Exponential and Logarithmic Functions

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1 Eponential and Logarithmic Functions Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) Chapter Outline. Eponential Functions. Logarithmic Properties. Graphs of Eponential Functions. Eponential and Logarithmic Equations. Logarithmic Functions.7 Eponential and Logarithmic Models. Graphs of Logarithmic Functions. Fitting Eponential Models to Data Introduction Focus in on a square centimeter of our skin. Look closer. Closer still. If ou could look closel enough, ou would see hundreds of thousands of microscopic organisms. The are bacteria, and the are not onl on our skin, but in our mouth, nose, and even our intestines. In fact, the bacterial cells in our bod at an given moment outnumber our own cells. But that is no reason to feel bad about ourself. While some bacteria can cause illness, man are health and even essential to the bod. Bacteria commonl reproduce through a process called binar fission, during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce ver quickl. Unlike humans and other comple organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours, as opposed to das or ears. [] For simplicit s sake, suppose we begin with a culture of one bacterial cell that can divide ever hour. Table shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! And if we were to etrapolate the table to twent-four hours, we would have over million! Hour Bacteria 0 Table In this chapter, we will eplore eponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closel related to eponential functions. Both tpes of functions have numerous real-world applications when it comes to modeling and interpreting data.. Todar, PhD, Kenneth. Todar s Online Te tbook of Bacteriolog. tbookofbacteriolog.net/growth_.html. This OpenSta book is available for free at 7

2 SECTION. Graphs of Eponential Functions Learning Objectives In this section, ou will: Graph eponential functions. Graph eponential functions using transformations.. Graphs of Eponential Functions As we discussed in the previous section, eponential functions are used for man real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a realworld situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things b seeing their pictorial representations, and that is eactl wh graphing eponential equations is a powerful tool. It gives us another laer of insight for predicting future events. Graphing Eponential Functions Before we begin graphing, it is helpful to review the behavior of eponential growth. Recall the table of values for a function of the form f () = b whose base is greater than one. We ll use the function f () =. Observe how the output values in Table change as the input increases b. 0 f () = _ _ _ Table Each output value is the product of the previous output and the base,. We call the base the constant ratio. In fact, for an eponential function with the form f () = ab, b is the constant ratio of the function. This means that as the input increases b, the output value will be the product of the base and the previous output, regardless of the value of a. Notice from the table that the output values are positive for all values of ; as increases, the output values increase without bound; and as decreases, the output values grow smaller, approaching zero. Figure shows the eponential growth function f () =. f (),,, (, ) (, ) (0, ) (, ) f () = The -ais is an asmptote. Figure Notice that the graph gets close to the -ais, but never touches it. The domain of f () = is all real numbers, the range is (0, ), and the horizontal asmptote is = 0. To get a sense of the behavior of eponential deca, we can create a table of values for a function of the form f () = b whose base is between zero and one. We ll use the function g() = _. Observe how the output values in Table change as the input increases b. This OpenSta book is available for free at

3 CHAPTER Eponential and logarithmic Functions g() = 0 _ _ Table Again, because the input is increasing b, each output value is the product of the previous output and the base, or constant ratio _. Notice from the table that the output values are positive for all values of ; as increases, the output values grow smaller, approaching zero; and as decreases, the output values grow without bound. Figure shows the eponential deca function, g() = _. g() = (, ) (, ) g(),, (, ), (0, ) The -ais is an asmptote. Figure The domain of g() = _ is all real numbers, the range is (0, ), and the horizontal asmptote is = 0. _ characteristics of the graph of the parent function f () = b An eponential function with the form f () = b, b > 0, b, has these characteristics: one-to-one function horizontal asmptote: = 0 domain: (, ) range: (0, ) -intercept: none -intercept: (0, ) increasing if b > decreasing if b < Figure compares the graphs of eponential growth and deca functions. f () = b b > f() (, b) (0, ) (0, ) Figure f() f () = b 0 < b < (, b) How To Given an eponential function of the form f () = b, graph the function.. Create a table of points.. Plot at least point from the table, including the -intercept (0, ).. Draw a smooth curve through the points.. State the domain, (, ), the range, (0, ), and the horizontal asmptote, = 0.

4 SECTION. Graphs of Eponential Functions Eample Sketching the Graph of an Eponential Function of the Form f () = b Sketch a graph of f () = 0.. State the domain, range, and asmptote. Solution Before graphing, identif the behavior and create a table of points for the graph. Since b = 0. is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asmptote = 0. Create a table of points as in Table. 0 f () = Table Plot the -intercept, (0, ), along with two other points. We can use (, ) and (, 0.). Draw a smooth curve connecting the points as in Figure. f() = 0. (, ) f() Figure (0, ) (, 0.) The domain is (, ); the range is (0, ); the horizontal asmptote is = 0. Tr It # Sketch the graph of f () =. State the domain, range, and asmptote. Graphing Transformations of Eponential Functions Transformations of eponential graphs behave similarl to those of other functions. Just as with other parent functions, we can appl the four tpes of transformations shifts, reflections, stretches, and compressions to the parent function f () = b without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the eponential function also maintains its general shape regardless of the transformations applied. Graphing a Vertical Shift The first transformation occurs when we add a constant d to the parent function f () = b, giving us a vertical shift d units in the same direction as the sign. For eample, if we begin b graphing a parent function, f () =, we can then graph two vertical shifts alongside it, using d = : the upward shift, g() = + and the downward shift, h() =. Both vertical shifts are shown in Figure. g() = + f () = h() = 0 Figure = = 0 = This OpenSta book is available for free at

5 CHAPTER Eponential and logarithmic Functions Observe the results of shifting f () = verticall: The domain, (, ) remains unchanged. When the function is shifted up units to g() = + : The -intercept shifts up units to (0, ). The asmptote shifts up units to =. The range becomes (, ). When the function is shifted down units to h() = : The -intercept shifts down units to (0, ). The asmptote also shifts down units to =. The range becomes (, ). Graphing a Horizontal Shift The net transformation occurs when we add a constant c to the input of the parent function f () = b, giving us a horizontal shift c units in the opposite direction of the sign. For eample, if we begin b graphing the parent function f () =, we can then graph two horizontal shifts alongside it, using c = : the shift left, g() = +, and the shift right, h () =. Both horizontal shifts are shown in Figure. g() = f () = h() = 0 = 0 Observe the results of shifting f () = horizontall: The domain, (, ), remains unchanged. The asmptote, = 0, remains unchanged. The -intercept shifts such that: Figure When the function is shifted left units to g() = +, the -intercept becomes (0, ). This is because + = (), so the initial value of the function is. When the function is shifted right units to h() =, the -intercept becomes 0, _ so the initial value of the function is _.. Again, see that = _, shifts of the parent function f () = b For an constants c and d, the function f () = b + c + d shifts the parent function f () = b verticall d units, in the same direction of the sign of d. horizontall c units, in the opposite direction of the sign of c. The -intercept becomes (0, b c + d). The horizontal asmptote becomes = d. The range becomes (d, ). The domain, (, ), remains unchanged.

6 SECTION. Graphs of Eponential Functions 7 How To Given an eponential function with the form f () = b + c + d, graph the translation.. Draw the horizontal asmptote = d.. Identif the shift as ( c, d). Shift the graph of f () = b left c units if c is positive, and right c units if c is negative.. Shift the graph of f () = b up d units if d is positive, and down d units if d is negative.. State the domain, (, ), the range, (d, ), and the horizontal asmptote = d. Eample Graphing a Shift of an Eponential Function Graph f () = +. State the domain, range, and asmptote. Solution We have an eponential equation of the form f () = b + c + d, with b =, c =, and d =. Draw the horizontal asmptote = d, so draw =. Identif the shift as ( c, d), so the shift is (, ). Shift the graph of f () = b left units and down units. 0 (0, ) (, ) 0 f () f () = + (, ) = Figure 7 The domain is (, ); the range is (, ); the horizontal asmptote is =. Tr It # Graph f () = +. State domain, range, and asmptote. How To Given an equation of the form f () = b + c + d for, use a graphing calculator to approimate the solution.. Press [Y=]. Enter the given eponential equation in the line headed Y =.. Enter the given value for f () in the line headed Y =.. Press [WINDOW]. Adjust the -ais so that it includes the value entered for Y =.. Press [GRAPH] to observe the graph of the eponential function along with the line for the specified value of f ().. To find the value of, we compute the point of intersection. Press [ND] then [CALC]. Select intersect and press [ENTER] three times. The point of intersection gives the value of for the indicated value of the function. Eample Approimating the Solution of an Eponential Equation Solve =.() +. graphicall. Round to the nearest thousandth. Solution Press [Y=] and enter.() +. net to Y =. Then enter net to Y =. For a window, use the values to for and to for. Press [GRAPH]. The graphs should intersect somewhere near =. For a better approimation, press [ND] then [CALC]. Select [: intersect] and press [ENTER] three times. The -coordinate of the point of intersection is displaed as.9. (Your answer ma be different if ou use a different window or use a different value for Guess?) To the nearest thousandth,.. This OpenSta book is available for free at

7 CHAPTER Eponential and logarithmic Functions Tr It # Solve = 7.(.).7 graphicall. Round to the nearest thousandth. Graphing a Stretch or Compression While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multipl the parent function f () = b b a constant a > 0. For eample, if we begin b graphing the parent function f () =, we can then graph the stretch, using a =, to get g() = () as shown on the left in Figure, and the compression, using a = _, to get h() = _ () as shown on the right in Figure. Vertical stretch g() = () f () = 0 (a) Vertical compression f () = h() = () 0 = 0 = 0 Figure (a) g() = () stretches the graph of f () = verticall b a factor of. (b) h() = () compresses the graph of f () = verticall b a factor of. (b) stretches and compressions of the parent function f ( ) = b For an factor a > 0, the function f () = a(b) is stretched verticall b a factor of a if a >. is compressed verticall b a factor of a if a <. has a -intercept of (0, a). has a horizontal asmptote at = 0, a range of (0, ), and a domain of (, ), which are unchanged from the parent function. Eample Graphing the Stretch of an Eponential Function Sketch a graph of f () = _. State the domain, range, and asmptote. Solution Before graphing, identif the behavior and ke points on the graph. Since b = _ is between zero and one, the left tail of the graph will increase without bound as decreases, and the right tail will approach the -ais as increases. Since a =, the graph of f () = _ will be stretched b a factor of. Create a table of points as shown in Table. 0 f () = 0. Table Plot the -intercept, (0, ), along with two other points. We can use (, ) and (, ).

8 SECTION. Graphs of Eponential Functions 9 Draw a smooth curve connecting the points, as shown in Figure 9. f() = 0 0 (, ) (0, ) (, ) f () = Figure 9 The domain is (, ); the range is (0, ); the horizontal asmptote is = 0. Tr It # Sketch the graph of f () = _ (). State the domain, range, and asmptote. Graphing Reflections In addition to shifting, compressing, and stretching a graph, we can also reflect it about the -ais or the -ais. When we multipl the parent function f () = b b, we get a reflection about the -ais. When we multipl the input b, we get a reflection about the -ais. For eample, if we begin b graphing the parent function f () =, we can then graph the two reflections alongside it. The reflection about the -ais, g() =, is shown on the left side of Figure 0, and the reflection about the -ais h() =, is shown on the right side of Figure 0. Reflection about the -ais Reflection about the -ais f () = f () = h() = = 0 = 0 g() = 0 Figure 0 (a) g() = reflects the graph of f () = about the -ais. (b) g() = reflects the graph of f () = about the -ais. reflections of the parent function f () = b The function f () = b reflects the parent function f () = b about the -ais. has a -intercept of (0, ). has a range of (, 0). has a horizontal asmptote at = 0 and domain of (, ), which are unchanged from the parent function. The function f () = b reflects the parent function f () = b about the -ais. has a -intercept of (0, ), a horizontal asmptote at = 0, a range of (0, ), and a domain of (, ), which are unchanged from the parent function. This OpenSta book is available for free at

9 0 CHAPTER Eponential and logarithmic Functions Eample Writing and Graphing the Reflection of an Eponential Function Find and graph the equation for a function, g (), that reflects f () = _ about the -ais. State its domain, range, and asmptote. Solution Since we want to reflect the parent function f () = _ about the -ais, we multipl f () b to get, g () = _. Net we create a table of points as in Table. 0 g() = Table Plot the -intercept, (0, ), along with two other points. We can use (, ) and (, 0.). Draw a smooth curve connecting the points: g() 0 = 0 (, ) 0 g() = (, 0.) (0, ) Figure The domain is (, ); the range is (, 0); the horizontal asmptote is = 0. Tr It # Find and graph the equation for a function, g(), that reflects f () =. about the -ais. State its domain, range, and asmptote. Summarizing Translations of the Eponential Function Now that we have worked with each tpe of translation for the eponential function, we can summarize them in Table to arrive at the general equation for translating eponential functions. Shift Translations of the Parent Function f () = b Translation Form Horizontall c units to the left Verticall d units up Stretch and Compress Stretch if a > Compression if 0 < a < Reflect about the -ais f () = b + c + d f () = ab f () = b Reflect about the -ais f () = b = _ b General equation for all translations Table f () = ab + c + d

10 SECTION. Graphs of Eponential Functions translations of eponential functions A translation of an eponential function has the form f () = ab + c + d Where the parent function, = b, b >, is shifted horizontall c units to the left. stretched verticall b a factor of a if a > 0. compressed verticall b a factor of a if 0 < a <. shifted verticall d units. reflected about the -ais when a < 0. Note the order of the shifts, transformations, and reflections follow the order of operations. Eample Writing a Function from a Description Write the equation for the function described below. Give the horizontal asmptote, the domain, and the range. f () = e is verticall stretched b a factor of, reflected across the -ais, and then shifted up units. Solution We want to find an equation of the general form f () = ab + c + d. We use the description provided to find a, b, c, and d. We are given the parent function f () = e, so b = e. The function is stretched b a factor of, so a =. The function is reflected about the -ais. We replace with to get: e. The graph is shifted verticall units, so d =. Substituting in the general form we get, f () = ab + c + d = e = e + The domain is (, ); the range is (, ); the horizontal asmptote is =. Tr It # Write the equation for function described below. Give the horizontal asmptote, the domain, and the range. f () = e is compressed verticall b a factor of _, reflected across the -ais and then shifted down units. Access this online resource for additional instruction and practice with graphing eponential functions. Graph Eponential Functions ( This OpenSta book is available for free at

11 CHAPTER Eponential and logarithmic Functions. Section EXERCISES Verbal. What role does the horizontal asmptote of an eponential function pla in telling us about the end behavior of the graph? Algebraic. The graph of f () = is reflected about the -ais and stretched verticall b a factor of. What is the equation of the new function, g()? State its -intercept, domain, and range.. The graph of f () = 0 is reflected about the -ais and shifted upward 7 units. What is the equation of the new function, g()? State its -intercept, domain, and range. 7. The graph of f () = _ _ + is shifted downward units, and then shifted left units, stretched verticall b a factor of, and reflected about the -ais. What is the equation of the new function, g()? State its -intercept, domain, and range.. What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraicall?. The graph of f () = _ is reflected about the -ais and compressed verticall b a factor of _. What is the equation of the new function, g()? State its -intercept, domain, and range.. The graph of f () = (.) is shifted right units, stretched verticall b a factor of, reflected about the -ais, and then shifted downward units. What is the equation of the new function, g()? State its -intercept (to the nearest thousandth), domain, and range. Graphical For the following eercises, graph the function and its reflection about the -ais on the same aes, and give the -intercept.. f () = _ 9. g() = (0.) 0. h() = (.7) For the following eercises, graph each set of functions on the same aes.. f () = _, g() = (), and h() = (). f () = (), g() = (), and h() = () For the following eercises, match each function with one of the graphs in Figure. B C D E _ A F Figure. f () = (0.9). f () = (.). f () = (0.). f () = (.) 7. f () = (.9). f () = (0.9)

12 SECTION. Section Eercises For the following eercises, use the graphs shown in Figure. All have the form f () = ab. B C D A E F Figure 9. Which graph has the largest value for b? 0. Which graph has the smallest value for b?. Which graph has the largest value for a?. Which graph has the smallest value for a? For the following eercises, graph the function and its reflection about the -ais on the same aes.. f () = _ (). f () = (0.7). f () = () + For the following eercises, graph the transformation of f () =. Give the horizontal asmptote, the domain, and the range.. f () = 7. h() = +. f () = For the following eercises, describe the end behavior of the graphs of the functions. 9. f () = () 0. f () = _. f () = () + For the following eercises, start with the graph of f () =. Then write a function that results from the given transformation.. Shift f () units upward. Shift f () units downward. Shift f () units left. Shift f () units right. Reflect f () about the -ais 7. Reflect f () about the -ais For the following eercises, each graph is a transformation of =. Write an equation describing the transformation This OpenSta book is available for free at

13 CHAPTER Eponential and logarithmic Functions For the following eercises, find an eponential equation for the graph Numeric For the following eercises, evaluate the eponential functions for the indicated value of.. g () = _ (7) for g().. f () = () for f ().. h() = _ + for h( 7). _ Technolog For the following eercises, use a graphing calculator to approimate the solutions of the equation. Round to the nearest thousandth. f () = ab + d.. 0 = _ 7. = _ _. = () + 9. = _ 0. 0 = () + + Etensions. Eplore and discuss the graphs of f () = (b) and g() = _ b. Then make a conjecture about the relationship between the graphs of the functions b and _ b for an real number b > 0.. Eplore and discuss the graphs of f () =, g() =, and h() = _. Then make a conjecture about the relationship between the graphs of the functions b and _ b n b for an real number n and real number b > 0.. Prove the conjecture made in the previous eercise.. Prove the conjecture made in the previous eercise.