Chapter 6: Eponential Functions Section 6.1 Chapter 6: Eponential Functions Section 6.1: Eploring Characteristics of Eponential Functions Terminology: Eponential Functions: A function of the form: y = a(b) Where a 0, b > 0, b 1 a is the coefficient, b is the base, and is the eponent. Eponential Functions and Their Characteristics Graph the following eponential function and interpret its characteristics. (a) f() = 10 Table: Graph: y 3 2 1 0 1 2 3 Characteristics: f() 1000 800 600 400 200-4 - 2 2 4-200 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Number of Turning Points: 135
Chapter 6: Eponential Functions Section 6.1 (b) g() = ( 1 2 ) Table: 3 2 1 0 1 2 3 f() Graph: 10 8 6 4 2-4 - 2 2 4-2 y Characteristics: 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Number of Turning Points: (c) h() = 10(2) Table: Graph: 3 2 1 0 1 2 3 f() 175 150 125 100 75 50 25 y Characteristics: - 4-2 2 4-25 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Number of Turning Points: 136
Chapter 6: Eponential Functions Section 6.1 (d) h() = 8 ( 1 4 ) Table: Graph: y 3 2 1 0 1 2 3 f() Characteristics: 560 480 400 320 240 160 80-4 - 2 2 4-80 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Number of Turning Points: SUMMARY: What effect does the value of b have on the graph and characteristics of the eponential function? What effect does the value of a have on the graph and characteristics of the eponential function? 137
Chapter 6: Eponential Functions Section 6.2 Section 6.2: Eploring Characteristics of Eponential Equations Terminology: Euler s Number: The symbol e is a constant known as Euler s number. It is an irrational number that equals 2.718 This number occurs naturally in some situations where a quantity increases continuously, such as increasing populations. Connecting the Characteristics of an Increasing Eponential Function to its Equation and Graph Predict the number of -intercepts, the y-intercept, the end behaviour, the domain, and the range of the following functions. Also determine whether this function is increasing or decreasing: (a) y = e 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Increasing or decreasing: (b) (c) y = 9 ( 2 3 ) 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Increasing or decreasing: y = 2(5) 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Increasing or decreasing: 138
Chapter 6: Eponential Functions Section 6.2 (d) y = 8 ( 3 4 ) 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Increasing or decreasing: (e) y = 1 2 (5 3 ) 1. Number of -intercept: 2. Coordinates of y-intercept: 3. End Behaviour: 4. Domain: 5. Range: 6. Increasing or decreasing: 139
Chapter 6: Eponential Functions Section 6.2 Matching an Eponential Equation with its Corresponding Graph Which eponential function matches each graph below? Provide your reasoning i) y = 3(0.2) ii) y = 4(3) iii) y = 4(0.5) iv) y = 2(4) 8 6 4 2 y y 5 4 3 2 1-4 - 3-2 - 1 1 2 3 4-2 - 4-3 - 2-1 1 2 3 4-1 - 2 y y 8 5 6 4 4 2 3 2 1-4 - 3-2 - 1 1 2 3 4-2 - 4-3 - 2-1 1 2 3 4-1 - 2 140
Chapter 6: Eponential Functions Section 6.3 Section 6.3: Solving Eponential Equations Laws of Eponents Remember from grade 10: Law 1: Product of Powers When we multiply two powers with the same base, we must add the values of the eponents. Law 2: Quotient of Powers a m a n = a m+n When we divide two powers with the same base, we must subtract the values of the eponents. Law 3: Power of a Power a m a n = am n or a m a n = a m n When an eponent is applied to a power, we must multiply the powers of the two eponents. Law 4: Power of a Product (a m ) n m n = a When an eponent is applied to the product of two numbers, we use the distributive property and apply that eponent to each of the factors. Law 5: Power of a Quotient (a b) m = a m b m When an eponent is applied to the quotient of two numbers, we use the distributive property and apply that eponent to both the numerator and denominator (divisor and dividend). ( a b ) m = am b m 141
Chapter 6: Eponential Functions Section 6.3 Law 6: Power of Zero When a base has an eponent of zero, the value of the power is one. Law 7: Power of One a 0 = 1 When a base does not have an eponent, it actually has an eponent of one. Law 8: Negative Eponents a = a 1 When a power has negative eponent, it can be rewritten as the reciprocal of the base with a positive eponent. Law 9: Rational Eponents a m = 1 a m or (a b ) m = ( b a ) m When a power has a rational eponent, its numerator represents the applied the eponent and the denominator represents the inde of the applied radical (root). a m n n = a m or a m n n = ( a ) m Solving Eponential Equations when Bases are Powers of One Another It is important to note that when you are dealing with eponential equations, if there is a single base on each side of the equation and both bases are equal, than the eponents must also be equal. The opposite is also true. Solve each Equation (a) 5 = 125 (b) 3 2 = 1 81 142
Chapter 6: Eponential Functions Section 6.3 (c) 9 = 27 (d) 4 +3 = 8 2 (e) 2(4) 2 = 1 32 (f) 5 +2 = 125 (g) 9 1 = 3 3 (h) 4 +5 = 64 2 Terminology: Half-Life Eponential Function: A function of the form y = a ( 1 2 ) h Where the base is 1 2, a 0, and h 0; the value of h is called the half-life because it corresponds to the point on the graph of the function where the function is half of its original value, a. 143
Chapter 6: Eponential Functions Section 6.3 Determining the Half-Life of an Eponential Function E1. When driving underwater, the light decreases as the depth of the diver increases. On a sunny day off the coast of Vancouver Island, a diving team recorded 100% visibility at the surface but only 25% visibility 10 m below the surface. Determine the half-life equation and use it to determine when the visibility will be half of that at the surface. E2. On another day, the same diving team noticed that the diving conditions were worse. They recorded 100% visibility at the surface and 3.125% visibility 8 m below the surface. Determine the half-life equation and use it to determine when the visibility will be half of that at the surface? 144
Chapter 6: Eponential Functions Section 6.3 Solving Eponential Equations When the Bases are Not Powers of One Another Solve the following eponential equation. Round your answer to one decimal place. (a) 2 +1 = 5 1 (b) 3 5 = 5 4 X=2.5 =7.9 Done using graphs NOTE: To do this you can either use trial and error or use your graphing calculator. For the graphing calculator use the following steps: 1. Use the right hand side of the equation to be y 1 = in the y= menu 2. Use the left hand side of the equation to be y 2 = in the y= menu 3. Hit graph, then manipulate the window until you can see the intersection of the two graphs 4. Hit 2nd and trace and select intersect. 5. Hit Enter three times and your answer will be the value that is given. 145
Chapter 6: Eponential Functions Section 6.3 Word Problems A cup of coffee cools eponentially over time after it is brought into a car. The cooling is described by the function shown where T is the temperature of the coffee in degrees Celsius with respect to time, m, in minutes. T(m) = 75 ( 1 m ) 20. Determine how long it would take to reach a temperature of 3 C? 5 The half-life of a certain drug in the bloodstream is 6 days. If a patient is given 480 mg, algebraically determine how long it will take for the amount of drug in the patient s body to reduce to 15 mg. Recall that the equation for the half-life of a function is A(t) = A o ( 1 t ) h. 2 146
Chapter 6: Eponential Functions Section 6.3 A laboratory assistant decided to observe the reproductive properties of a new strain of bacteria. The assistant started observing a population of 300 bacteria and noted that the bacteria population doubled every 5 minutes. This can be modeled by P(t) = 300(2) t 5, determine the population after 2 hours. A university student studied and recorded the population of a bacterial culture every 30 minutes. The eponential function that models the bacteria population during the study is P(t) = 50(2) 30, t use it to find the bacteria population 180 minutes after the study began. 147
Chapter 6: Eponential Functions Section 6.4 Section 6.4: Eponential Regression Creating Graphical and Algebraic Models of Given Data E1. The population of Canada from 1871 to 1971 is shown in the table below. In the third column, the values have been rounded. (a) Using graphing technology, create a graphical model and an algebraic eponential model for the data. State the regression equation. (b) Assuming that the population growth continued at the same rate to 2011, estimate the population in 2011. Round your answer to the nearest million. 148
Chapter 6: Eponential Functions Section 6.4 Solving a Problem Using an Eponential Regression Model E1. Sonja did an eperiment to determine the cooling curve of water. She placed the same volume of hot water in three identical cups. Then she recorded the temperature of the water in each cup as it cooled over time. Her data for three trials is given as follows. (a) Construct a scatter plot to display the data. Determine the equation of the eponential regression function that models Sonja s data. (b) Estimate the temperature of the water 15 min after the eperiment began. Round your answer to the nearest degree. (c) Estimate when the water reached a temperature of 30 C. Round your answer to the nearest minute. 149
Chapter 6: Eponential Functions Section 6.4 E2. Emma did the same eperiment, but performed only one trial. Her data is given below. Time (min) 0 5 10 20 30 45 Temperature (ºC) 90 78 68 52 38 26 (a) Construct a scatter plot to display Emma s data. Determine the equation of the eponential regression function that models her data. (b) Compare the characteristics of the eponential regression function for Emma s data with the characteristics of the function for Sonja s data. How are the graphs the same? How are they different? (c) At what time, to the nearest minute, did the water reach 51 C in Emma s eperiment? How much longer did it take than in Sonja s eperiment? 150
Chapter 6: Eponential Functions Section 6.5 Section 6.5: Financial Applications Involving Eponential Functions Terminology: Compound Interest: The interest earned on both the original amount that was invested and any interest that has accumulated over time. The formula for compound interest is: A(n) = P(1 + i) n Where A(n) represents the future value, P represents the principal, i represents the interest rate per compounding period, and n represents the number of compounding periods. Future Value: The amount that an investment will be worth after a specific period of time. Principal: The original amount of money that is invested or borrowed. Compounding Period: The time over which interest is calculated and paid on an investment. (NOTE: /a means per annum or per year) Simple Interest: The interest earned only on the original amount that was invested, any interest that has accumulated will not affect the earning of future interest. The formula for simple interest is: A = P(1 + rt) Where Arepresents the future value, P represents the principal, r represents the interest rate per annum, and t represents the time in years. 151
Chapter 6: Eponential Functions Section 6.5 Simple Interest E. Jayden decides to invest $1000 of his money in a savings account that offers to pay 5%/a simple interest. (a) Determine the equation that models this situation (b) Determine how much his investment would be worth after 5 years. (c) Determine how much his investment would be worth after 10 years. E2. Morgan invests $3200 into a savings account that offers 2.5%/a simple interest. (a) Determine the equation that models this situation (b) Determine how much his investment would be worth after 5 years. (c) Determine how much his investment would be worth after 10 years. Compounded Interest E1. Jayden instead decides to invest $1000 of his money in a savings account that offers to pay 5%/a compounded annually. (a) Determine the equation that models this situation (b) Determine how much his investment would be worth after 5 years. 152
Chapter 6: Eponential Functions Section 6.5 (c) Determine how much his investment would be worth after 10 years. (d) Referring to you answers to this eample and the simple interest eample on the previous page, which is a better choice to invest your money, simple interest or compounded interest. Eplain. E. Brittany invested $2500 in an account that pays 3.5% /a compounded monthly. (a) Determine the equation that models this situation (b) How much would her investment be worth after 4 years (48 months)? (c) Use graphing technology to estimate when the investment will grow to $3000. E2. After her account reached a value of $3000, Brittany reinvested into an account that provided 5%/a, compounded semi-annually. (a) Determine the equation that models this situation. (b) How much would the investment be worth after 8 years? (c) Use graphing technology to determine when the account will be worth $7667. 153
Chapter 6: Eponential Functions Section 6.5 Modelling Appreciation and Depreciation with an Ep. Function Terminology: Appreciation: The increase in a value over time. Appreciation can be modelled using the standard eponential equation: A(n) = P(1 + i) n Depreciation: The decrease in a value over time. Depreciation can be modelled using a slightly modified eponential equation: A(n) = P(1 i) n Where A(n) = future value, P = initial value, and i is the interest per annum E1. Gina brought a camera for her studio two years ago. Her accountant told her that the camera will have a depreciation rate of 20% per year. When she originally purchased her camera it cost her $2000. (a) Determine the equation that models the situation. (b) Determine how much the camera would be worth 4 years after it was purchased. 154
Chapter 6: Eponential Functions Section 6.5 E2. Ali inherited a set of rare silver coins from his great-grandfather. An appraiser told Ali that the coins were currently valued at $1000 and their appreciation rate will likely be 2.5%/a. (a) Determine the equation that would model this situation. (b) Using your equation, determine the value of the coins after 16 years. E3. Hector paid $ 20 000 for a new car, it is estimated to depreciate 5% annually. (a) Determine the equation that models this situation (b) What would the value of the car be after 7 years? E4. A baseball card valued at $100 appreciates 8%/a. (a) Determine the equation that models this situation (b) What would the value of the card be after 12 years? 155
Chapter 6: Eponential Functions Section 6.5 Determining an Eponential Equation from a Table Time 0 2 4 6 8 Amount 1200 300 75 18.75 4.6875 X -3 0 3 6 9 Y 15 30 60 120 240 Time -10-5 0 5 10 Volume 1 10 100 1000 10000 156