Chapter 8 Notes SN AA U2C8

Size: px

Start display at page:

Download "Chapter 8 Notes SN AA U2C8"

Darleen Gardner
5 years ago
Views:

1 Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of their graphs. Another tpe of graph that deals with eponents is that of eponential functions. An eponential growth function is one of the form = ab where a is some constant, b is some constant greater than one, and is some variable. (We will discuss eponential deca functions in the net section.) To see what eponential growth functions look like, let s tr graphing = 3. To do so, let s make a table with values that fit it Notice that as ou get more positive values for, the number keeps getting bigger and bigger, toward infinit. Also, as ou get more and more negative numbers for, the number is approaching zero. (The value of will never be negative since taking a positive number to an power will never result in a negative number.) This means the following is true: Since the graph will never move beond the line = 0 (since that is a limit), the line = 0 is a horizontal asmptote, a line the graph approaches as one moves awa from the origin. If we graph these points, we get a graph that looks like the following: Notice that the graph has all positive and negative -values. Also notice that the graph can have all -values greater than zero. This means the following is true: Domain: Range: The -ais ( = 0) is an asmptote of the graph. The graph passes through the -ais at (0, ), which is the same as (0, a). This point is the -intercept. We could to choose to graph a variation of this graph. For eample, let s graph = 3. page SN AA U2C8

2 Notice that as ou get more positive values for, the number keeps getting smaller and smaller, toward negative infinit. Also, as ou get more and more negative numbers for, the number is approaching zero. (The value of will never be positive because of the negative in front of the eponential epression.) This means the following is true: Since the graph will never move beond the line = 0 (since that is a limit), the line = 0 is again a horizontal asmptote. If we graph these points, we get a graph that looks like the following: Notice that the graph has all positive and negative -values. Also notice that the graph can have all -values less than zero. This means the following is true: Domain: Range: The -ais ( = 0) is an asmptote of the graph. The graph passes through the -ais at (0, ), which is the same as (0, a). This point is the -intercept. We can also work on other aspects of these graphs. For eample, what happens when one adds or subtracts numbers from the eponents, or what would happen when one adds or subtracts from the entire function? Eample : Graph the function = State the domain and range. Before finding points, realize some of the transformations we have done before in Chapter 7 will appl here. Because we are subtracting from the value, the normal graph of 2 will be shifted to the right b unit. Because we are adding 4 to the entire function, the graph will be shifted up b 4 units. Because we are multipling the eponential function b a factor of 2, it will be stretched verticall b a factor of 2. page 2 SN AA U2C8

3 Now, let us find some points Notice that as ou get more positive values for, the number keeps getting bigger and bigger, toward infinit. Also, as ou get more and more negative numbers for, the number is approaching four. (It would normall approach zero ecept ou are adding 4 to that value now.) This means the following is true: Since the graph will never move beond the line = 4 (since that is a limit), the line = 4 is a horizontal asmptote. If we graph these points, we get a graph that looks like the following: Notice that the graph has all positive and negative -values. Also notice that the graph can have all -values greater than 4. This means the following is true: Domain: Range: Eponential growth functions can be used in man different fields, from biolog concerning cell growth to finance dealing with compound interest. Compound interest can be calculated the following wa: A P r n nt where r is the annual rate of interest (epressed as a decimal), n is the times per ear the interest is compounded, P is the starting principal (amount or mone with which one starts), and A is the final amount after t ears. NOTE: Note that r n is the growth factor (what is inside the parentheses). Also, the percent increase is whatever is being added to the in parentheses as a percent. For eample, if r/n was 0.43, the percent increase would be 43%. Eample 2: From 97 to 995, the average number of transistors on a computer chip can be modeled b n = 2300(.59) t where t is the number of ears since 97. #: Identif the initial amount, the growth factor, and the annual percent increase. page 3 SN AA U2C8

4 #2: Estimate the number of transistors on a computer chip in 998. We can also use the compound interest formula to determine how much we should invest to get some amount of mone. Eample 3: You want to have $2500 after 2 ears. Find the amount ou should deposit if the account pas 2.25% annual interest compounded monthl. We learned about eponential growth in which the base b of the eponential function was greater than one. This was because multipling b an number larger than will make a number bigger. If ou multipl b a number less than but greater than 0 (so that 0 < b < ), ou have what is called an eponential deca function. Function: Eponential growth functions: Eponential deca functions: Eample 4: Tell whether the function represents eponential growth or eponential deca. #: f () : #2: f () 87 : Graphing these kinds of functions are like graphing the eponential growth functions we did in the last section. Eample 5: Graph the function 2 3. State the domain and range. page 4 SN AA U2C8

5 Domain: Range: Eample 6: 2 Graph the function = (0.25) + 3. State the domain and range Domain: Range: Eponential deca functions can be used in man different fields, from biolog concerning cell growth to finance dealing with decomposition and depreciation. Depreciation and deca can be modeled b the following equation: a r t where r is the percent decrease (epressed as a decimal), t is the amount of ears (or other time period as indicated), a is the starting amount, and is the final amount. NOTE: Note that r is the deca factor (what is inside the parentheses). Also, the percent decrease is whatever is being subtracted from in parentheses as a percent. For eample, if r/n was 0.43, the percent decrease would be 43%. page 5 SN AA U2C8

6 Eample 7: You bu a new car for $22,000. The value of the car decreases b 2.5% each ear. Write an eponential deca model for the value of the car. Use the model to estimate the value after 3 ears. During last term and this term, we have learned about man different special kinds of numbers:, i, integers, etc. In this section, we will learn about a special number e, which is called a natural base or the Euler number (a number discovered b Leonhard Euler). What is e? If we were to graph the eponential function approach a specific number as gets larger and larger. where is a positive number, it will 0 = = = = = = As ou can see, there is a limit of the function as approaches infinit: lim e Note that e is a number greater than one, so taking e to a positive value of will result in a function that is an eponential growth function. Also notice that if e is taken to a negative value of, this is an eponential deca function. In other words, f() = ae n is an eponential growth function if n represents a positive number. f() = ae n is an eponential deca function if n represents a negative number. Eample 8: Tell whether the function is an eample of eponential growth or eponential deca. #: f () 5e 3 : #2: f () 4 e2 : page 6 SN AA U2C8

7 Properties of eponents also appl to the number e. Eample 9: Simplif the epression. #25: 00e : #29: 3 27e 6 : You can evaluate e to different powers on our TI-83 Plus as well. How?. Press the ellow 2 nd button on our calculator and press the LN button. (Notice above the LN button it reads e.) 2. Put in the power of e. Close the parentheses following this. 3. Press ENTER and get our answer. Eample 0: Simplif the epression. #33: e 3 : #37: e /4 : Also realize graph functions of e is just like graphing eponential growth and deca functions. Eample : Graph the function 3 e2. State the domain and range Domain: Range: page 7 SN AA U2C8

8 One application of the Euler number deals with what is called continuousl compounded interest. The more frequentl interest is compounded, the more quickl the amount in an account increases. The formula for continuousl compounded interest uses the number e. Formula for continuousl compounded interest: A = Pe rt A: amount in account P: principal (starting amount) r: rate of annual interest (epressed as a decimal, so 9.2% would be 0.092) t: time in ears Eample 2: How much mone will ou have after 0 ears if ou invest $500 at 9.5% compounded continuousl? Like ou did with lines, parabolas, and other equations, if ou know some points on a graph of an eponential growth or deca function, ou can create equations for eponential growth and deca functions. Just remember: an eponential function (growth or deca) has the form of = ab. Eample 3: Write an eponential function = ab for a graph that includes the points (4, 8) and (6, 32). Section 8-3: Logarithmic Functions as Inverses We know how to find 4 2. You simpl take 2 4 s and multipl them together to get 6. However, if ou know that 4 = 0, how do ou find? You find the value of using what is called a logarithm. Definition of logarithm with base b: If b and are positive numbers and b, the logarithm of with base b is shown b log b = and means that: log b = if and onl if b =. Eample : Rewrite the equation in eponential form. #: log 5 5 : #2: log : page 8 SN AA U2C8

9 Note that there are special logarithm values. If b is a positive real number and b, log b = 0 because b 0 = and log b b = because b = b Eample 2: Evaluate the epression without using a calculator. #: log : #2: log : #3: log /5 25 : The most common logarithms are the common logarithm, or log which is log 0, and the natural logarithm, or ln which is log e. Your calculator can do both of these. Eample 3: Use a calculator to evaluate the epression. Round the result to three decimal places. #: log 0.3: #2: ln 50: B the definition of what a logarithm is, if f() = b, then the inverse is g() = log b. For eample, g(f()) = log b b = and f(g()) = b log b. Eample 4: Simplif the epression. #: 35 log 35 : #2: log : Talking about inverses, recall that, to create an inverse, ou simpl have to switch the and and solve for. Eample 5: Find the inverse of the function. #: log / 4 : page 9 SN AA U2C8

10 #2: ln( 2) : We can also graph logarithmic functions just like we graphed functions with e and other eponential growth and deca functions. Although ou know most all of these alread from similar graphs, here are some reminders about the characteristics of the graph of = log b ( h) + k: The line = h is a vertical asmptote since plugging in h for will make the log of 0. Since b > 0, ou can never get what is in the parentheses to be zero. The domain is alwas > h, and the range is alwas all real numbers. If b >, the graph moves up to the right. If 0 < b <, the graph moves down to the right. Eample 6: Graph the function log 5 ( 4). State the domain and range. 0 Domain: Range: Section 8-4: Properties of Logarithms You learned how to solve logarithms in the last section. Tr these out to get read for this section: log 0 0 log 0 log log log log Now, take a look at some of the properties that come from these. log log 000 log (0 0 0) log 0 log 0 log 0 3 log log 000 log 0 3 3log 0 3() 3 0 log log 000 log 0 log page 0 SN AA U2C8

11 The eamples above illustrate three properties of logarithms. Properties of Logarithms Let b, u, and v be positive numbers where b. Propert Name Propert Eample Eample: Product log Propert b uv log b u log b v Quotient Propert Power Propert log b u v log b u log b v log b u n n log b u Eample: Eample: Eample : Use log and log 5.76 to approimate the value of the epression. #: log 25 : #2: log 3 : Sometimes it is helpful, when calculating, to simpl epand a term so there is no fraction, eponent, or combined epression onl the simplest number inside the logarithmic function itself. Eample 2: Epand the epression. #: ln 22 : #2: log 3 25 : #3: ln 3 4 : 3 On the other hand, sometimes it is helpful to condense an epression (make it into one epression). Eample 3: Condense the epression. #: 4 log log 6 2: page SN AA U2C8

12 #2: log log 3 : #3: 4 log 8 2 log log 4 5 : The stor problems in this section all involve these properties. Remember one thing: tr condensing or epanding the epressions to get our answers. Section 8-5: Eponential and Logarithmic Equations Section 8-6: Natural Logarithms Up to this point, the onl wa ou have been able to work with logarithmic bases other that 0 or e has been to have numbers that come out perfect with eponents. Given ou will deal with not-so-nice numbers and ones that are not so eas (grrrrrrrrrrrrrrrrrrrrrrrrr!!!!!!), ou need to find a wa to find logarithms of different bases. Formula log c u log u log c log c u ln u ln c Change-Of-Base Formula Let u, b, and c be positive numbers with b and c. Then, log c u log b u log b c Eample Eample: log 2 log 7 2 log(2) / log(7).277 log 7 Eample: 5 ln 5 log ln(5 / 6) / ln(9) ln 9 Up to this point, ou have been simplifing single logarithmic or eponential epressions. However, in this section, ou actuall be solving entire equations. In one of the previous lessons, ou learned that if each side has the same base, each side must also have the same eponent. For eample, if 0 = 0 5, ou know that = 5. For b > 0 and b, if b = b. To solve an equation when eponents have in them, tr getting the base to be the same, if possible. page 2 SN AA U2C8

13 Eample : Solve the equation. #: : #2: : Sometimes ou just can t get the same base on both sides. One thing to remember is that what ou do to one side, ou must do to the other. For eample, if ou want to get to an eponent, take the logarithm of the base that has the eponent to cancel out the base. Eample 2: Solve the equation. #: 2 5 : #2: : Solving logarithmic equations is ver similar: If ou have the same logarithm and base on each side, ou can set equal what is in the logarithms. If ou have onl one logarithm in the equation, take the base of the logarithm to the logarithm to get to the inside. (i.e., if log b, b log b ) Eample 3: Solve the equation. #: log 8 (4 + ) = log 8 (2 + 5): #2: ln ln 2 : page 3 SN AA U2C8

14 #3: 5 2ln3 5: #4: 6.5log : Back when we were discussing the Euler number before, ou could calculate how much mone ou would earn if ou were compounding the interest on it continuousl. Now that ou can solve for eponents, ou could work that in reverse to calculate how much time ou would need to earn that mone from interest compounded continuousl. Eample 4: Dr. Hartman loves investing mone and has $000 to invest in an account which compounds interest continuousl at an annual rate of 4%. How long will it take his account to increase to $3000? (Round to the nearest tenth of a ear.) page 4 SN AA U2C8

where a 0 and the base b is a positive number other

7. Graph Eponential growth functions No graphing calculators!!!! EXPONENTIAL FUNCTION A function of the form than one. a b where a 0 and the base b is a positive number other a = b = HA = Horizontal Asmptote: