Transformations of Exponential Functions. Investigate the effects of transformations on the graphs and equations of exponential functions.

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1 . Transformations of Eponential Functions YOU WILL NEED graphing calculator GOAL Investigate the effects of transformations on the graphs and equations of eponential functions. INVESTIGATE the Math Recall the graph of the function f () 5. It is an increasing function. It has a -intercept of. Its asmptote is the line 5. =? If f() 5, how do the parameters a, k, d, and c in the function g() 5 af(k( d)) c affect the size and shape of the graph of f()? Tech Support You can adjust to these settings b pressing ZOOM and. ZDecimal. A. Use our graphing calculator to graph the function f () 5. Use the window settings shown. B. Predict what will happen to the function f () 5 if it is changed to g() 5 or h() 5 p() 5 or q() 5. Transformations of Eponential Functions

2 C. Cop and complete the table b graphing the given functions, one at a time, as Y. Keep the graph of f () 5 as Y for comparison. For each function, sketch the graph on the same grid and describe how its points and features have changed. Description of Function Sketch Table of Values Changes of New Graph g() h() p() q() D. Describe the tpes of transformations ou observed in part C. Comment on how the features and points of the original graph were changed b the transformations. E. Predict what will happen to the function f () 5 if it is changed to g() 5 3( ) h() 5.5( ) j() 5( ) F. Create a table like the one in part C using the given functions in part E. Graph each function one at a time, as Y. Keep the graph of f () 5 as Y for comparison. In our table, sketch the graph on the same grid, complete the table of values, and describe how its points and features have changed. Chapter Eponential Functions 5

3 G. Describe the tpes of transformations ou observed in part F. Comment on how the features and points of the original graph were changed b the transformations. H. Predict what will happen to the function f () 5 if it is changed to g() 5 h() 5.5 j() 5 I. Create a table like the one in part C using the given functions in part H. Graph each function one at a time, as Y. Keep the graph of f () 5 as Y for comparison. In our table, sketch the graph, complete the table of values, and describe how its points and features have changed. J. Describe the tpes of transformations ou observed in part I. Comment on how the features and points of the original graph were changed b such transformations. K. Choose several different bases for the original function. Eperiment with different kinds of transformations. Are the changes in the function affected b the value of the base? L. Summarize our findings b describing the roles that the parameters a, k, d, and c pla in the function defined b f () 5 ab k(d ) c. Reflecting M. Which transformations change the shape of the curve? Eplain how the equation is changed b these transformations. N. Which transformations change the location of the asmptote? Eplain how the equation is changed b these transformations. O. Do the transformations affect f () 5 b in the same wa the affect f () 5, f () 5, f () 5, f () 5!, and f () 5? Eplain.. Transformations of Eponential Functions

4 APPLY the Math EXAMPLE Using reasoning to predict the shape of the graph of an eponential function Use transformations to sketch the function 5(3 ). State the domain and range. J.P. s Solution = 3 = 3 = (3 ) I began b sketching the graph of 5 3. Three of its ke points are (, ), (, 3), and (, 3). The asmptote is the -ais, 5. The function I reall want to graph is 5(3 ). The base function, 5 3, was changed b multipling all -values b, resulting in a vertical stretch of factor and a reflection in the -ais. Subtracting from results in a translation of units to the right. I could perform these two transformations in either order, since one affected onl the -coordinate and the other affected onl the -coordinate. I did the stretch first. With vertical stretches and reflection in the -ais (multipling b graphed in red), m ke points had their -values doubled: (, ) S (, ), (, 3) S (, ), and (, S 3) (, 3) The asmptote 5 was not affected. With translations (subtracting, graphed in black), the ke points changed b adding to the -values: (, ) S (, ), (, ) S (5, ), and (, S 3) (3, 3) This shifted the curve units to the right. The asmptote 5 was not affected., = ( 3 ) The domain of the original function, 5 [ R, was not changed b the transformations. The range, determined b the equation of the asmptote, was. for the original function. There was no vertical translation, so the asmptote remained the same, but, due to the reflection in the -ais, the range changed to 5 [ R,. Chapter Eponential Functions 7

5 EXAMPLE Connecting the graphs of different eponential functions Use transformations to sketch the graph of 5 3. Ilia s Solution = I began b sketching the graph of the base curve, 5. It has the line 5 as its asmptote, and three of its ke points are (, ), (, ), and (, ). I factored the eponent to see the different transformations clearl: 5 () 3 The -values were multiplied b, resulting in a horizontal compression of factor, as well as a reflection in the -ais. There were two translations: units to the left and 3 units up. I applied the transformations in the proper order. = 3 = The table shows how the ke points and the equation of the asmptote change: = Point or Horizontal Stretch Horizontal Vertical Asmptote and Reflection Translation Translation (, ) (, ) (, ) (, ) 5 (, ) (, ) 5 (,) (, ) (, ) 5 (, ) (, 7) (, 3 ) 5 3 There was one stretch and one reflection, each of which applied onl to the -coordinate: a horizontal compression of factor and a reflection in the -ais (shown in red). There were two translations: units to the left and 3 units up (shown in black).. Transformations of Eponential Functions

6 EXAMPLE 3 Communicating the relationship among different eponential functions Compare and contrast the functions defined b f () 5 9 and g() 5 3. Pinder s Solution: Using Eponent Rules f () (3 ) g() Both functions are the same. 9 is a power of 3, so, to make it easier to compare 9 with 3, I substituted 3 for 9 in the first equation. B the power-of-a-power rule, f() has the same equation as g(). Kareem s Solution f() = 9 = 3 Both functions are the same. f() 5 9 is an eponential function with a -intercept of and the line 5 as its asmptote. Also, f() 5 9 passes through the points (, 9) and (, 9). g() 5 3 is the base function 5 3 after a horizontal compression of factor. This means that the ke points change b multipling their -values b. The point (, 3) becomes (.5, 3) and (, 9) becomes (, 9). When I plotted these points, I got points on the curve of f(). Chapter Eponential Functions 9

7 EXAMPLE Connecting the verbal and algebraic descriptions of transformations of an eponential curve An eponential function with a base of has been stretched verticall b a factor of.5 and reflected in the -ais. Its asmptote is the line 5. Its -intercept is (, 3.5). Write an equation of the function and discuss its domain and range. Louise s Solution 5 a k(d) c 5.5( ) c 5.5( ) 5.5( () ) 5.5() The original domain is 5 [ R. The transformations didn t change this. The range changed, since there was a vertical translation. The asmptote moved up units along with the function, so the range is 5 [ R.. I began b writing the general form of the eponential equation with a base of. Since the function had been stretched verticall b a factor of.5, a 5.5. The function has also been reflected in the -ais, so k 5. There was no horizontal translation, so d 5. Since the horizontal asmptote is 5 the function has been translated verticall b units, so c 5. I substituted 5 into the equation and calculated the -intercept. It matched the stated -intercept, so m equation seemed to represent this function. 5. Transformations of Eponential Functions

8 In Summar Ke Ideas In functions of the form g() 5 af(k( d)) c, the constants a, k, d, and c change the location or shape of the graph of f(). The shape is dependent on the value of the base function f() 5 b, as well as on the values of a and k. Functions of the form g() 5 af(k( d)) c can be graphed b appling the appropriate transformations to the ke points of the base function f() 5 b, one at a time, following the order of operations. The horizontal asmptote changes when vertical translations are applied. Need to Know In eponential functions of the form g() 5 a b k(d) c: If a., a vertical stretch b a factor of a occurs. If, a,, a vertical compression b a factor of a occurs. If a is also negative, then the function is reflected in the -ais. If k., a horizontal compression b a factor of Z Z k occurs. If, k,, a horizontal stretch b a factor of Z Z k occurs. If k is also negative, then the function is reflected in the -ais. If d., a horizontal translation of d units to the right occurs. If d,, a horizontal translation to the left occurs. If c., a vertical translation of c units up occurs. If c,, a vertical translation down occurs. You might have to factor the eponent to see what the transformations are. For eample, if the eponent is, it is easier to see that there was a horizontal stretch of and a horizontal translation of to the left if ou factor to ( ). When transforming functions, consider the order. You might perform stretches and reflections followed b translations, but if the stretch involves a different coordinate than the translation, the order doesn t matter. The domain is alwas 5 [ R. Transformations do not change the domain. The range depends on the location of the horizontal asmptote and whether the function is above or below the asmptote. If it is above the asmptote, its range is. c. If it is below, its range is, c. CHECK Your Understanding. Each of the following are transformations of f () 5 3. Describe each transformation. a) g() c) b) g() d) g() 5 3 (3 ) g() For each transformation, state the base function and then describe the transformations in the order the could be applied. a) f () 53( ) c) h() 5 7(.5 ) b) g() 5 a d) k() b 3 Chapter Eponential Functions 5

9 3. State the -intercept, the equation of the asmptote, and the domain and range for each of the functions in questions and. PRACTISING. Each of the following are transformations of h() 5 ( ). Use words to describe the sequence of transformations in each case. a) g() 5a b b) g() 5 5a b (3) c) g() 5a b Let f () 5. For each function that follows, K state the transformations that must be applied to f () state the -intercept and the equation of the asmptote sketch the new function state the domain and range a) g() 5.5f () c) g() 5f ( ) b) h() 5f (.5 ) d) h() 5 f (.5 ). Compare the functions f () 5 and g() A cup of hot liquid was left to cool in a room whose temperature was C. C The temperature changes with time according to the function T(t) 5 Q t Use our knowledge of transformations to sketch this R 3. function. Eplain the meaning of the -intercept and the asmptote in the contet of this problem.. The doubling time for a certain tpe of east cell is 3 h. The number of cells after t hours is described b N(t) 5 N 3, t where N is the initial population. a) How would the graph and the equation change if the doubling time were 9 h? b) What are the domain and range of this function in the contet of this problem? 9. Match the equation of the functions from the list to the appropriate graph at the top of the net page. a) f () 5a c) b 3 g() 5a 5 b 3 b) d) h() 5 a 5 5 a b 3 b 3 5. Transformations of Eponential Functions

10 i) iii) ii) iv) Each graph represents a transformation of the function f () 5. Write an equation for each one. a) b). State the transformations necessar (and in the proper order) to transform T f () 5 5 to g() 5 ( ). Etending. Use our knowledge of transformations to sketch the function f () Use our knowledge of transformations to sketch the function g() 5 Q 3 R.5.. State the transformations necessar (and in the proper order) to transform m() 5Q 3 to n() 5Q 9 R. R Chapter Eponential Functions 53