Use Properties of Exponents

Transcription

1 4. Georgia Performance Standard(s) MMAa Your Notes Use Properties of Eponents Goal p Simplif epressions involving powers. VOCABULARY Scientific notation PROPERTIES OF EXPONENTS Let a and b be real numbers and let m and n be integers. Product of Powers Propert a m p a n 5 a Power of a Power Propert (a m ) n 5 a Power of a Product Propert (ab) m 5 a b Negative Eponent Propert a m 5, a Þ 0 Zero Eponent Propert a 0 5, a Þ 0 Quotient of Powers Propert a m } a n 5 a, a Þ 0 Power of a Quotient Propert a } b m 5, b Þ 0 Eample a. (6 ) b. 45 } c Evaluate a numerical epression Checkpoint Evaluate or simplif the epression } 8 4 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

2 Your Notes Eample Use scientific notation in real life Iceland Iceland covers about square kilometers and has a population of approimatel people. About how man people are there per square kilometer? Solution Population } Land area 5 Divide population b land area. 5 Quotient of powers propert ø Use a calculator. 5 Zero eponent propert There are about people per square kilometer. Eample 3 Simplif epressions a. (5 ) 3 } Power of a product propert Power of a power propert 5 Quotient of powers propert 5 Simplif eponents. 5 Zero eponent propert 5 Negative eponent propert b. a 4 } b 5 Power of a quotient propert 5 5 Power of a power propert Negative eponent propert Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 5

3 Your Notes Checkpoint Simplif or evaluate the epression. Tell which properties of eponents ou used.. ( )(7 3 0 ) 3. a b } a 5 b 3 Eample 4 Compare real-life volumes Beach Ball The radius of a beach ball is about 5.6 times greater than the radius of a baseball. How man times as great as the baseball s volume is the beach ball s volume? Let r represent the radius of the baseball. 4 } Beach ball's volume 3 π( ) 3 The volume of a }} Baseball's volume 5 4 }3 πr 3 sphere is } 4 3 πr } 3 π Power of a 4 } 3 πr 3 product propert 5 Quotient of powers 5 Zero eponent propert ø The beach ball s volume is about the baseball s volume. Approimate power. times as great as Homework Checkpoint Complete the following eercise. 4. Rework Eample 4 where the radius of a beach ball is about 6 times the radius of a baseball. 6 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

4 Name Date LESSON 4. Practice Evaluate the power Evaluate the epression. Tell which properties of eponents ou used p (3) 4 (3). (5 ) 3. (7 0 ) p } (0 3 ) 3 6. } (5) 6 } } } 9 0. } 5 Write the number in scientific notation.. 57, ,983,000,000, ,600,000,000,000, Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 7

5 Name Date LESSON 4. Practice continued Write the answer in scientific notation. 7. ( )( ) 8. ( )( ) 9. ( ) 30. ( ) } } Simplif the epression. Tell which properties of eponents ou used. 33. b 4 p b p (s 7 ) 36. (5) 37. } z9 3z m } m } n 40. } 4m n 4. Earth Science The total volume of water on Earth is about 36,000,000 cubic miles. Write this number in scientific notation. 4. National Debt On August, 005, the national debt of the United States was about $7,870,000,000,000. The population of the United States at that time was about 97,000,000. If the national debt was divided evenl among everone in the countr, how much would each person owe? Write our answer in scientific notation. 8 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

6 4. Perform Function Operations and Composition Georgia Performance Standard(s) MMA5d Your Notes Goal p Perform operations with functions. VOCABULARY Power function Composition Eample Add and subtract functions Let f() 5 3 and g() 55. Find the following. a. f() g() b. f() g() c. the domains of f g and f g Solution a. f() g() 5 3 (5 ) 5 b. f() g() 5 3 (5 ) 5 c. The functions f and g each have the same domain:. So, the domains of f g and f g also consist of. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 9

7 Your Notes Eample Multipl and divide functions Let f() 5 7 and g() 5 6. Find the following. a. f() p g() b. f() } g() c. the domains of f p g and f } g Solution a. f() p g() 5 (7)( 6 ) 5 b. f() } g() 5 c. The domain of f consists of, and the domain of g consists of. So, the domain of f p g consists of. Because g(0) 5, the domain of } f g is restricted to. Checkpoint Let f() and g() 5 3. Perform the indicated operation. State the domain.. f() g(). f() g() 3. f() p g() 4. f() } g() 0 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

8 Your Notes Eample 3 Find compositions of functions Let f() 5 6 and g() Find the following. a. f(g()) b. g(f()) c. f(f()) d. the domain of each composition Solution a. f(g()) 5 f(3 5) 5 b. g(f()) 5 g(6 ) 5 c. f(f()) 5 f(6 ) 5 d. The domain of f(g()) consists of ecept 5 because g 5 0 is not in the. (Note that f(0) 5, which is.) The domains of g(f()) and f(f()) consist of ecept 5, again because. Checkpoint Complete the following eercise. 5. Let f() and g() 5 3. Find (a) f(g()), (b) g(f()), (c) f(f()), and (d) the domain of each composition. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics

9 Your Notes Eample 4 Solve a multi-step problem Computer Store You purchase a computer with a price of $700. The computer store applies a store fler coupon of $00 and a % promotional discount. Use composition to find the final price of the purchase when the coupon is applied before the discount. Use composition to find the final price of the purchase when the discount is applied before the coupon. Step Write functions for the discounts. Let be the original price, f() be the price after the $00 coupon, and g() be the price after the % promotional discount. Function for the $00 coupon: f() 5 Function for the % discount: g() 5 5 Step Compose the functions. $00 coupon is applied first: g(f()) 5 5 % discount is applied first: f(g()) 5 5 Step 3 Evaluate the functions g(f()) and f(g()) when g(f(700)) f(g(700)) The final price is when the $00 coupon is applied before the % discount. The final price is when the % discount is applied before the $00 coupon. Checkpoint Complete the following eercise. Homework 6. Rework Eample 4 for an original price of $800, a $0 coupon, and a 5% discount. Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

10 Name Date LESSON 4. Practice Let f() 5, g() 5 3, and h() 5 3. Perform the indicated operation. State the domain.. f() g(). f() h() 3. h() g() 4. f() g() 5. h() f() 6. g() h() Let f() 5 4 3, g() 5 4, and h() 5 6. Perform the indicated operation. State the domain. 7. f() p g() 8. f() p h() 9. h() p g() 0. f() } g(). h() } f(). h() } g() Let f() 5 3, g() 5, and h() 5 }. Find the indicated value f(g()) 4. h(g(4)) 5. f(h(6)) 6. g( f()) 7. h( f(3)) 8. g(g()) Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3

11 Name Date LESSON 4. Practice continued Let f() 5, g() 5 5, and h() 5 } 4. Perform the indicated operation. 9. f(g()) 0. g(h()). f(h()). g( f()) 3. h( f()) 4. g(g()) Let f() 5, g() 5 3, and h() 5 }. State the domain of the operation. 5. f() g() 6. h() f() 7. h() p g() 8. g() } f() 9. h(g()) 30. f(g()) 3. Profit A compan estimates that its cost C and revenue R can be modeled b the functions C() ,000 and R() 5.5 where is the number of units produced. The compan s profit P is modeled b P() 5 R() C(). Find the profit equation and determine the profit when 500,000 units are produced. 4 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

12 4.3 Use Inverse Functions Georgia Performance Standard(s) MMA5a, MMA5b, MMA5c, MMA5d Goal p Find inverse functions. VOCABULARY Inverse relation Your Notes Inverse functions nth root of a INVERSE FUNCTIONS Functions f and g are inverse functions of each other provided: f(g()) 5 and g(f()) 5 The function g is denoted b f, read as f inverse. HORIZONTAL LINE TEST The inverse of a function f is also a function if and onl if no horizontal line intersects the graph of f. Function Not a function Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 5

13 Your Notes Eample Verif that functions are inverses Verif that f() and f () 5 } } 4 are 7 7 inverse functions. Show that f(f ()) 5. Show that f (f()) 5. f (f()) 5 f (7 4) f(f ()) 5 f } 7 4 } Checkpoint Find the inverse of the function. Then verif that our result and the original function are inverses.. f() 53 5 Eample Find the inverse of a function of the form 5 a } Consider the function f() 5 } 3. Determine whether the inverse of f is a function. Then find the inverse. Graph the function. Notice that no horizontal line intersects the graph more than once. The inverse of f is a function. To find an equation for f, complete the following steps. 5 3 } Replace f() with. 5 } Switch and. 5 3 Multipl each side b. 5 Divide each side b. The inverse of f is f () 5. 6 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

14 Your Notes Checkpoint Complete the following eercise.. Find the inverse of f() 5 4 }. Eample 3 Find the inverse of f() 5 4, 0. Then graph f and f. f() 5 4 Find the inverse of a quadratic function Write original function. 5 4 Replace f() with. Switch and. Divide each side b 4. Take square roots of each side. The domain of f is restricted to negative values of. So, the range of f must also be restricted to negative values, and therefore the inverse is f () 5. (If the domain were restricted to 0, ou would choose f () 5.) Checkpoint Complete the following eercise. 3. Find the inverse of f() 5 9, 0. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 7

15 Your Notes Eample 4 Find the inverse of a power function Consider the function f() Determine whether the inverse of f is a function. Then find the inverse. Solution Graph the function f. Notice that no intersects the graph more than once. So, the inverse of f is a. To find an equation for f, complete the following steps. f() Write original function Replace f() with. Switch and. Divide each side b. Take seventh root of each side. The inverse of f is f () 5. Checkpoint Find the inverse of the function. 4. f() g() 5 } 3 5 Homework 8 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

16 Name Date LESSON 4.3 Practice Find the inverse relation Find an equation for the inverse relation } } 3 } 3 Graph the function. Then use the horizontal line test to determine whether the inverse of f is a function. 9. f() f() 5 4. f() 5 5 } Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 9

17 Name Date LESSON 4.3 Practice continued Verif that f and g are inverse functions.. f() 5 ; g() 5 3. f() 5 3; g() 5 } 3 4. f() 5 3 ; g() 5 Ï 3 } 5. f() 5 4 ; g() 5 } 4 } 4 6. f() 5 } ; g() 5 } 7. f() 5 } 3 ; g() 5 } } 6 In Eercises 8 and 9, use the following information. Conversion The formula to convert miles m to kilometers k is.609m 5 k. 8. Write the inverse function, which converts kilometers to miles. 9. How man miles is 40 kilometers? Round our answer to two decimal places. In Eercises 0 and, use the following information. Geometr The formula C 5 πr gives the circumference of a circle of radius r. 0. Write the inverse function, which gives the radius of a circle of circumference C.. What is the radius of a circle with a circumference of 4 inches? Round our answer to two decimal places. 30 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

18 4.4 Graph Eponential Growth Functions Georgia Performance Standard(s) MMAb, MMAc Your Notes Goal p Graph and use eponential growth functions. VOCABULARY Eponential function Eponential growth function Growth factor Asmptote End behavior Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3

19 Your Notes Eample Graph 5 b for b > Graph 5 3. Analze the graph. Find the average rate of change over the interval. Step Make a table of values. 0 Step Plot the points from the table. Step 3 Draw from left to right, a smooth curve that begins just the -ais, passes through the plotted points, and moves. Step 4 Eamine the graph. The graph intersects the -ais at point. So the -intercept of 5 3 is. You can see from the graph that as the value of approaches ` the value of the function approaches but never reaches. So the function has no zeros or -intercepts. As the value of approaches `, the value of the function approaches. Also, the function is increasing on the interval. Using the points, } 9, and (, 9), the average rate of change over the interval is, or about. Checkpoint Complete the following eercise.. Graph 5 and find the average rate of change over the interval. 3 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

20 Your Notes Eample Graph 5 ab for b > Graph the function 5 } 4 p 6. Solution Plot 0, and,. Then, from left to right, draw a curve that begins just the -ais, passes through the two points, and moves. Eample 3 Graph 5 ab h k for b > Graph 5 p 3. State the domain and range. Solution Begin b sketching the graph of 5 p 3, which passes through (0, ) and (, ). Then translate the graph and. The graph s asmptote is the line. The domain is all real, and the range is. Checkpoint Complete the following eercise.. Graph the function 5 p 4 3. State the domain and range. Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 33

21 Name Date LESSON 4.4 Practice Match the function with its graph.. f() 5. f() 5 3. f() 5 4( ) 4. f() 5 } ( ) 5. f() 5 } ( ) 6. f() 5 4( ) A. B. C. (0, ) (, ) (0, ) (, ) ( 0, ) (, ) 4 D. (, 8) 4 (0, 4) E. (0, 4) (, 8) F. ( 0, ) (, ) 34 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

22 Name Date LESSON 4.4 Practice continued Graph the function. 7. f() f() 5 3 p 3 9. f() 5 5 Graph the function. State the domain and range. Describe the end behavior. Find the rate of change over the interval. 0. f() 5. f() 5 3. f() 5 Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 35

23 4.5 Graph Eponential Deca Functions Georgia Performance Standard(s) MMAb, MMAc, MMAe Your Notes Goal p Graph and use eponential deca functions. VOCABULARY Eponential deca function Deca factor Eample Graph 5 b for 0 < b < Graph 5 }. Analze the graph. Find the average rate of change over the interval. Step Make a table of values. 0 Step Plot the points from the table. Step 3 Draw from right to left, a smooth curve that begins just the -ais, passes through the plotted points, and moves. Step 4 Eamine the graph. The -intercept of the function is. As the value of approaches ` the value of the function approaches. As the value of approaches `, the value of the function approaches. Also, the function is decreasing on the interval. Using the points (, 4), and, } 4, the average rate of change over the interval is, or about. 36 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

24 Your Notes Checkpoint Complete the following eercise.. Graph 5 } 4 and find the average rate of change over the interval. 4 Eample Graph 5 ab for 0 < b < Graph the function 5 3 } 4. Plot (0, ) and,. Then, from right to left, draw a curve that begins just the -ais, passes through the two points, and moves to the left. Eample 3 Graph 5 ab h k for 0 < b < Graph 5 3 } 5. State the domain and range. Solution Begin b sketching the graph of 5 3 } 5, which passes through (0, ) and,. Then translate the graph and. Notice that the graph passes through (, ) and,. The graph s asmptote is the line. The domain is, and the range is. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 37

25 Your Notes Checkpoint Complete the following eercise.. Graph the function. State the domain and range. 5 3 } 3 3 Eample 4 Find a value after depreciation Televisions A new television costs $00. The value of the television decreases b % each ear. Write an eponential deca model giving the television s value (in dollars) after t ears. Estimate the value after ears. Solution The initial amount is a 5 and the percent decrease is r 5. So, the model is: 5 a( r) t Write eponential deca formula. 5 Substitute for a and r. 5 Simplif. When t 5, the television s value is 5 00(0.79) 5. Checkpoint Complete the following eercise. Homework 3. Rework Eample 4, with a % decrease each ear. 38 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

26 Name Date LESSON 4.5 Practice Tell whether the function represents eponential growth or eponential deca.. f() 5 } 3 4. f() 5 } f() f() 5 } (3 ) 5. f() f() 5 } 3 } 3 Match the function with its graph. 7. f() 5 } 8. f() 5 } 9. f() 5 3 } 0. f() 5 } 3 }. f() 5 3 }. f() 5 } 3 } A. 3 (, ) (0, 3) B. (0, ) (, ) C. ( 0, 3 ) (, 6 ) D. E. F. ( 0, 3 ) (, 6 ) (0, 3) 3 (, ) (0, ) (, ) Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 39

27 Name Date LESSON 4.5 Practice continued Graph the function. State the domain and range. Describe the end behavior of the graph. Find the rate of change over. 3. f() 5 } 3 4. f() 5 } 3 5. f() 5 } f() 5 } 7. Depreciation You bu a new computer and accessories for $00. The value of the computer decreases b 30% each ear. Write an eponential deca model giving the computer s value V (in dollars) after t ears. What is the value of the computer after four ears? 40 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

28 4.6 Solve Eponential Equations and Inequalities Georgia Performance Standard(s) MMAd Your Notes Goal p Solve eponential equations and inequalities. VOCABULARY Eponential equation Eponential inequalities in one variable Eample Solve b equating eponents Solve Write original equation. ( ) 5 ( ) Rewrite each power with base. 5 Power of a power propert 5 Propert of equalit 5 Solve for. The solution is. CHECK Substitute the solution into the original equation Substitute for. 5 Solution checks. Checkpoint Complete the following eercise.. Solve Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 4

29 Your Notes Eample Solve an eponential equation b graphing Solve 5 } 4 3. Graph 5 and 5 } 4 3 in the same coordinate plane. The graphs intersect onl once, when 5. So, is the onl solution. Checkpoint Complete the following eercise.. Solve 4 5 }. Eample 3 Solve 3 4z 9 z z 9 z 3 Write original inequalit. ( ) 4z ( ) z 3 Rewrite each power with base. Solve an eponential inequalit Power of a power propert Because f() 5 3 is an increasing function, f( ) f( ) implies that. z Solve for z. CHECK Check that the solution is reasonable b substituting several values into the original inequalit. Subsitute z 5. Subsitute z ()? (4)? Because z 5 is a solution of the inequalit and z 5 is not, the soution z is reasonable. 4 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

30 Your Notes Checkpoint Complete the following eercise. 3. Solve 4. Eample 4 Solve an eponential inequalit b graphing Solve 3 3 > Graph and in the same coordinate plane. The graphs intersect onl once, when 5. The graph of is the graph of when. The solution is. Checkpoint Complete the following eercise. 4. Solve 4 < 0.5. Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 43

31 Name Date LESSON 4.6 Practice Solve the eponential equation Solve the eponential inequalit Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

32 Name Date LESSON 4.6 Practice continued Solve the eponential equation using a graph. Round our answer to the nearest hundredth Solve the eponential inequalit using a graph. Round our answer to the nearest hundredth } Virus An infectious virus is defined b its infectivit, or how contagious the virus is to humans. The number of people (in thousands) epected to contract the virus within 6 months is modeled b 5.04(8.35) where is the infectivit rating of the virus. How low must the infectivit rating be so no more than 00,000 people become infected within 6 months? How high must the infectivit rating be so more than 700,000 become infected within 6 months? Round our answers to the nearest hundredth. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 45

33 4.7 Define and Use Sequences and Series Georgia Performance Standard(s) MMA3d Your Notes Goal p Recognize and write rules for number patterns. VOCABULARY Sequence Terms Series Summation notation Sigma notation SEQUENCES A sequence is a function whose domain is a set of integers. If a domain is not specified, it is understood that the domain starts with. The values in the range are called the of the sequence. Domain: n The relative position of each term Range: a a a 3 a 4... a n Terms of the sequence A sequence has a limited number of terms. An sequence continues without stopping. Finite sequence:, 4, 6, 8 Infinite Sequence:, 4, 6, 8,... A sequence can be specified b an equation, or. For eample, both sequences above can be described b the rule a n 5 n or f(n) 5 n. 46 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

34 Your Notes Eample Write terms of sequences Write the first si terms of a n 5 n. a 5 5 st term a 5 5 nd term a rd term a th term a th term a th term Eample Write rules for sequences Describe the pattern, write the net term, and write a rule for the nth term of the sequence (a), 4, 9, 6,... and (b) 0, 7, 6, 63,.... Solution a. You can write the terms as,,,,.... The net term is a A rule for the nth term is a n 5. b. You can write the terms as,,,,.... The net term is a A rule for the nth term is a n 5. Checkpoint Complete the following eercises.. Write the first si terms of the sequence f(n) 5 3n 7.. For the sequence 3, 9, 7, 8,..., describe the pattern, write the net term, and write a rule for the nth term. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 47

35 Your Notes SERIES In the series 4 i 5 i, i is the, is the, and 4 is the. Eample 3 Write series using summation notation Write the series using summation notation. a b. } 8 } 7 } Solution a. Notice that the first term is 3(), the second is, the third is, and the last is. So, a i 5 where i 5,, 3,...,. The lower limit of summation is and the upper limit of summation is. The summation notation for the series is. b. Notice that for each term, the denominator is a perfect cube. So, a i 5 where i 5,, 3, 4,.... The lower limit of summation is and the upper limit of summation is. The summation notation for the series is. Checkpoint Write the series using summation notation Homework Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

36 Name Date LESSON 4.7 Practice Write the first si terms of the sequence.. a n 5 n. a n 5 3n 3. a n 5 3n 4. f(n) 5 n 3 5. f(n) 5 0 } n 6. f(n) 5 n } n For the sequence, describe the pattern, write the net term, and write a rule for the nth term. 7. 4, 8,, 6, ,, 5, 9, , 6, 9,, , 9, 7, 8,.... }, }, 3 }, 4 },.... } 3, } 9, } 7, } 8, , 0.,.3,.4, , 4., 8.4, 6.8,... Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 49

37 Name Date LESSON 4.7 Practice continued Graph the sequence. 5. 6, 4,, 0, 6. 0., 0., 0.4, 0.8, , 3,, } 3, } 9 a n a n a n n 0.3 n n Write the series using summation notation (3) (7) Find the sum of the series.. 5 n. n 5 7 i 5 3i k} k Amoebas A Petri dish contains three amoebas. An amoeba is a microorganism that reproduces using the process of fi ssion, b simpl dividing itself into two smaller amoebas. Once the new amoebas mature, the will go through the same process. a. Write the terms of the sequence describing the first four generations of the amoeba in the Petri dish. b. Write a rule for the nth term of the sequence. c. Find the 0th term of the sequence and describe in words what this term represents. 50 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

38 4.8 Analze Arithmetic Sequences and Series Georgia Performance Standard(s) MMA3d, MMA3e Your Notes Goals p Stud arithmetic sequences and series. VOCABULARY Arithmetic sequence Common difference Arithmetic series RULE FOR AN ARITHMETIC SEQUENCE The nth term of an arithmetic sequence with first term a and common difference d is given b: a n 5 a (n )d THE SUM OF A FINITE ARITHMETIC SERIES The sum of the first n terms of an arithmetic series is: S n 5 n a a n } In words, S n is the of the terms, b. Eample Identif arithmetic sequences Tell whether the sequence 5, 3,,, 3,... is arithmetic. Find the differences of consecutive terms. a a 5 5 a 3 a 5 5 a 4 a a 5 a Each difference is, so the sequence arithmetic. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 5

39 Your Notes Checkpoint Tell whether the sequence is arithmetic.. 3, 7,, 7, 0,... Eample Write a rule for the nth term Write a rule for the nth term of the sequence, 9, 6, 3,.... Then find a 9. Solution The sequence is arithmetic with first term a 5 and common difference d 5 5. So, a rule for the nth term is: a n 5 a (n )d Write general rule. 5 (n ) Substitute for a and d. 5 Simplif. The 9th term is a Eample 3 Find the sum of an arithmetic series Find the sum of the first 30 terms of the arithmetic series The series is arithmetic with first term a 5 and common difference d 5 5. So, a rule for the nth term is: a n 5 a (n )d Write general rule. 5 (n ) Substitute for a and for d. 5 Simplif. The 30th term is a The sum of the first 30 terms is: S a a 30 } Write rule for S Substitute for a and for a Simplif. The sum of the first 30 terms is. 5 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

40 Your Notes Checkpoint Complete the following eercises.. Write a rule for the nth term of the sequence 9, 5,, 3,.... The find a. 3. Find the sum of the first 0 terms of the arithmetic series Eample 4 Find a formula for the sum of a series Find a formula for the sum of the series (n ) n (n ). The sequence is arithmetic with first term a 5 and common difference d 5 5. So, a rule for the nth term is: a n 5 a (n )d 5 (n ) 5 The sum of the first n terms is: S n 5 n a a n } Write rule for S n. 5 Substitute for a and for a n. S n represents the partial sum of the first n terms of this series. Note that S n is a quadratic function that can be written as S n 5. Homework Checkpoint Complete the following eercise. 4. Find a formula for the partial sum of the series (5n 5) 5n. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 53

41 Name Date LESSON 4.8 Practice Tell whether the sequence is arithmetic. Eplain wh or wh not.., 4, 7, 0, 3,...., 3, 7, 5, 3, ,,, 3, 5, , 4, 0, 4, 8,... 5.,, 3, 4, 5, }, } 3, } 4, } 5, } 6,... Write a rule for the nth term of the arithmetic sequence. Then find a , 0, 3, 6, , 9, 0,,, ,,, 3, 5, d 5, a 5 5. d 5 4, a d 5 5, a Write a rule for the nth term of the arithmetic sequence that has the two given terms. 3. a 8 5, a a 3 5, a a 5 5 5, a a 5 5, a a 5 0, a a , a Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

42 Name Date LESSON 4.8 Practice continued Find the sum of the arithmetic series i 5 (i ) i. i 5 i 5 (i ). 7 i 5 (3i 4) 3. 8 i 4. i 5 0 i 5 6 (5 i) Write a rule for nth term of the sequence whose graph is shown. 5. a n 6. a n 7. a n (, 7) (, 5) (3, 3) (4, ) (5, ) n (5, 6) (4, 4) (3, ) (, 0) n (, ) (, 5) (, 4) (3, 3) (4, ) (5, ) n 8. Weightlifting You are tring to find the maimum weight that ou can lift in a weightlifting eercise. You start with a single lift of 5 pounds. Then ou increase the weight b pounds and tr again. You repeat this procedure until ou reach a weight that ou are unable to lift. a. Write a rule for the total weight of our nth lifting attempt. b. You are unable to lift the weight on our sith lift. So, based on our fifth lift, what is the maimum amount of weight that ou can lift in this eercise? c. Find the sum of the weights lifted in our five successful lifts. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 55

43 4.9 Analze Geometric Sequences and Series Georgia Performance Standard(s) MMAf Your Notes Goal p Stud geometric sequences and series. VOCABULARY Geometric sequence Common ratio Geometric series RULE FOR A GEOMETRIC SEQUENCE The nth term of a geometric sequence with first term a and common ratio r is given b: a n 5 a r n Eample Identif geometric sequences Tell whether the sequence, 4, 6, 64, 56,... is geometric. Find the ratios of consecutive terms. If the ratios are constant then the sequence is geometric. a } a 5 4 } 5 4 a 3 } a 5 5 a 4 } a3 5 5 a 5 } a4 5 5 Each ratio is, so the sequence geometric. Checkpoint Complete the following eercise.. Tell whether the sequence 5, 8, 64, 8,... is geometric. 56 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

44 Your Notes Eample Write a rule for the nth term Write a rule for the nth term of the sequence 97, 34, 08, 36,.... Then find a 0. Solution The sequence is geometric with first term a 5 and common ratio r 5 5. So, a rule for the nth term is: a n 5 a r n Write general rule. 5 n Substitute for a and r. The 0th term is a Eample 3 Write a rule given a term and common ratio One term of a geometric sequence is a The common ratio is r 5 3. (a) Write a rule for the nth term. (b) Graph the sequence. a. Use the general rule to find the first term. a n 5 a r n Write general rule. 5 a ( ) Substitute for a n, r, and n. 5 a Solve for a. So, a rule for the nth term is: a n 5 a r n Write general rule. 5 Substitute for a and r. b. Create a table of values for the a n sequence. Notice that the points lie on an eponential curve. 0 n n 3 a n n 4 5 a n Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 57

45 Your Notes Checkpoint Write a rule for the nth term of the geometric sequence. Then find a 9.. 4, 8, 56,, a , r 5 3 THE SUM OF A FINITE GEOMETRIC SERIES The sum of the first n terms of a finite geometric series with common ratio r Þ is: S n 5 a rn } r Eample 4 Find the sum of a geometric series Find the sum of the geometric series 3 i 5 3(4) i. a 5 5 Identif first term. r 5 Identif common ratio. S 3 5 a r3 } r Write rule for S Substitute and simplif. Homework Checkpoint Complete the following eercise. 4. Find the sum of the geometric series i 5 7(5) n. 58 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

46 Name Date LESSON 4.9 Practice Tell whether the sequence is geometric. Eplain wh or wh not.., 4, 8, 6, 3,...., 0, 00, 000, 0,000,... 3., 3, 9, 7, 8, , 8,, 0.5, 0.5, , 5,, } 5, } 5, } 4, } 4, }, } 36, } 08,... Write a rule for the nth term of the geometric sequence. Then find a , 8, 6, 3, , 500, 50, 5, ,, 48, 9,... Write a rule for the nth term of the geometric sequence. Then graph the first five terms of the sequence. 0. r 5, a 5. r 5 3, a 5 5. r 5 } 3, a 5 54 a n a n a n n n n Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 59

47 Name Date LESSON 4.9 Practice continued Write a rule for the nth term of the geometric sequence that has the two given terms. 3. a 5 4, a 5 4. a 5, a a 3 5 3, a a 3 5, a 6 5 } 7 7. a 5 0, a a 5 0, a } 4 Find the sum of the geometric series i 5 () i 0. 5 i 5 (3) i. 8 i 5 0.5() i. 5 } i (0)i 3. 5 i } i 4. 6 i } 5 i 5. Production A compan plans to increase production of a product b 0% each ear over the net ears. The compan will produce 70,000 units net ear (ear ). a. Write a rule giving the number of units produced b the compan in ear n. b. Find the numbers of units produced in ears 4, 8, and. c. Find the total number of units produced over the net ears. 60 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

48 Words to Review Give an eample of the vocabular word. Scientific notation Power function Composition Inverse relation Inverse function nth root of a Eponential function Eponential growth function Growth factor Asmptote End behavior Eponential deca function Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 6

49 Deca factor Eponential equation Eponential inequalit in one variable Sequence Terms Series Summation notation Sigma notation Arithmetic sequence Common difference Arithmetic series Geometric sequence Common ratio Geometric series 6 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.