Modeling with Exponential and Logarithmic Functions

---- Modeling with Exponential and Logarithmic Functions For use with Exploration 6.7 Essential Question How can you recognize polynomial, exponential, and logarithmic models? EXPLORATION: Recognizing Different Types of Models Go to BigldeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Match each type of model with the appropriate scatter plot. Use a regression program to find a model that fits the scatter plot. a. linear (positive slope) b. c. quadratic linear (negative slope) )< 1,- 1) d. cubic C e. ( '1 exponential E '1 i 0 f. logarithmic B =- ,, ....... rI - B. A. __ I 2 1 4 1_6_ x_\ D. Yl_j ·6-·42 E. '-/ 2 184 Algebra 2 4 6 _!J'.. f !I� ± ------------------------- Date ---- Modeling with Exponential and Logarithmic Functions (continued) EXPLORATION: Exploring Gaussian and Logistic Models Go to BigldeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Two common types of functions that are related to exponential functions are given. Use a graphing calculator to graph each function. Then determine the domain, range, intercept, and asymptote(s) of the function. a. Gaussian Function: e-x f (X) 2 b. Logistic Function: f ( X) 1 1 .... Dom�,)"\ I�\ rt'\-ls 1+� r. -ft '( �+- y;o >') /+Syfl)IJto� S �+- Communicate Your Answer y:-o y�, How can you recognize polynomial, exponential, and logarithmic models? �� $ 2, 1 i, >< Algebra 2 185 Notetaking with Vocabulary For use after Lesson 6.7 r In your own words, write the meaning of each vocabulary term. · conunon ratio , finite differences ,· } point-slope form (y = y�=m(y_-�1 M ;: 5(ope... '/ �_:. f1.t.. y � vt-1\ i,.e. XI :::- � y_ ---·--'l-1 -y..2- 186 Algebra 2 -·2.. {,J "? � 0 2. r l � v-\ ,'2. I "t)" I t / (tif ,,� �I ,;.-, -2- Al 4l, (I 0 � - ,__I .._ I" fo.,J 1 .J, l\ .l,L·\ }- "I ..._,.;:; ,I I f 1.-1 + "{ Cc,n 5 iStt 11 + pe::. � n ';,�cottd d�ff.eM,·tce ht.is " '- e) f�,f, C, �IA� ftA 11tCn" D ,,, ,�,) � � � � '--4 fo1 ,,+ 1 ,-. ' vt. l e. v y. :) ( op� - Pc,1 m 1, '\ Y\ I ( Notetaking with Vocabulary (continued) Extra Practice In Exercises 1 and 2, determine the type of function represented by the table. Explain your reasoning. 1 • 6 34 ➔ ... � 7 47 '-..,__:;J )r"J..-/'\) -5 -5l 8 -3 Co��l�½t\;t 9 79 62 V \_)> f-1. ... .2 l.,\ loAS•���f ·"'1 27 5 "--- y. t, 119 -\--Z.. ' f / 98 � � -1 11 C {';> t'd 10 J "1 l "'\ .vz.. 2. +\ I ; I I vI I I I c'd 3 �rt, � 5 +lt.41°' ; ,k o. f ,).,t i � Pl � l!--.dr,.,t{e, frA �elf OV\ V �W" 5 5.,. 243 5 \ , Algebra 2 t p ,1p>', I I FCA 1<1,t,1i� � 187 In Exercises 3-6, write an exponential function y == abx whose graph passes through the Notetaking with Vocabulary (continued) given points. 3. f;, 12), (3,10·8) ><. ,; 'X1 y :, ,::'ih (:') (�1, 12), (3, 32) \...-,/ 2. -=-°' ,b >< 1;,.-;.a1b • rb lo.�::. �·b IO'lr:. 1'2. 61. b ... I -;:. et • 32. � ( ,')b b(2)==/� b , l /ib = ot j I' -: 2 b 2(2-'):.'1 :. b �J 2::. � ·b I/{. z_\) -:r(�� b z. \ b� aj :: � b fz... ,.., '· ��-�·-•�.;�� ... 2(±):.t; z.(�') �'1 r �� 188 Algebra 2 K y :. L./ '.,=-1 -� r� "t�,, 4-_!_ (-2, 2), �1, 0.25) 6. b -=- b "'1 7. An Olympic swimmer starts selling a new type of goggles. The table shows the number y of goggles sold during a 6-month period. 1 28 Months, x Goggles sold, y I 2 47 4 79 3 64 5 6 107 97 b. Create a scatterplot of the data pairs (x, 1n y) to show that an exponential model should be a good fit for the original data pairs (x, y). Write a function that models the data. .. a. 9_�te a scatterplot of the data. 0) 5 �/, L·\ D � 1--/ I lJ -, > lj {., ,·� -, 14 l"'-0 �)(\,\ �. � r I I() l l I -i.rnowtl'\ �---4----'--f---'�......... C:, 1,,- i ,,, \ s IO T (\ tO I � �L-�'7 1.:-1 5 ') ) �t � -7 y - �-s 3 � . 2c, & ,x --. 2 b � rI +-s .33> +-�. s3 ,..r,, �'X. -?,. Ob, 1 y')= y= !,0�2.. e\ .2�&-- ')( ) :::, - ( e. Y e!,olot. d. Use each model to predict the number of goggles sold after 1 year. -:- I t- M o v-- fk.,1 y :. , 0'1 C, (t. �o7) ' 12. � .ot{, (21-1/1'28) Algebra 2 189