on the symbol immediately before it, Thus in ( ), the eponent only acts on the base,. (another way to look at this is ( ) ( ) ( ). Hopefully one realizes why we ignore the unnecessary symbols. The last two graphing activities were designed to look at values of where or. These are the only values of that are acceptable in an eponential function. Let s eplain why cannot be negative. Consider the function ( ), where, we then have ( ) ( ). Since is a variable which can become any real number then it is possible that. When we have ( ) ( ). As we know the square root of negative two has no real answer. Thus, there are many times that an eponential function will break down with no answer so that we define the eponential function to be the statement, an eponential function is ( ) where or, so that negative numbers are not allowed. This leaves two values for left to discuss. We do not allow to equal zero or one. Why not? To answer this question, graph both ( ) ( ) and ( ) ( ). Describe the graph and eplain what function should be used to describe these functions. With a basic understanding of an eponential function, let s add another constant so that we are now looking at ( ). How does multiplying the basic eponential function by a constant affect the function? To find this out, graph the three functions below and describe the characteristics. Also, play close attention to how the value of affects the problem. ( ) ( ) 5( ) - - - - - -
Boise State, 4 The effect is: For many eponential functions the variable is time (we typically replace in this case). The value of the function at is what we are currently referring to as. In this type of situation, we call the initial value. Before moving on, let s discuss growth and decay problems with a constant percent rate of change. The value of the base has two sets of values: ) ). When, the graph is going to increase as it did in the first set of graphs. In this case we say that the problem has a constant percent growth. When, the graph decreases. In this case, we say that the problem has a constant percent decay. We have been working with compound interest formula ( ) The base of this function is ( ) and the initial value is. Mathematicians prefer simplicity whenever possible. Compound interest can be approimated using a continuous interest formula. Continuous interest implies that we are constantly compounding. This formula is: Where is still the amount at time is still the principle amount in decimal form (or the initial amount started with), still the interest rate and is still the time. So what is is called the natural base and is an irrational number with To see how close these two formulas are, compute the following interest problems accurately using the compound interest formula and then compute the approimate answer using the continuous formula. (Note that continuous compounding is a slight over estimate. is Most calculators have an so that you can just use this key rather than remembering that When using the key in this problem, make sure that you use parentheses, so that both the rate and time are in the eponent. Enter rate and time as follows: ( )
Boise State, 4 ) Initial principle = $, rate =6%, time = years. Compounding = monthly ) Initial principle = $3, rate =8%, time = years. Compounding = quarterly 3) Initial principle = $, rate =7%, time = years. Compounding = monthly