3. Properties of Eponential Functions Capacitors are used to store electric potential energ. When a capacitor in a resistor-capacitor (RC) circuit is discharged, the electric potential across the capacitor decas eponentiall over time. This sort of circuit is used in a variet of electronic devices, such as televisions, computers, and MP3 plaers. Engineers and technicians who design and build such devices must have a solid understanding of eponential functions. Man situations can be modelled using functions of the form f () 5 ab, where a and b, b 1. How do the values of a and b affect the properties of this tpe of function? Tools computer with The Geometer s Sketchpad or grid paper or graphing calculator interval an unbroken part of the real number line is either all of or has one of the following forms: < a, > a, a, a, a < < b, a b, a < b, a < b, where a, b, and a < b Connections It is important to be careful around discarded electrical equipment, such as television sets. Even if the device is not connected to a power source, stored electrical energ ma be present in the capacitors. Investigate How can ou discover the characteristics of the graph of an eponential function? A: The Effect of b on the Graph of = ab Start with the function f () 5. In this case, a 5 1. 1. a) Graph the function. b) Describe the shape of the graph.. Use algebraic and/or graphical reasoning to justif our answers to the following. a) What are the domain and the range of the function? b) What is the -intercept? c) Is there an -intercept? d) Over what interval is the function i) increasing? ii) decreasing? 3. Change the value of b. Use values greater than. a) Compare each graph to the graph of 5. Describe how the graphs are alike. How do the differ? b) Describe how the value of b has affected the characteristics listed in step. c) Eplain wh a value of b greater than has this effect on the graph. 178 MHR Functions 11 Chapter 3
If ou are using The Geometer s Sketchpad, ou can set b as a parameter whose value ou can change dnamicall: From the Graph menu, choose New Parameter. Set the name as b and its initial value to. Click OK. From the Graph menu, choose Plot New Function. Click on the parameter b, and then click on ^ and OK. You can change the value of b in three was: Click on parameter b and press and on the keboard to increase or decrease the value of b in 1-unit increments. Right-click on parameter b and choose Edit Parameter to enter a specific value. Right-click on parameter b and choose Animate Parameter. Use the buttons on the Motion Controller to see the effects of changing b continuousl.. Change the value of b again. This time, use values between and 1. a) How has the graph changed? b) Describe how the values of b affect the characteristics listed in step. c) Eplain wh a value of b between and 1 has this effect on the graph. Connections In Section 3.3, ou saw how to deal with rational eponents. The definition of an eponent can be etended to include real numbers, and so the domain of a function like f () = is all real numbers. Tr to evaluate π using a calculator. 5. What happens to the graph when ou set b 5 1? Eplain this result. 6. Reflect Summarize how the values of b affect the shape and characteristics of the graph of f () 5 b. B: The Effect of a on the Graph of = ab Use the function f () 5 a. In this part of the Investigate, keep b 5, and eplore what happens as ou change the value of a. 1. Set a 5 1. This gives the original graph of f () 5. Eplore what happens when a) a 1 b) a 1 c) a. Reflect Write a summar of the effects of various values of a on the graph of the function f () 5 a. Include the following characteristics: domain, range, - and -intercepts, and intervals of increase and decrease. Eplain wh changing the value of a has these effects. Sketch diagrams to support our eplanations. Technolog Tip With a graphing calculator, ou can var the line stle to see multiple graphs simultaneousl. Refer to the Technolog Appendi on pages 96 to 516. 3. Properties of Eponential Functions MHR 179
Connections You encountered horizontal and vertical asmptotes in Chapter 1 Functions. One of the interesting features of an eponential function is its asmptotic behaviour. Consider the function f () 5. If ou keep looking left at decreasing values of, ou will see that the corresponding -value of the function gets closer and closer to, but never reaches, the -ais. In this case, the -ais is an asmptote. Eample 1 Analse the Graph of an Eponential Function Graph each eponential function. Identif the domain range - and -intercepts, if the eist intervals of increase/decrease asmptote a) 5 ( 1_ ) b) 5 3 Solution a) 5 ( 1_ ) Method 1: Use a Table of Values Select negative and positive values of that will make it eas to compute corresponding values of. 16 1 8 1 1 3 1_ 1_ ( 1_ ) = ( _ 1 ) = 16 Use the table of values to graph the function. 16 1 8 = ( ) 1 18 MHR Functions 11 Chapter 3
Method : Use a Graphing Calculator Use a graphing calculator to eplore the graph of this function. The function is defined for all values of. Therefore, the domain is { }. The function has positive values for, but never reaches zero. Therefore, the range is {, }. The graph never crosses the -ais, which means there is no -intercept. The graph crosses the -ais at. Therefore, the -intercept is. The graph falls to the right throughout its domain, so the -values decrease as the -values increase. Therefore, the function is decreasing over its domain. As the -values increase, the -values get closer and closer to, but never reach, the -ais. Therefore, the -ais, or the line 5, is an asmptote. b) 5 3 The domain is { }. All function values are negative. Therefore, the range is {, }. There is no -intercept. The -intercept is 1. The graph rises throughout its domain. Therefore, the function is increasing for all values of. The -ais, whose equation is 5, is an asmptote. Connections Could ou use transformations to quickl sketch the graph of = 3? You will eplore this option in Section 3.5. 3. Properties of Eponential Functions MHR 181
Eample Write an Eponential Equation Given Its Graph Write the equation in the form 5 ab that describes the graph shown. Solution Read some ordered pairs from the graph. 5 8 36 3 (3, 5) 5 8 36 3 18 1 6 18 (, 18) 1 6 (1, 6) (, ) Note that as changes b 1 unit, increases b a factor of 3, confirming that this function is an eponential function. 1 6 18 3 5 Change in 3 3 3 Since each successive value increases b a factor of 3, this function must have b 5 3. Since all points on this graph must satisf the equation 5 ab, substitute the coordinates of one of the points, and the value of b, to find the value of a. Pick a point that is eas to work with, such as (1, 6). Substitute 5 1, 5 6, and b 5 3: 5 ab 6 5 a 3 1 6 5 a 3 a 5 Therefore, the equation that describes this curve is 5 3. 18 MHR Functions 11 Chapter 3
Eample 3 Write an Eponential Function Given Its Properties A radioactive sample has a half-life of 3 das. The initial sample is mg. a) Write a function to relate the amount remaining, in milligrams, to the time, in das. b) Restrict the domain of the function so that the mathematical model fits the situation it is describing. Solution a) This eponential deca can be modelled using a function of the form A() 5 A ( _ 1 ), where is the time, in half-life periods; A is the initial amount, in milligrams; and A is the amount remaining, in milligrams, after time. Start with mg. After ever half-life, the amount is reduced b half. Substituting A 5 into this equation gives A() 5 ( 1_ ). This epresses A as a function of, the number of half-lives. To epress A as a function of t, measured in das, replace with t_ 3. The half-life of this material is 3 das. Therefore, the number of elapsed half-lives at an given point is the number of das divided b 3. t_ A(t) 5 ( 1_ ) 3 This equation relates the amount, A, in milligrams, of radioactive material remaining to time, t, in das. b) A graph of this function reveals a limitation of the mathematical model. The initial sample size, at t 5, was mg. It is not clear that the function has an meaning before this time. Since it is onl certain that the mathematical model fits this situation for non-negative values of t, it makes sense to restrict its domain: A(t) 5 ( 1_ t_ 3 ) for {t, t }. A 8 16 1 8 t A(t) = ( 1 ) 3 t A 8 16 1 8 t 3. Properties of Eponential Functions MHR 183
Ke Concepts The graph of an eponential function of the form 5 ab is i ncreasing if a and b 1 d ecreasing if a and b 1 d ecreasing if a and b 1 i ncreasing if a and < b 1 The graph of an eponential function of the form 5 ab, where a and b, has domain { } range {, } a horizontal asmptote at 5 a -intercept of a The graph of an eponential function of the form 5 ab, where a and b, has domain { } range {, } a horizontal asmptote at 5 a -intercept of a You can write an equation to model an eponential function if ou are given enough information about its graph or properties. Sometimes it makes sense to restrict the domain of an eponential model based on the situation it represents. Communicate Your Understanding C1 a) Is an eponential function either alwas increasing or alwas decreasing? Eplain. b) Is it possible for an eponential function of the form 5 ab to have an -intercept? If es, given an eample. If no, eplain wh not. ( _1 ) C Consider the eponential functions f () 5 1 and g() 5 1(). a) Which function has a graph with range i) {, }? ii) {, }? Eplain how ou can tell b inspecting the equations. b) Which function is i) increasing? ii) decreasing? Eplain how ou can tell b inspecting the equations. C3 Describe what is meant b asmptotic behaviour. Support our eplanation with one or more sketches. 18 MHR Functions 11 Chapter 3 Functions 11 CH3.indd 18 6/1/9 :5:5 PM
A Practise For help with questions 1 to 3, refer to Eample 1. 1. Match each graph with its corresponding equation. a) b) c) 8 8 8 8 A 5 B 5 ( 1_ ) C 5 1_ D 5. a) Sketch the graph of an eponential function that satisfies all of these conditions: domain { } range {, } -intercept 5 function increasing b) Is this the onl possible graph? Eplain. 3. a) Sketch the graph of an eponential function that satisfies all of these conditions: domain { } range {, } -intercept function decreasing b) Is this the onl possible graph? Eplain. For help with questions and 5, refer to Eample.. Write an eponential equation to match the graph shown. 16 1 (, 16) d) 8 (1, 8) (, ) 8 8 3. Properties of Eponential Functions MHR 185
5. Write an eponential equation to match the graph shown. For help with question 6, refer to Eample 3. 6. A radioactive sample with an initial mass of 5 mg has a half-life of das. B a) Which equation models this eponential deca, where t is the time, in das, and A is the amount of the substance that remains? A A 5 5 t_ B A 5 5 ( 1_ ) t C A 5 5 ( 1_ ) D A 5 5 t_ b) What is the amount of radioactive material remaining after 7 das? t_ Connect and Appl 7. Graph each function and identif the i) domain ii) range iii) - and -intercepts, if the eist iv) intervals of increase/decrease v) asmptote a) f () 5 ( 1_ ) b) 5 1.5 c) 5 ( 1_ 3 ) 8 56 8 3 16 8 (, ) (1, 1) (, 6) 8. a) Graph each function. i) f () 5 ii) r() 5 _ Connecting b) Describe how the graphs are alike. How do the differ? c) Compare the asmptotes of these functions. What do ou observe? 9. a) Graph each function. i) g() 5 ( 1_ ) ii) r() 5 _ b) Describe how the graphs are alike. How do the differ? c) Compare the asmptotes of these functions. What do ou observe? 1. a) Predict how the graphs of the following functions are related. i) f () 5 3 ii) g() 5 ( 1_ 3 ) b) Graph both functions and check our prediction from part a). c) Use algebraic reasoning to eplain this relationship. 11. The graph shows the voltage drop across a capacitor over time while discharging an RC circuit. At t 5 s, the circuit begins to discharge. a) What is the domain of this function? b) What is the range? Representing Voltage Drop (V) Reasoning and Proving Problem Solving Communicating Selecting Tools Reflecting c) What is the initial voltage drop across the capacitor? d) What value does the voltage drop across the capacitor approach as more time passes? e) Approimatel how long will it take the voltage drop to reach 5% of the initial value? V 3 1 1 3 t Time (milliseconds) 186 MHR Functions 11 Chapter 3
1. A flwheel is Reasoning and Proving rotating under Representing friction. The Problem Solving number, R, of Connecting revolutions per minute after Communicating t minutes can be determined using the function R(t) 5 (.75) t. a) Eplain the roles of the numbers.75 and in the equation. b) Graph the function. c) Which value in the equation indicates that the flwheel is slowing? d) Determine the number of revolutions per minute after i) 1 min ii) 3 min C Etend 13. Use Technolog Refer to question 11. The equation that models this situation is given b V 5 V b t_ RC, where V is the voltage drop, in volts; V is the initial voltage drop; t is the time, in seconds; R is the resistance, in ohms (Ω); and C is the capacitance, in farads (F). For this circuit, R 5 Ω and C 5 1 µf. Note that 1 µf 5. 1 F. a) Determine the value of the base, b. b) Eplain our method. Selecting Tools Reflecting c) Graph the function using a graphing calculator or graphing software. Use the window settings shown. d) What are the domain and range of this function? e) Eplain how and wh the domain and range are restricted, as illustrated in the graph of question 11. 1. Suppose a square-based pramid has a fied height of 5 m. a) Write an equation, using rational eponents where appropriate, to epress the side length of the base of a squarebased pramid in terms of its volume. b) How should ou limit the domain of this function so that the mathematical model fits the situation? c) What impact does doubling the volume have on the side length of the base? Eplain. 15. Suppose that a shelf can hold clindrical drums with a fied height of 1 m. a) Write a simplified equation, using rational eponents where appropriate, to epress the surface area in terms of the volume for drums that will fit on the shelf. b) Find the surface area and diameter of a drum with a volume of.8 m 3. c) What are the restrictions on the domain of the function used in this model? d) Graph the function for the restricted domain. 16. Math Contest Find all solutions to 3 1 5. 17. Math Contest Consider the function 5 1 ( 1_ ) 3. The -intercept is b and the -intercept is a. The sum of a and b is A 11 B 6 C 7 D 18 18. Math Contest A number is between and 3. When this number is subtracted from its cube, the result is 13 8. When the same number is added to its cube, the answer is A 13 88 B 13 85 C 13 86 D 13 8 3. Properties of Eponential Functions MHR 187