13.1 Exponential Growth Functions
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1 Name Class Date 1.1 Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > 1 related to the graph of f () = b? Resource Locker Eplore 1 Graphing and Analzing f () = and f () = 1 An eponential function is a function of the form ƒ () = b, where the base b is a positive constant other than 1 and the eponent is a variable. Notice that there is no single parent eponential function because each choice of the base b determines a different function. A Complete the input-output table for each of the parent eponential functions below. f () = p () = Houghton Mifflin Harcourt Publishing Compan B Graph the parent functions ƒ () = and p () = 1 b plotting points Module 1 65 Lesson 1
2 C What is the domain of each function? D What is the range of each function? Domain of ƒ () = : Range of ƒ () = : Domain of p () = 1 : Range of p () = 1 : E What is the -intercept of each function? -intercept of ƒ () = : -intercept of p () = 1 : F What is the trend of each function? In both ƒ () = and p () = 1, as the value of increases, the value of increases/ decreases. Reflect 1. Will the domain be the same for ever eponential function? Wh or wh not?. Will the range be the same for ever eponential function in the form ƒ() = b, where b is a positive constant? Wh or wh not?. Will the value of the -intercept be the same for ever eponential function? Wh or wh not? Eplain 1 Graphing Combined Transformations of f () = b Where b > 1 A given eponential function g () = a ( b - h ) + k with base b can be graphed b recognizing the differences between the given function and its parent function, ƒ () = b. These differences define the parameters of the transformation, where k represents the vertical translation, h is the horizontal translation, and a represents either the vertical stretch or compression of the eponential function and whether it is reflected across the -ais. You can use the parameters in g () = a ( b - h) + k to see what happens to two reference points during a transformation. Two points that are easil visualized on the parent eponential function are (, 1) and (1, b). In a transformation, the point (, 1) becomes (h, a + k) and (1, b) becomes (1 + h, ab + k). The asmptote = for the parent function becomes = k. Houghton Mifflin Harcourt Publishing Compan Module 1 66 Lesson 1
3 The graphs of ƒ () = and p () = 1 are shown below with the reference points and asmptotes labeled. f () = p () = 1 f() p() 6 6 (1, ) (, 1) = (, 1) = Eample 1 State the domain and range of the given function. Then identif the new values of the reference points and the asmptote. Use these values to graph the function. A g () = - ( - ) + 1 The domain of g () = - ( - ) + 1 is - < <. = 1 The range of g () = - ( - ) + 1 is < 1. Eamine g () and identif the parameters (, -) a = -, which means that the function is reflected across the -ais and verticall stretched b a factor of. h =, so the function is translated units to the right (, -5) f() k = 1, so the function is translated 1 unit up. The point (, 1) becomes (h, a + k). Houghton Mifflin Harcourt Publishing Compan (h, a + k) = (, - + 1) = (, - ) (1, b) becomes (1 + h, ab + k). (1 + h, ab + k) = (1 +, - () + 1) = (, ) = (, - 5) The asmptote becomes = k. = k = 1 Plot the transformed points and asmptote and draw the curve. Module 1 67 Lesson 1
4 B q () = 1.5 ( 1 - ) - 5 The domain of q () = 1.5 (1 - ) - 5 is The range of q () = 1.5 (1 - ) - 5 is Eamine q () and identif the parameters... a = so the function is stretched verticall b a factor of 1.5. h = k = so the function is translated units to the right. so the function is translated 5 units down. The point (,1) becomes (h, a + k). (h, a + k) = (, 1.5-5) = 1 (1, b) becomes (1 + h, ab + k). (1 + h, ab + k) = (1 +, 1.5 (1) - 5) = The asmptote becomes = k. = k = Plot the transformed points and asmptote and draw the curve. - Your Turn. g () = ( + ) q () = - _ 5 (1 + ) Houghton Mifflin Harcourt Publishing Compan Module 1 6 Lesson 1
5 Eplain Writing Equations for Combined Transformations of f() = b Where b > 1 Given the graph of an eponential function, ou can use our knowledge of the transformation parameters to write the function rule for the graph. Recall that the asmptote will give the value of k and the -coordinate of the first reference point is h. Then let 1 be the -coordinate of the first point and solve the equation 1 = a + k for a. Finall, use a, h, and k to write the function in the form g () = a (b -h ) + k. Eample Write the eponential function that will produce the given graph, using the specified value of b. Verif that the second reference point is on the graph of the function. Then state the domain and range of the function in set notation. A Let b =. The asmptote is = 1, showing that k = 1. The first reference point is ( - 1_, - 1_ ). This shows that h = - 1_ and that a + k = - 1_. Substitute k = 1 and solve for a. a + k = - 1_ a + 1 = - 1_ a = - _ h = - 1_ k = (- 1, -1) -1 g() = 1 1 (, -5) Substitute these values into g () = a ( b -h ) + k to find g (). Houghton Mifflin Harcourt Publishing Compan g () = a ( b -h ) + k = - _ ( + 1_ ) + 1 Verif that g ( _ ) = - 5_. g ( _ ) = - _ ( _ + 1_ ) + 1 = - _ ( 1 ) + 1 = - _ () + 1 = _ - _ = - 5_ The domain of g () is - < < +. The range of g () is < 1. Module 1 69 Lesson 1
6 B Let b = 1. = 6 The asmptote is =, showing that k =. The first reference point is (-,.). This shows that h = that a + k =. Substitute for k and solve for a. and (-,.) q() a + k = a + = - (-, -1) a = h = k = Substitute these values into q () = a ( b - h ) + k to find q (). q () = a ( b - h ) + k = ( 1 - ) + Verif that q (-) = -1. q (-) = ( ) + = ( 1 ) + = + = The domain of q () is. The range of q () is. Your Turn Write the eponential function that will produce the given graph, using the specified value of b. Verif that the second reference point is on the graph of the function. Then state the domain and range of the function in set notation. 6. b = g() (, 1 ) (, -1) = - 5 Houghton Mifflin Harcourt Publishing Compan Module 1 6 Lesson 1
7 Eplain Modeling with Eponential Growth Functions An eponential growth function has the form ƒ(t) = a (1 + r) t where a > and r is a constant percent increase (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 increases b r ƒ(t) on the interval [t, t + 1]. The base 1 + r of an eponential growth function is called the growth factor, and the constant percent increase r, in decimal form, is called the growth rate. Eample Find the function that corresponds with the given situation. Then use the graph of the function to make a prediction. A Ton purchased a rare guitar in for $1,. Eperts estimate that its value will increase b 1% per ear. Use a graph to find the number of ears it will take for the value of the guitar to be $6,. Write a function to model the growth in value for the guitar. ƒ (t) = a (1 + r) t = 1, (1 +.1) t = 1, (1.1) t Use a graphing calculator to graph the function. Use the graph to predict when the guitar will be worth $6,. Use the TRACE feature to find the t-value where ƒ (t) 6,. So, the guitar will be worth $6, approimatel 1.9 ears after it was purchased. B At the same time that Ton bought the $1, guitar, he also considered buing another rare guitar for $15,. Eperts estimated that this guitar would increase in value b 9% per ear. Determine after how man ears the two guitars will be worth the same amount. Write a function to model the growth in value for the second guitar. Houghton Mifflin Harcourt Publishing Compan g (t) = a (1 + r) t t = ( 1 + ) t = ( ) Use a graphing calculator to graph the two functions. Use the graph to predict when the two guitars will be worth the same amount. Use the intersection feature to find the t-value where g (t) =. So, the two guitars will be worth the same amount ears after. Module 1 61 Lesson 1
8 Reflect 7. In part A, find the average rates of change over the intervals (, ), (, ), and (, 1). Do the rates increase, decrease, or sta the same? Your Turn Find the function that corresponds with the given situation. Then graph the function on a calculator and use the graph to make a prediction.. John researches a baseball card and finds that it is currentl worth $.5. However, it is supposed to increase in value 11% per ear. In how man ears will the card be worth $6? Elaborate 9. How are reference points helpful when graphing transformations of ƒ () = b or when writing equations for transformed graphs? 1. Give the general form of an eponential growth function and describe its parameters. 11. Essential Question Check-In Which transformations of f () = b change the function s end behavior? Which transformations change the function s -intercept? Houghton Mifflin Harcourt Publishing Compan Module 1 6 Lesson 1
9 Evaluate: Homework and Practice Describe the effect of each transformation on the parent function. Graph the parent function and its transformation. Then determine the domain, range, and -intercept of each function. 1. ƒ () = and g () = ( ). ƒ () = and g () = -5 ( ) Online Homework Hints and Help Etra Practice ƒ () = and g () = +. ƒ () = and g () = + 5 Houghton Mifflin Harcourt Publishing Compan Module 1 6 Lesson 1
10 5. ƒ () = 1 and g () = (1 ) 6. ƒ () = 1 and g () = - (1 ) ƒ () = 1 and g () = 1 -. ƒ () = 1 and g () = ) 9. Describe the graph of g () = ( - in terms of ƒ () =. Houghton Mifflin Harcourt Publishing Compan + 7) 1. Describe the graph of g () = 1 ( + 6 in terms of ƒ () = 1. Module 1 6 Lesson 1
11 State the domain and range of the given function. Then identif the new values of the reference points and the asmptote. Use these values to graph the function. 11. h () = ( + ) k () = -.5 ( - 1 ) ƒ () = ( 6-7 ) - Houghton Mifflin Harcourt Publishing Compan Module 1 65 Lesson 1
12 1. ƒ () = - ( + 1 ) h () = - 1_ ( ) - _ p () = ( - ) Houghton Mifflin Harcourt Publishing Compan Module 1 66 Lesson 1
13 Write the eponential function that will produce the given graph, using the specified value of b. Verif that the second reference point is on the graph of the function. Then state the domain and range of the function in set notation. 17. b = (, 7.) (-1, 5.) g() = b = 1 q() = (-,.7) Houghton Mifflin Harcourt Publishing Compan - -6 (-1, 1) Module 1 67 Lesson 1
14 Find the function that corresponds with the given situation. Then graph the function on a calculator and use the graph to make a prediction. 19. A certain stock opens with a price of $.59. Over the first three das, the value of the stock increases on average b 5% per da. If this trend continues, how man das will it take for the stock to be worth $6?. Sue has a lamp from her great-grandmother. She has it appraised and finds it is worth $1. She wants to sell it, but the appraiser tells her that the value is appreciating b % per ear. In how man ears will the value of the lamp be $? 1. The population of a small town is 15,. If the population is growing b 5% per ear, how long will it take for the population to reach 5,? Houghton Mifflin Harcourt Publishing Compan Image Credits: milosljubicic/shutterstock Module 1 6 Lesson 1
15 . Bill invests $ in a bond fund with an interest rate of 9% per ear. If Bill does not withdraw an of the mone, in how man ears will his bond fund be worth $5? H.O.T. Focus on Higher Order Thinking. Analze Relationships Compare the end behavior of g () = and f () =. How are the graphs of the functions similar? How are the different?. Eplain the Error A student has a baseball card that is worth $6.5. He looks up the appreciation rate and finds it to be.5% per ear. He wants to find how much it will be worth after ears. He writes the function ƒ (t) = 6.5 (.5) t and uses the graph of that function to find the value of the card in ears. Houghton Mifflin Harcourt Publishing Compan Image Credits: RisingStar/Alam According to his graph, his card will be worth about $99. in ears. What did the student do wrong? What is the correct answer? Module 1 69 Lesson 1
16 Lesson Performance Task Like all collectables, the price of an item is determined b what the buer is willing to pa and the seller is willing to accept. The estimated value of a 19 Tucker automobile in ecellent condition has risen at an approimatel eponential rate from about $5, in December 6 to about $1,, in December 1. a. Find an equation in the form V (t) = V (1 + r) t, where V is the value of the car in dollars in December 6, r is the average annual growth rate, t is the time in ears since December 6, and V (t) is the value of the car in dollars at time t. (Hint: Substitute the known values and solve for r.) b. What is the meaning of the value of r? c. If this trend continues, what would be the value of the car in December 17? Houghton Mifflin Harcourt Publishing Compan Module 1 65 Lesson 1
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