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1 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: A horizontal line that the graph approaches but does not touch or cross* Horizontal Asmptote Domain: Range: Ordered Pairs, 0,,, EXPONENTIAL GROWTH b: b is a called the growth factor A function of the form a b where a > 0 and the base b > a: a is the -intercept of the graph, written (0, a) When b is a number greater than, the graph will be an eponential growth function. In an eponential growth function, we see the values growing over time b the same multiple, called the growth factor. b must be greater than, and a must be greater than 0 in order for the graph to be eponential growth When a is a number greater than, the values will be more stretched out, causing the graph to stretch. When a is a number between 0 and, the values will be closer together, causing the graph to shrink. If ou were looking at the table and did not know the equation, how could ou determine the growth factor? a = b = HA = Domain: Range: Ordered Pairs, 0,,,
2 Eponential Growth with Shifts (translations): a b h k a = intercept before the shifts take place h = the value that is being added or subtracted from. It affects all of the values of the problem b shifting them LEFT or RIGHT. b = growth factor k = whatever number is being subtracted or added to the OUTSIDE of the function. K affects al the values of the graph b shifting them UP or DOWN. If h is being subtracted in the problem, ou will add it to each of our values (RIGHT). If h is being added in the problem, subtract it from each of our values (LEFT). If k is positive, add it to each value (UP). If k is negative, subtract it from each value (DOWN). Eample: a = h = ONLY affects our values Describe shift: b = k = ONLY affects our values Describe shift: Step : Identif a, b, h, k. Step : Find the and values for Step : Appl h to each of the values. Step : Appl k to each one of the values. Step 5: Determine and graph the HA. Step 6: Create ordered pairs and graph. Step 7: Determine the Domain and Range. (no shifts) HA = Domain: Range: Appl h if necessar Appl k if necessar, 0,,,
3 Eample: a = h = ONLY affects our values Describe shift: b = k = ONLY affects our values Describe shift: Step : Identif a, b, h, k. Step : Find the and values for Step : Appl h to each of the values. Step : Appl k to each one of the values. Step 5: Determine and graph the HA. Step 6: Create ordered pairs and graph. Step 7: Determine the Domain and Range. (no shifts) HA = Domain: Range: Appl h if necessar Appl k if necessar, 0,,, EXPONENTIAL GROWTH MODELS When a real-life quantit increases b a fied percent each ear (or other time period), the amount of the quantit () after t ears can be modeled b the equation: a r where a is the initial amount of the quantit and r is the percent of increase, written as a decimal. Eample: In the last ears, an initial population of 8 buffalo in a state park grew b about 7% per ear. a. a. Write an eponential growth model giving the number of buffalo after t ears. b. About how man buffalo were in the park after 7 ears? t a r b t t NOTE: For the stor problems, formulas will be provided on the quiz and test, but will NOT be labeled. You must know what each letter represents in the formula, as well as when to use the formula!
4 Eponential Growth functions are also used in real-life scenarios involving compound interest. Compound interest is paid on an initial investment, called the principal, AND on previousl earned interest. So, we are tring to figure out how much mone this principal and its interest are worth after interest is calculated on the initial principal, then added to the principal and recalculated, n times per ear, over t number of ears. Consider an initial principal (P) deposited in an account that pas interest at an annual rate (r), epressed as a decimal, compounded n times per ear. The amount (A) in the account after t ears is given b the equation: A P r n nt Eample: You deposit $900 in an account that pas.5% annual interest. Find the balance after ear if the interest is compounded monthl. A P r n nt 7. Graph Eponential deca functions No graphing calculators!! EXPONENTIAL DECAY A function of the form a b where a > 0 and the base b is between 0 and b: b is a called the deca factor a: a is the -intercept of the graph, written (0, a) When b is a number between 0 and, the graph will When a is a number greater than, the values will be an eponential deca function. In an eponential be more stretched out, causing the graph to stretch. deca function, we see the values decreasing over When a is a number between 0 and, the values time b the same multiple, called the deca factor. will be closer together, causing the graph to shrink. a = b = DECAY FACTOR HA = Domain: Range: Ordered Pairs,, 0,,
5 Eponential DECAY with Shifts (translations): a b h k a = intercept before the shifts take place h = the value that is being added or subtracted from. It affects all of the values of the problem b shifting them LEFT or RIGHT. b = deca factor k = whatever number is being subtracted or added to the OUTSIDE of the function. K affects al the values of the graph b shifting them UP or DOWN. If h is being subtracted in the problem, ou will add it to each of our values (RIGHT). If h is being added in the problem, subtract it from each of our values (LEFT). If k is positive, add it to each value (UP). If k is negative, subtract it from each value (DOWN). Eample: a = h = ONLY affects our values b = k = ONLY affects our values Step : Identif a, b, h, k. Step : Find the and values for Step : Appl h to each of the values. Step : Appl k to each one of the values. Step 5: Determine and graph the HA. Step 6: Create ordered pairs and graph. Step 7: Determine the Domain and Range. (no shifts) HA = Domain: Range: Appl h if necessar Appl k if necessar,, 0,,
6 Eample: a = h = ONLY affects our values b = k = ONLY affects our values Step : Identif a, b, h, k. Step : Find the and values for Step : Appl h to each of the values. Step : Appl k to each one of the values. Step 5: Determine and graph the HA. Step 6: Create ordered pairs and graph. Step 7: Determine the Domain and Range. (no shifts) HA = Domain: Range:,, 0,, EXPONENTIAL DECAY MODELS When a real-life quantit decreases b a fied percent each ear (or other time period), the amount of the quantit after t ears cab be modeled b the equation: a r where a is the initial amount of the quantit and r is the is the percent of decrease, written as a decimal. Eample: A new television costs $0. The value of the television decreases b % each ear. a. Write an eponential deca model giving the television's value (in dollars) after t ears. b. Estimate the value after ears. a. t a r b. t
7 e is an irrational (non-repeating, not-terminating) constant. Because it is non-repeating and non-terminating, we e have a button in our calculator that represents e, so that we do not have to round it ever time we want to use it. e is often referred to as the natural base. One reason is because of where we see it represented in nature. In nature, man populations grow eponentiall, in a manner that is best modeled b e. In general, e helps us model quantities from nature to mone that grow and deca continuousl. We will see e used as a base. EXAMPLE: The population of bacteria in a certain culture at time t is given b, where t is measured in hours. From the equation, we can see that 0 bacteria are initiall present. After hours, where t =, there are bacteria present. After 0 hours, roughl bacteria are present. Find e on our calculator and use it to evaluate the following values. Round answers to two decimal places. ( nd LN, close eponent in parenthesis) 5 e = 7.9 e = e = 9.6 Instead of a number, use e as a base to graph eponential growth and deca functions. 7. USE FUNCTIONS INVOLVING e e a = b = r = Growth or Deca HA = Domain: Range: Ordered Pairs,, 0,, Natural base e Functions A function on the form r a e is called a natural base eponential function. If a > 0 and r > 0, the function is an eponential GROWTH function. If a > 0 and r < 0, the function is an eponential DECAY function.
8 Eample: e a = b = r = h = k = Growth or Deca Describe shift: Describe shift: Step : Identif a, b, r, h, k. Step : Find the and values for (no shifts) e Step : Appl h to each of the values. Step : Appl k to each one of the values. Step 5: Determine and graph the HA. Step 6: Create ordered pairs and graph. Step 7: Determine the Domain and Range. HA = Domain: Range:,, 0,, 0.5 Eample: e a = b = r = h = k = Growth or Deca Describe shift: Describe shift: Step : Identif a, b, r, h, k. Step : Find the and values for (no shifts) e Step : Appl h to each of the values. Step : Appl k to each one of the values. Step 5: Determine and graph the HA. Step 6: Create ordered pairs and graph. Step 7: Determine the Domain and Range. 0.5 HA = Domain: Range:,, 0,,
9 Use our eponent rules to simplif epressions with e. e 6 e 9 e 8 e 6 e 9e e e 5 9 (e ) 6e 6 e 8e 5 6 9e CONTINUOUSLY COMPOUNDED INTEREST When interest is compounded continuousl, the amount A in an account after t ears is given b the formula A Pe rt where P is the principal and t is the annual interest rate epressed as a decimal. NOTE: In our other interest model, the interest was compounded a certain amount of times. You deposit $800 in an account that pas 6.5% annual interest compounded continuousl. What is the balance after ears?
10 7. Evaluate Logarithms and Graph Logarithmic Functions EXPONENTIAL FORM QUESTION: What is the result of raising a base of to the third power? ANSWER: After, we see the answer is 8. 8 b Base Eponent Result Base Eponent Result LOGARITHMIC FORM B definition a arithmic situation and an eponential situation are inverses of one another, so the question needs to change Before: QUESTION: What is the result of raising a base of to the third power? 8 Base Eponent Result Rewrite the equation in eponential form. (DO THE LOG LOOP) 9 After: QUESTION: Base raised to what power gives ou 8? Pronounced: Log base of 8 is The answer to the arithm is the eponent needed to take a base of to 8. The answer to a arithm is alwas an eponent, and that s how the question is different from eponential form. Note: the base must be a positive number The LOG LOOP s job is to take us from unfamiliar form to the ver familiar eponential form. Evaluate the arithm. (TO FIND ITS VALUE, DO THE LOOP, THEN FIND THE EXPONENT NEEDED)
11 A common arithm is one with a base of. It is denoted as or simpl. Your calculator is CALCULATOR designed to find the value of common arithms, so when ou press the button, our calculator is assuming that ou are tping in something that has a base of. Use a calculator to evaluate the arithm. Round our answer to three decimal places. 8 is the same as e gets its own special arithm. Antime we take the e, we can replace this notation with ln. Base e ln stands for the arithm of natural base e. To avoid saing that entire phrase we also call ln the natural for short. This also makes taking the of e etremel eas to put in the calculator. ln e Use a calculator to evaluate the arithm. Round our answer to three decimal places. e 0. is the same as ln 0..0 e 7 INVERSE We have alread studied and manipulated several tpes of inverses. Squaring and square rooting, cubing and cube rooting, and we know that that a form and its inverse can be changed from one to the other b switching the place of the and and resolving for. We also know that inverses reflect over the line on the coordinate plane. (For a refresher graph and on our calculator.) Because arithmic form and eponential form are inverses of one another, we can convert from one to its inverse (the other) b following the same method. Find the inverse of the function. (switch the place of the and, then resolve for ) 6 Have : LOG FORM Want: It s inverse: EXP. FORM ln Have : Want: First invert: 6 Now let s change it into the correct form. How can we turn a into an eponential??? THE LOOP!!
12 7. DAY #! 6 Have : Want: 6 Have : Want: INVERSE Simplif Think about inverses: or or 6. What do inverses do when one is applied to the other? 6 Consider the problems below. In each problem we see the form and eponential form of the same base. 9 8 nothing cancels.hmm 7 7 () When ou have a base 6, right net to an eponential base 6, the cancel each other. The answer is 8! 6 When ou have an eponential base 7, right net to a base 7, the cancel each other. The answer is. GRAPHING. Plot some convenient points b thinking about the loop. raised to =. Appl an necessar shifts.. Determine the vertical asmptote, b setting the value and its contained ( ) = 0. Draw a curve through the points. 5. State the domain and range. VA = Domain: Range: Ordered Pairs,,,
13 h = VA = Domain: Range: Ordered Pairs,,, h = k = VA = Domain: Range: Ordered Pairs,,,
14 7.5 da # Appl Properties of arithms NOTE: In the following properties, the base cannot be negative, nor can it =. Also, the numbers ou are taking the of must be positive. PRODUCT PROPERTY We are adding two s of the same base. To create our answer, we start b writing down a with the same base as the s in the problem. We will then multipl what we were taking the s of = 8 = QUOTIENT PROPERTY We are subtracting two s of the same base. 0 5 = To create our answer, we start b writing down a with the same base as the s in the problem. We will then divide what we were taking the of in each of the problem (first divided b second) 6 = POWER PROPERTY We are multipling a b a number. To create our answer, we start b writing down a with the same base as the in the problem. We will then take the number being multiplied and bring it to the eponent position of the number or epression that we re taking the of. We then simplif if possible. 7 = = ln = 5 6 = What we are doing is learning was to CONDENSE arithmic epressions. In other words, we will be given an epression and asked to simplif it as much as possible. These properties give us the rules for doing so.
15 Condense the following arithmic epressions. ) Alwas start with the numbers that are multipling b the s. Move those numbers to the eponent position and simplif before moving on. ) Move from left to right and use the properties to condense the s. ln ln ln ln ln 0 ln ln ln ln ln
16 Your calculator is designed to find the value of common arithms, so when ou press the button, CHANGE OF BASE THEOREM (CALCULATOR) our calculator is assuming that ou are tping in something that has a base of. For all other arithms that do not have a base of, we have to appl the change of base theorem and put them into base so the can be entered into the calculator. Eample: Use a calculator to evaluate the arithm. Round our answer to three decimal places in calculator : 56/ 6
17 7.5 Da # EXPANSION We can use the properties of s to condense arithms, and we can also use them to epand a arithm. In other words, given the answer to a problem, ou will recreate the original problem that created that answer. Epand the epression. (THINK: WHAT ARE THE NUMBERS AND VARIABLES IN THE ANSWER DOING? WHAT ADDITION OR SUBTRACTION PROBLEM WOULD HAVE CREATED THAT? ) How could we have gotten connected to? Was it an addition problem between s or a subtraction problem? ln 5 ln ln 5 7 ln You will know ou are done with the problem when each has onl one variable or number behind it, and that number or variable does not have an eponent!!!
18 Epand the epression using 7 Use:.585 and and/or. Then find its value using the following information:
19 7.6 Solve eponential and arithmic equations (Note: all rounding in this section will be done to three decimal places.) SOLVING EXPONENTIAL EQUATIONS What ou ll see: equations in which the variable is stuck up in the eponent What ou want: to get the variable out of the eponent so it can be solved for Method:. Equating eponents. Taking the arithm of both sides. Taking the natural arithm of both sides Method : EQUATING EXPONENTS When it can be used: This method can be used when the bases on each side of the equal sign are the same or when the bases are not the same, but can be rewritten to become the same. Supporting Propert:
20 Method : TAKING THE LOGARITHM OF BOTH SIDES Steps:. Isolate the base that contains the eponent.. Take that arithm of both sides.. Solve what remains. Round to the hundredths place.. Check for etraneous solutions Method : TAKING THE NATURAL LOGARITHM (LN) OF BOTH SIDES Steps:. Isolate the base that contains the eponent.. Take that natural of both sides. Solve what remains. Round to the hundredths place.. Check for etraneous solutions. e e e 8 6
21 7.6 Da # SOLVING LOGARITHMIC EQUATIONS What ou ll see: equations with arithms on one or both sides What ou want: to simplif the arithms as much as possible and solve for the variable Method:. Equating arithms. Converting from arithmic to eponential form Method : EQUATING LOGARITHMS When it can be used: After each side of the equation has been simplified (all product, quotient, and power properties applied), this method can be used when each side is left with a single. Supporting Propert: Steps:. Starting on one side at a time, use the product, quotient, and power properties to do all addition, subtraction, mult.. Check to make sure that each side contains a single.. B propert of equalit, cancel the s.. Solve what remains. Round to the hundredths place. 5. Check for etraneous solutions ln 9 ln7 8 ln ln.5
22 Method : CONVERTING FROM LOGARITHMIC TO EXPONENTIAL FORM (THE LOOP) When it can be used: After each side of the equation has been simplified (all product, quotient, and power properties applied), this method can be used when there is onl one on one side. Steps:. Starting on one side at a time, use the product, quotient, and power properties to do all addition, subtraction, mult. of s. If the equation onl contains one, make sure that the is completel alone.. Check to make sure that onl one side contains a.. Use the loop to rewrite the as an eponent.. Solve what remains (ou ma have to factor). 5. Check for etraneous solutions. 6 ln
23 7.7 Write and Appl Eponential and Power Functions Write an eponential function = ab whose graph passes through (, ) and (, 80). Step : Substitute the coordinates of the two given points into = ab. Step : Solve for a in the first equation to obtain a, and then substitute this epression for a into the second equation. Step : Because b =, it follows that a = So, =. Determine whether an eponential model is appropriate for the data. If so, find the eponential model (equation/function) for the data. Savings: The table shows the amount () in a savings account () ears after the account was opened To determine whether this information would be best modeled b an eponential equation, we will make a table of the points (, ln ). Once we create the table, we will make a scatterplot of these points in the calculator. If the modeled points lie along a linear pattern, then we ll know that an eponential model is best for the original data ln = 5.5 Enter the original data into our calculator and use the Eponential Regression feature to get the eponential model. =
24 Write a power function = a b whose graph passes through (, ) and (6, ). Step : Substitute the coordinates of the two given points into = a b Step : Solve for a in the first equation to obtain a, and then substitute this epression for a into the second equation. Step : Because b =, it follows that a = So, =. Determine whether a power model is appropriate for the data. If so, find the power model (equation/function) for the data. Birds: The table shows the tpical wingspan (), in feet, and the tpical weights (), in pounds, for several tpes of birds.. cuckoo crow curlew goose vulture Wingspan, Weight, To determine whether this information would be best modeled b a power function, we will make a table of the points (ln, ln ). Once we create the table, we will make a scatterplot of these points in the calculator. If the modeled points lie along a linear pattern, then we ll know that a power model is best for the original data. Enter the original data into our calculator and use the Power Regression feature to get the power model. =
25 TI-8 and TI-8 Eponential Regression. Press STAT.. Arrow over to CALC.. Choose #0 EpReg.. You should see EpReg on the screen. 5. Press ENTER. The screen will show ou the coefficients and constant of the eponential equation. Power Regression. Press STAT.. Arrow over to CALC.. Choose A PwrReg.. You should see PwrReg on the screen. 5. Press ENTER. The screen will show ou the coefficients and constant of the power equation.
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