13.2 Exponential Decay Functions

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1 Name Class Date 13. Eponential Deca Functions Essential Question: How is the graph of g () = a b h + k where < b < 1 related to the graph of f () = b? Eplore 1 Graphing and Analzing f () = ( 1 and f () = ( ) Resource Locker 1) Eponential functions with bases between and 1 can be transformed in a manner similar to eponential functions with bases greater than 1. Begin b plotting the parent functions of two of the more commonl used bases: _ and 1 B To begin, fill in the table in order to find points along the function ƒ () = ( _ ). You ma need to review the rules of the properties of eponents, including negative eponents. What does the end behavior of this function appear to be as increases? 1. f () = ( _ ) Plot the points on the graph and draw a smooth curve through them Houghton Mifflin Harcourt Publishing Compan Complete the table for ƒ () = ( 1). Plot the points on the graph and draw a smooth curve through them f () = ( ) - - Module Lesson

2 F Fill in the following table of properties: f () = ( _ ) f () = ( 1 1 ) Domain - < < Range End behavior as f () f () End behavior as - f () f () -intercept G Both of these functions [decrease/increase] throughout the domain. Of the two functions, ƒ () = ( ) decreases faster. Reflect 1. Make a Conjecture Look at the table of properties for the functions. What do ou notice? Make a conjecture about these properties for eponential functions of the form ƒ () = ( _ n ), where n is a constant. Eplain 1 Graphing Combined Transformations of f () = b Where < b < 1 When graphing transformations of ƒ () = b where < b < 1, it is helpful to consider the effect of the transformation on two reference points, (, 1) and ( -1, _ b ), as well as the effect on the asmptote, =. The table shows these reference points and the asmptote = for ƒ () = b and the corresponding points and asmptote for the transformed function, g () = a b -h + k. Houghton Mifflin Harcourt Publishing Compan Module Lesson

3 f () = b g () = a b -h + k First reference point (, 1) (h, a + k) Second reference point ( -1, b) ( h - 1, _ a b + k ) Asmptote = = k Eample 1 The graph of a parent eponential function is shown. Use the reference points and the asmptote shown for the parent graph to graph the given transformed function. Then describe the domain and range of the transformed function using set notation. g () = 3 ( ) - - Identif parameters: a = 3 b = Find reference points: h = k = - (h, a + k) = (, 3 - ) = (, 1) ( h - 1, _ a b + k ) ( = - 1, _ 3 _ - = (1, ) ) Find the asmptote: = - Plot the points and draw the asmptote. Then connect the points with a smooth curve that approaches the asmptote without crossing it. Domain: - < < Range: > - = - - = - Houghton Mifflin Harcourt Publishing Compan B g () = - ( 1) Identif parameters: a = b = h = k = Find reference points: (h, ) = (-, ) = (-, 7) = ( h - 1, a_ b + k ) = (- -1, _ + 8) (, ) Module Lesson

4 Find the asmptote: = 8 Plot the points and draw the asmptote. Then connect the points with a smooth curve that approaches the asmptote without crossing it. Domain: Range: - - = Reflect. Which parameters make the domain and range of g () differ from those of the parent function? Write the transformed domain and range for g () in set notation. Your Turn Graph the given transformed function. Then describe the domain and range of the transformed function using set notation. 3. g () = 3 ( 3 ) Houghton Mifflin Harcourt Publishing Compan Module Lesson

5 Eplain Writing Equations for Combined Transformations of f () = b where < b < 1 Given a graph of an eponential function, g () = ab - h + k, the reference points and the asmptote can be used to identif the transformation parameters in order to write the function rule. Eample Write the function represented b this graph and state the domain and range using set notation. = - (-3, 1) - - (-, -) - Find k from the asmptote: k =. The first reference point is at (-3,1). Equate point value with parameters-based epression. (-3, 1) = (h, a + k) Use the -coordinate to solve for h. h = -3 Use the -coordinate to solve for a. a = 1 - k = -3 Houghton Mifflin Harcourt Publishing Compan The second reference point is at (-, -). Equate point value with parameters-based epression. (-, -) = ( h - 1, Equate -coordinate with parameters. Solve for b. g () = -3 _ ( ) + Domain: - < < Range: < _-3 b + = - _-3 b = -6 b = _-3-6 = a_ b + k ) Module Lesson

6 B (1, ) = -1 1 (, - 1 ) 3 Find k from the asmptote: k =. The first reference point is at (, ), so (, - ) = ( h, ) h = a = - k = The second reference point is at (, ), so (, ) = (h -1, ) _ b -1 = _ b = _ b = _ 5 = - g () = ( 1) - Domain: - < < Range: -1 Houghton Mifflin Harcourt Publishing Compan Module Lesson

7 Reflect. Compare the -intercept and the asmptote of the function shown in this table to the function plotted in Eample A g () _ _ 16 1 _ Compare the -intercept and the asmptote of the function shown in this table to the function plotted in Eample B g () Your Turn Write the function represented b this graph and state the domain and range using set notation. 6. Houghton Mifflin Harcourt Publishing Compan = - Module Lesson

8 Eplain 3 Modeling with Eponential Deca Functions An eponential deca function has the form ƒ (t) = a (1 - r) t where a > and r is a constant percent decrease (epressed as a decimal) for each unit increase in time t. That is, since ƒ(t + 1) = (1 r) ƒ(t) = ƒ(t) r ƒ(t), the value of the function decreases b r ƒ (t) on the interval [t, t + 1]. The base 1 r of an eponential deca function is called the deca factor, and the constant percent decrease r, in decimal form, is called the deca rate. Eample 3 Given the description of the deca terms, write the eponential deca function in the form f (t) = a (1 - r) t and graph it with a graphing calculator. The value of a truck purchased new for $8, decreases b 9.5% each ear. Write an eponential function for this situation and graph it using a calculator. Use the graph to predict after how man ears the value of the truck will be $5. Purchased new for $8,... a = 8,...decreases b 9.5% each ear. r =.95 Substitute parameter values. Simplif. V T (t) = 8, ( ) t V T (t) = 8, (.95 ) t Graph the function with a graphing calculator. Use WINDOW to adjust the graph settings so that ou can see the function and the function values that are important. Find when the value reaches $5 b finding the intersection between V T ( t ) = 8, (.95) t and V T ( t ) = 5 on the calculator. The intersection is at the point (17.6, 5), which means after 17.6 ears, the truck will have a value of $5. Houghton Mifflin Harcourt Publishing Compan Image Credits: Transtock Inc./Superstock Module Lesson

9 B The value of a sports car purchased new for $5, decreases b 15% each ear. Write an eponential function for the depreciation of the sports car, and plot it along with the previous eample. After how man ears will the two vehicles have the same value if the are purchased at the same time? Purchased new for $5,... = 5,...decreases b 15% each ear. r = Substitute parameter values. V c ( t ) = (1 - ) Simplif. V c ( t ) = 5, ( ) Add this plot to the graph for the truck value from Eample A and find the intersection of the two functions to determine when the values are the same. The intersection point is (, ). t t After ears, the values of both vehicles will be $. Reflect 7. What reference points could ou use if ou plotted the value function for the sports car on graph paper? Confirm that the graph passes through them using the calculate feature on a graphing calculator. 8. Using the sports car from eample B, calculate the average rate of change over the course of the first ear and the second ear of ownership. What happens to the absolute value of the rate of change from the first interval to the second? What does this mean in this situation? Houghton Mifflin Harcourt Publishing Compan Module Lesson

10 Your Turn 9. On federal income ta returns, self-emploed people can depreciate the value of business equipment. Suppose a computer valued at $765 depreciates at a rate of 3% per ear. Use a graphing calculator to determine the number of ears it will take for the computer s value to be $35. Elaborate 1. Which transformations of ƒ () = _ ( ) or ƒ () = ( 1 1 ) change the function s end behavior? 11. Which transformations change the location of the graph s -intercept? 1. Discussion How are reference points and asmptotes helpful when graphing transformations of ƒ () = _ ( ) or ƒ () = ( 1 1 ) or when writing equations for transformed graphs? 13. Give the general form of an eponential deca function based on a known deca rate and describe its parameters. 1. Essential Question Check-In How is the graph of ƒ () = b used to help graph the function g () = a b - h + k? Houghton Mifflin Harcourt Publishing Compan Module Lesson

11 Evaluate: Homework and Practice Describe the transformation(s) from each parent function and give the domain and range of each function. 1. g () = ( + 3. g () = ) ( 1) + Online Homework Hints and Help Etra Practice 3. g () = - ( 1) g () = 3 ( ) Graph the given transformed function. Then describe the domain and range of the transformed function using set notation. 5. g () = - ( ) Houghton Mifflin Harcourt Publishing Compan Module Lesson

12 6. g () = ( ) g () = ( 3 ) g () = -3 ( ) Houghton Mifflin Harcourt Publishing Compan Module Lesson

13 Write the function represented b each graph and state the domain and range using set notation (3, 7) -1 - (, 3) 6 = = 3 Houghton Mifflin Harcourt Publishing Compan -6 - (-, 1) - - (-5, -3) - Module Lesson

14 Write the eponential deca function described in the situation and use a graphing calculator to answer each question asked. 11. Medicine A quantit of insulin used to regulate sugar in the bloodstream breaks down b about 5% each minute after the injection. A bodweight-adjusted dose is generall 1 units. How long does it take for the remaining insulin to be half of the original injection? 1. Paleontolog Carbon-1 is a radioactive isotope of carbon that is used to date fossils. There are about 1.5 atoms of carbon-1 for ever trillion atoms of carbon in the atmosphere, which known as 1.5 ppt (parts per trillion). Carbon in a living organism has the same concentration as carbon-1. When an organism dies, the carbon-1 content decas at a rate of 11.% per millennium (1 ears). Write the equation for carbon-1 concentration (in ppt) as a function of time (in millennia) and determine how old a fossil must be that has a measured concentration of. ppt. 13. Music Stringed instruments like guitars and pianos create a note when a string vibrates back and forth. The distance that the middle of the string moves from the center is called the amplitude (a), and for a guitar, it starts at.75 mm when a note is first struck. Amplitude decas at a rate that depends on the individual instrument and the note, but a deca rate of about 5% per second is tpical. Calculate the time it takes for an amplitude of.75 mm to reach.1 mm. Houghton Mifflin Harcourt Publishing Compan Image Credits: (t) Eliana Aponte/Reuters/Corbis; (b) Joshua David Treisner/Shutterstock Module Lesson

15 H.O.T. Focus on Higher Order Thinking 1. Analze Relationships Compare the graphs of ƒ () = ( Which of the following properties are the same? Eplain. a. Domain b. Range c. End behavior as increases d. End behavior as decreases ) and g () =. 15. Communicate Mathematical Ideas A quantit is reduced to half of its original amount during each given time period. Another quantit is reduced to one quarter of its original amount during the same given time period. Determine each deca rate, state which is greater, and eplain our results. 16. Multiple Representations Eponential deca functions are written as transformations of the function ƒ () = b, where < b < 1. However, it is also possible to use negative eponents as the basis of an eponential deca function. Use the properties of eponents to show wh the function ƒ () = - is an eponential deca function. Houghton Mifflin Harcourt Publishing Compan 17. Represent Real-World Problems You bu a video game console for $5 and sell it 5 ears later for $1. The resale value decas eponentiall over time. Write a function that represents the resale value, R, in dollars over the time, t, in ears. Eplain how ou determined our function. Module Lesson

16 Lesson Performance Task Sodium- is a radioactive isotope of sodium used as a diagnostic aid in medicine. It undergoes radioactive deca to form the stable isotope magnesium- and has a half-life of about 15 hours. This means that, in this time, half the amount of a sample mass of sodium- decas to magnesium-. Suppose we start with an initial mass of 1 grams sodium-. a. Use the half-life of sodium- to write an eponential deca function of the form m Na (t) = m (1 - r) t, where m is the initial mass of sodium-, r is the deca rate, t is the time in hours, and m Na (t) is the mass of sodium- at time t. What is the meaning of r? b. The combined amounts of sodium- and magnesium- must equal m, or 1, for all possible values of t. Show how to write a function for m Mg (t), the mass of magnesium- as a function of t. c. Use a graphing calculator to graph m Na (t) and m Mg (t). Describe the graph of m Mg (t) as a series of transformations of m Na (t). What does the intersection of the graphs represent? Mass of Na/Mg (g) Houghton Mifflin Harcourt Publishing Compan Time in Minutes (t) Module Lesson