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1 Chapter Review FREQUENTLY ASKED Questions Q: How can ou identif an eponential function from its equation? its graph? a table of values? A: The eponential function has the form f () 5 b, where the variable is an eponent. The shape of its graph depends upon the parameter b. 1 = = 1 ( ) 1 Stud See Lesson.5. Tr Chapter Review Questions 9 and If b. 1, then the curve increases as If, b, 1, then the curve increases. decreases as increases. In each case, the function has the -ais (the line 5 ) as its horizontal asmptote. A differences table for an eponential function shows that the differences are never constant, as the are for linear and quadratic functions. The are related b a multiplication pattern. First Second 5 3 Differences Differences Chapter Eponential Functions 65

2 Stud See Lesson.6, Eamples 1,, and 3. Tr Chapter Review Questions 11 and 1. Stud See Lesson.7, Eamples 1,, 3 and. Tr Chapter Review Questions 13 to 17. Q: How can transformations help in drawing the graphs of eponential functions? A: Functions of the form g () 5 af (k( d )) 1 c can be graphed b appling the appropriate transformations to the ke points and asmptotes of the parent function f () 5 b, following an appropriate order often, stretches and compressions, then reflections, and finall translations. In functions of the form g () 5 ab k(d ) 1 c, the constants a, k, d, and c change the location or shape of the graph of f (). The shape of the graph of g() depends on the value of the base of the function, f () 5 b. a represents the vertical stretch or compression factor. If a,, then the function has also been reflected in the -ais. k represents the horizontal stretch or compression factor. If k,, then the function has also been reflected in the -ais. c represents the number of units of vertical translation up or down. d represents the number of units of horizontal translation right or left. Q: How can eponential functions model growth and deca? How can ou use them to solve problems? A: Eponential functions can be used to model phenomena ehibiting repeated multiplication of the same factor. Each formula is modelled after the eponential function 5 ab The amount in the future The present value or initial amount The growth/deca factor: if growth, then b. 1; if deca, then, b, 1 The number of periods of growth or deca When solving problems, list these four elements of the equation and fill in the data as ou read the problem. This will help ou organize the information and create the equation ou require to solve the problem. Here are some eamples: Growth Cell division (doubling bacteria, east cells, etc.): P(t) 5 P () t D Population growth: P(n) 5 P (1 1 r) n Growth in mone: A(n) 5 P(1 1 i) n Deca Radioactivit or half-life: N(t) 5 1a 1 b t H Depreciation of assets: V(n) 5 V (1 r)n Light intensit in water: V(n) 5 1(1 r) n 66 Chapter Review

3 PRACTICE Questions Lesson. 1. If. 1, which is greater, or? Wh? Are there values of that make the statement. true? Eplain.. Write each as a single power. Then evaluate. Epress answers in rational form. (7) 3 (7) 1 ( 3 ) 6 ( ) () () 3 e) (11) 9 a b f) a (3)7 (3) 3 (5) 3 (5) 6 b 5 3 (3 ) 3 Lesson.3 3. Epress each radical in eponential form and each power in radical form.! 3 7 (!p) Evaluate. Epress answers in rational form. a 3 (! 3 7) 5 b a 16.5 e) 5 b (1).5 f)! Simplif. Write with onl positive eponents. a 3 ( 3 a ) b. b. e) 5 c a c 6b f) c m 1.5 (! 5 3)(! 6 6) 5 " 6 (() 3 ) d 5 d 11 (d 3 ) ( (e ) 7 ) ((f 1 6 ) 6 5 ) 1 6. Eplain wh!a 1 b!a 1!b, for a. and b.. Lesson. 7. Evaluate each epression for the given values. Epress answers in rational form. (5) () 3 ; 5 m 5 m 5 (m) 3 ; w(3w ) (w) ; w 53 (9) 5 (3 1 ) 3; e) (6( ) 3 ) 1 ; 5 ( ) 3 (6) f) 5 1 (3 1 ) 3 ;. Simplif. Write each epression using onl positive eponents. All variables are positive. Lesson.5! a 6 b 5 e) Å a b 3 m 3 n f) m 7 n 3 " 16 ( 6 ) 6 ( ) 11 ((.5 ) 3 ) 1. " 6 ( 3 ) ( 3 ) 9. Identif the tpe of function (linear, quadratic, or eponential) for each table of values Chapter Eponential Functions 67

4 e) f) Identif each tpe of function (linear, quadratic, or eponential) from its graph. Lesson For each eponential function, state the base function, 5 b. Then state the transformations that map the base function onto the given function. Use transformations to sketch each graph. 5 a 1 b () 1 1 5(3) (5) The eponential function shown has been reflected in the -ais and translated verticall. State its -intercept, its asmptote, and a possible equation for it Chapter Review

Lesson.7 13. Complete the table. Eponential Initial Growth Growth or Value (- or Deca Function Deca? intercept) Rate V(t) 5 1(1.) t P(n) 5 3(.95) n A() 5 5(3) 15.

A hot cup of coffee cools according to the equation T(t) 5 69a 1 b t 3 1 1 where T is the temperature in degrees Celsius and t is the time in minutes.

5 Lesson Complete the table. Eponential Initial Growth Growth or Value (- or Deca Function Deca? intercept) Rate V(t) 5 1(1.) t P(n) 5 3(.95) n A() 5 5(3) 15. The value of a car after it is purchased depreciates according to the formula V(n) 5 (.75) n where V(n) is the car s value in the nth ear since it was purchased. Q(n) 5 6a 5 b n 1. A hot cup of coffee cools according to the equation T(t) 5 69a 1 b t where T is the temperature in degrees Celsius and t is the time in minutes. Which part of the equation indicates that this is an eample of eponential deca? What was the initial temperature of the coffee? Use our knowledge of transformations to sketch the graph of this function. Determine the temperature of the coffee, to the nearest degree, after min. e) Eplain how the equation would change if the coffee cooled faster. f) Eplain how the graph would change if the coffee cooled faster. What is the purchase price of the car? What is the annual rate of depreciation? What is the car s value at the end of 3 ears? What is its value at the end of 3 months? e) How much value does the car lose in its first ear? f) How much value does it lose in its fifth ear? 16. Write the equation that models each situation. In each case, describe each part of our equation. the percent of a pond covered b water lilies if the cover one-third of a pond now and each week the increase their coverage b 1% the amount remaining of the radioactive isotope U if it has a half-life of ears the intensit of light if each gel used to change the colour of a spotlight reduces the intensit of the light b % 17. The population of a cit is growing at an average rate of 3% per ear. In 199, the population was 5. Write an equation that models the growth of the cit. Eplain what each part of the equation represents. Use our equation to determine the population of the cit in 7. Determine the ear during which the population will have doubled. Suppose the population took onl 1 ears to double. What growth rate would be required for this to have happened? Chapter Eponential Functions 69

Chapter 8 Notes SN AA U2C8

Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of