Name..Class. Date. Expnential Functins, Grwth and Decay Essential questin: What are the characteristics f an expnential junctin? In an expnential functin, the variable is an expnent. The parent functin isf(x) If, where b is any real number greater than 0, except 1. CC.9-12,F.IE7e EXAMPLE - Graphing f(x) = If fr b > 1 Graph f(x) = 2*. A B Cmplete the table f values belw. Plt the pints n the graph and cnnect the pints with a smth curve. -3 f (x) = 2" -2 4-1 I j. 10 -f -t-m-4- Q. O U ^REFLECT \. What happens t/(x) as x increases withut bund? What happens t/(x) as x decreases withut bund? j. Des the graph intersect the x-axis? Explain hw yu knw. 1 c. What are the dmain and range f Chapter 4 191 Lessn 1
CC.9~12.F.IF.7e E 1 A M P I, ii. Grapmrag ffcr) = Graph f(x) = (1)* A Cmplete the table f values belw. B Plt the pints n the graph and cnnect the pints with a smth curve. y I i -3-2 -1 i D I --ff'- ' 4 * ( REFLECT ^\. What happens t/(^) as x increases withut bund? What happens t/(x) as x decreases withut bund? X f i \. Hw d the dmain and range ff(x) 1^1 cmpare t the dmain and range f W = 2'? 2c, What d yu ntice abut the y-intercepts f the graphs ff(x) = I ^ J and f(x) = 2X1 Why des this make sense?. What transfrmatin can yu use t btain the graph ff(x) = f ^1 frm the graph f/(.x) - 2*? Chapter 4 192 Lessn 1
CC9-12.F.IR7e ENGAGE Recgnizing Types f Expnential Functins A functin f the frm/(x) = V is an expnential grwth functin if b > 1 and an expnential decay functin if 0 < b < I. Expnential Grwth f(x) = b* f r b > 1 Expnential Decay f(x) = b* fr 0 <b< y I f"reflect'";\. Describe the end behavir f an expnential grwth functin. 3b. Describe the end behavir f an expnential decay functin. 3e. Explain why the pint (1, b} is always n the graph ff(x) = b*. Q. 3d. Explain why the pint (0, 1) is always n the graph f/(x) = bx. O U -. Aief(x) = 3X and g(x) = 5"x bth expnential grwth functins r bth expnential decay functins? Althugh they have the same end behavir, hw yu d think their graphs differ? Explain yur reasning. I 31. Are/(jc) = fi] andg(x) = f ) bth expnential grwth functins r bth expnential decay functins? Althugh they have the same end behavir, hw d yu think their graphs differ? Explain yur reasning. Chapter 4 193 Lessn 1
CC.9-12.FLE.3 E JC F L 0 11 Cmparing Onea?, Cuibk, aaid Cmpare each f the functins f(x) = x + 3 and g(x) = x3 t the expnential functin h(x) = 3* fr x > 0. A Cmplete the table f values fr the three functins. X f (x) = x + 3 g(x) = x3 Ai(x) = 3' 0 3 0 1 1 2 3 4 5 B The graph f h(x) 3X is shwn n the crdinate grid belw. Graph /(X) = x + 3 n the same grid. C The graph f h(x) = 3* is shwn n the crdinate grid belw. Graph g(x) x? n the same grid. X 2 4 6 10 8 10 [ REFLECTJ\. Hw d the values f h(x) cmpare t thse f/(x) and g(x) as x increases n withut bund? Chapter 4 194 Lessn 1
PRACTICE Tell whether the functin describes an expnential grwth functin r an expnential decay functin. Explain hw yu knw withut graphing. 2. 3. 4. = gf 5. In an expnential functin, f(x) = If, b is nt allwed t be 1. Explain why this restrictin exists. 6. Cmplete the table fr/(jc) = 4*. Then sketch the graph f the functin. -1 f(x) u 01 7. Cmplete the table fr/(x) = f ]. Then sketch the graph f the functin. -3 f(x) I -2-1 Chapter 4 195 Lessn 1
8. Cmpare the graph ff(x) = 2X t the graph f g(x) = x2. 9. Enter the functins/(jc) = 10* and g(x) = f^rl int yur graphing calculatr. a. Lk at a table f values fr the tw functins. Fr a given x-value, hw d the crrespnding functin values cmpare? b. Lk at graphs f the tw functins. Hw are the tw graphs related t each ther? le graph f an expnential functin f(x) b* is shwn. a. Which f the labeled pints, (0, 1) r (1, 5), allws yu t determine the value f b! Why desn't the ther pint help? - b. What is the value f bl Explain hw yu knw. " -i - E 6-, 2- I i 1i \ i-'! ^0..._ 4-..-.L I/ ; AI /»( _,5 JLi ii i >,1 -! tu - _,._ 11. Given an expnential functin y = V, when yu duble the value f x, hw des the value f y change? Explain. 12. Given an expnential functin y = b*, when yu add 2 t the value f x, hw des the value f y change? Explain. 13. Errr Analysis A student says that the functin/(:c) = f~j is an expnential decay functin. Explain the student's errr. 10 n 3 14. One methd f cutting a lng piece f string int smaller pieces is t make individual cuts, s that 1 cut results in 2 pieces, 2 cuts result in 3 pieces, and s n. Anther methd f cutting the string is t fld it nt itself and cut the flded end, then fld the pieces nt themselves and cut their flded ends at the same time, and cntinue t fld and cut, s that 1 cut results in 2 pieces, 2 cuts result in 4 pieces, and s n. Fr each methd, write a functin that gives the number p f pieces in terms f the number c f cuts. Which functin grws faster? Why? Chapter 4 196 Lessn 1
Name,. Class. Date. mm^m^^^^e^^^k^m^^^^^g^m^^a,^^^^^^ Tell whether the functin shws grwth r decay. Then graph. 1. g(x) = -(2)x 2. Mx) = -0.5(0.2)x /.10-5 -4-3 -2 -i I i 1 4i 1 2 3 4 5 X -5-4 -3-2 -1-1 2 3 4 5-50 -50 3.y(x) = -2(0.5)x 4. p(x) = 4(1.4)x y. ' : ' y -5-4 -3-2 -1 1 2 3 4 5 - j i i-30- E u ra Slve. 4-20- 4-30- -4-20- 4-30- --40- -1-0- 4-2Q- -430- -40- -50 5. A certain car's value depreciates abut 15% each year. This is mdeled by the functin V(t) = 20,000(0.85)' where $20,000 is the value f a brand-new mdel. 5 -< 30,000 25,000 20,000 y. -3 -j -1 0 ; : -h 1 2 3 4 5 ' ; ; i! a. Graph the functin. b. Suppse the car was wrth $20,000 in 2005. What is the first year that the value f this car will be wrth less than half f that value? 15,000 10,000 5,000 1 2 3 4 5 6 7 ' >» X 9 10 Chapter 4 197 Lessn 1
Prblem Slving Justin drve his pickup truck abut 22,000 miles in 2004. He read that in 1988 the average residential vehicle traveled abut 10,200 miles, which increased by abut 2.9% per year thrugh 2004. 1. Write a functin fr the average mileage, m(t), as a functin f t, the time in years since 1988. 2. Assume that the 2.9% increase is valid thrugh 2008 and use yur functin t cmplete the table t shw the average annual miles driven. Year t m(t) 1988 0 10,200 1992 4 1996 2000 2004 2008 3. Did Justin drive mre r fewer miles than the average residential vehicle driver in 2004? by hw much (t the nearest 100 miles)? 4. Later Justin read that the annual mileage fr light trucks increased by 7.8% per year frm 1988 t 2004. a. Write a functin fr the average miles driven fr a light truck, n(t), as a functin f t, the time in years since 1988. He assumes that the average number f miles driven in 1988 was 10,200. b. Graph the functin. Then use yur graph t estimate the average number f miles driven (t the nearest 1000) fr a light truck in 2004. 400 00^ 35000 30000 } f c. Did Justin drive mre r fewer miles than the average light truck driver in 2004? by hw much? 20000 0 2 4 6 8 10 12 14 16 18 20 9 3 Justin bught his truck new fr $32,000. Its value decreases 9.0% each year. Chse the letter fr the best answer. 5. Which functin represents the yearly value f Justin's truck? A f(t) = 32,000(1 +0.9)' B f(t) = 32,000(1-0.9)f C f(t) = 32,000(1 + 0.09)f D f(t) = 32,000(1-0.09)' 6. When will the value f Justin's truck fall belw half f what he paid fr it? F In 6 years G In 8 years H In 10 years J In 12 years Chapter 4 198 Lessn 1