) = 12(7)

Transcription

1 Chapter 6 Maintaining Mathematical Proficienc (p. 89). ( ) + 9 = (7) + 9 = (7) = = = 7 8 = = = = = = 0 = (0 ) = (0 6) = ( 6) = ( 6) = ( 6) = 6 + ( 6) =. Because 8 = 6, 6 = 8 = 8.. represents the negative square root. Because =, = =. 6. represents the negative square root. Because =, = =. 7. ± represents the positive or negative square root. Because =, ± = ± = ±. 8. The first term is, and the common difference is. a n = a + (n )d a n = + (n ) a n = + n() () a n = + n a n = n The first term is 6, and the common difference is. a n = a + (n )d a n = 6 + (n )( ) a n = 6 + n( ) ( ) a n = 6 n + a n = n The first term is, and the common difference is 7. a n = a + (n )d a n = + (n )( 7) a n = + n( 7) ( 7) a n = 7n + 7 a n = 7n + 9. es; no; The product of two perfect squares can be represented b m n = (mm)(nn) = (mn)(mn) = (mn). If m and n are integers, their product is also an integer. So, (mn) is an integer. There are man counterexamples illustrating that the quotient of two perfect squares does not have to be a perfect square, such as 9. Chapter 6 Mathematical Practices (p. 90). Year Population So, the population in the tenth ear is rabbits.. Row Sum So, the sum of the numbers in the tenth row is. 6. Explorations (p. 9). a. i. ( )( ) = ( ) ( ) = = ii. ( )( ) = ()( ) = = 6 iii. ( )( ) = ( )( ) = = 8 iv. (x )(x 6 ) = (x x)( x x x x x x) = x x x x x x x x = x 8 In each example, the exponent of the product is the sum of the exponents of the factors. So, a general rule is a m a n = a m+n. Copright Big Ideas Learning, LLC Algebra 0

2 b. i. = ii. iii. x6 iv. = = = = x = x x x x x x = x x x x x x = = = 0 = = x In each example, the exponent of the quotient is the difference of the exponents of the powers that are being divided. So, a general rule is am a n = am n. c. i. ( ) = ( )( )( )( ) = ( )( )( )( ) = = 8 ii. (7 ) = (7 )(7 ) = (7 7 7) (7 7 7) = = 7 6 iii. ( ) = ( )( )( ) = ( )( )( ) = = 9 iv. (x ) = (x )(x ) = (x x x x)(x x x x) = x x x x x x x x = x 8 In each example, the exponent of the single power is the product of the other two exponents. So, a general rule is (a m ) n = a mn. d. i. ( ) = ( )( ) = = = ii. ( ) = ( )( )( ) = = = iii. (6a) = (6a)(6a) = 6 a 6 a = 6 6 a a = 6 a iv. (x) = (x)(x) = x x = x x = x So, to find a power of a product, find the power of each factor and multipl, or (ab) m = a m b m. e. i. ( ) = ( ) ( ) = = ) = ( ) ( ) ( ) = 06 Algebra Copright Big Ideas Learning, LLC ii. ( iii. ( ) x = ( ) x ( ) x ( x iv. ( a b) = ( a b) ( a b) ( a b) ( a = ) = x x x = x b) = a a a a b b b b = a b So, to find the power of a quotient, find the power of the numerator and the power of the denominator and divide, or ( a b ) m = am b m.. Sample answer: Tr several examples to find a pattern. Then express the pattern using variables.. Because each side of the large cube has the same length as 9 small cubes, an expression for the number of small cubes in the large cube is = Monitoring Progress (pp. 9 9). ( 9) 0 =. = = 7. 0 = = =. x = 0 x = 9x = 0 +( 6) = 0 = 0 = x 9 x 9 = x 9+( 9) = x 0 = 7. 8 = 8 = = = 6 7 = = 9. (6 ) = 6 ( )( ) = 6 = 6

3 0. (w ) = w = w 60. (0) = (0) = 0 = 000. ( n ) = ( ) n = 0 n = 0. ( k ) = (k ) = (k ) = k = k 0. ( 6c 7 ) (6c) = 7 = 7 (6c) = 7 6 c = 9 6c. The base of a clinder is a circle. The formula for the area of a circle is A = πr. Also, r = h. A = πr = π ( h ) = π ( h ) = πh. So, two expressions that represent the area of a base of the clinder are πr and πh = = = = = 9. 0 n ) = π ( h So, Pluto orbits the Sun 9. 0, or 90,000, times while the Sun completes one orbit around the center of the Milk Wa. 6. Exercises (pp ) Vocabular and Core Concept Check. Sample answer: First, use the Product of Powers Propert to simplif the expression inside the parentheses to. Then, use the Power of a Power Propert to simplif the entire expression to 8. Then, use the definition of negative exponents to produce the final answer, 8 = 6,6.. Use the Power of a Product Propert when a product of factors is in parentheses, and the whole product is being raised to a power. In order to use this propert, find the power of each factor and then multipl the powers.. Use the Quotient of Powers Propert when powers with the same base are being divided. In order to use the propert, find the difference of the exponents of the numerator and denominator. The answer is the common base raised to this difference.. The one that is different is Simplif 6. This answer is 6 = 8. The other three expressions are each equal to +6 = 9. Monitoring Progress and Modeling with Mathematics. ( 7) 0 = 6. 0 = 7. = = 6 8. ( ) = ( ) = = 9. = 0 = = 0 = = 6 = 6 = 6 7 =.. ( 8) =. x 7 = x 7. 0 = ( 8) = x 0 = 9 = 9 6. c 8 d 0 = c 8 = c 8 7. m = n r s = 9. a 0 b 7 m = m = b7 s r = s 9r = b7 6 p q = 7 q 9 9 p 8 = 9q9 p z 0 x = 8 x = 8 x7 6. x 0 = x7 6 z = z 0 0 x = z0 x. 6 = 6 = = 6. ( 6)8 ( 6) = ( 6)8 = ( 6) = 6. ( 9) ( 9) = ( 9) + 6. = + = 0 = 7. (p 6 ) = p 6 = p = ( 9) = 66 = 6z0 x Copright Big Ideas Learning, LLC Algebra 07

4 8. (s ) = s = s = s = 6 8+ = 6 = 6 = ( 7) = ( 7) ( 7). x x x = x x = x x = x + = x = ( 7) +( ) = ( 7) = ( 7) = =. z8 z = z8+ z z = z0 z = z0 = z = 0 7+ = 0 So, the magnified length of the object is 0, or 00, meter (which is the same as centimeter).. a b 8ab = 8 a b = a b microns So, the length of the computer chip is a b microns.. The Product of Powers Propert should be used because powers with the same base are being multiplied. So, the product should have a base of, not. = + = 9 6. In the second step, the Quotient of Powers Propert should be used because powers with the same base are being divided. The exponents should be subtracted, not divided. x x x = x+ x = x8 x = x8 = x 7. ( z) = ( ) z = z 8. (x) = (x) = x = 6x 9. ( 6 n) 6 = n n = 6 = n 6 0. ( t ) = ( t) = ( ) t 9 = t 9 = t 9. (s 8 ) = (s 8 ) = (s 8 ) = s 8 = s 0. ( p ) = ( ) (p ) = p = p 9. ( w 6 ) = ( ) (w ) 6 = 6 ( ) (w ) = 6 w = 6 w 6 6. ( r ) 6 = 6 (r 6 ) 6 = (r 6 ) 6 6 = 6 (r 6 ) 6 = 6 r 6 6 = 6r 6. B, C, D; V = πr = π(s ) = π (s ) = π 8 s6 = πs6 An expression equivalent to πs6 represents the volume of the sphere. ( πs 6 )( ) = (πs 6 )( ) = πs6 (s) πs = s πs = πs s = πs6 None of the other expressions are equivalent to πs6. So, the answers are B, C, and D, because these expressions simplif to the volume of the sphere, which is πs6. 6. t = x D = (0 ) (0 ) = 0 0 = = = 0 8+ = 0 It takes about 0, or 0.000, second for the ink to diffuse micrometer. 7. ( x x x ) = ( + x ) + = ( 7 x ) x ) = ( 8. ( s t 7 = (7 ) (x ) = ( 7 ) (x ) = 67 8x = 68 8x s t ) = ( s s t t ) 7 = ( s+ t ) +7 = ( s7 ) (t ) = ( ) (s 7 ) t = ( s7 t ) = 8s7 t = 8s t 08 Algebra Copright Big Ideas Learning, LLC

5 9. ( m n m n ) ( mn 0 0. ( x 0 x ) ( x m ) ( mn 9n ) = ( m n x 8 ) = ( n m ) ( mn ) = (n ) (m ) (mn ) ( ) 9 ) = n m n m = n m n 6 m 6 n +6 = m n 0 = 6 9 m n 0 = m = ( x x ) ( 8 = ( x + ) ( +8 = (x ) ( 0 x ) = x 0 (x ) = 8 x0 0 x = 8 x0 0 x = 8x 0 x ) +. ( 0 )(. 0 ) = (.) (0 )(0 ) =. 0 +( ) =. 0 The product is. 0, or (6. 0 )(8 0 9 ) = 6.(8) (0 )(0 9 ) = = = = The product is , or 8,800,000.. (6. 07 ) (.6 0 ) = = 0 7 = 0 The quotient is 0, or 00. m x m x x ). (.9 0 ) ( ) = = 0. 0 ( 8) = = 0 0 = 0 + = 0 The quotient is 0, or = ( ) On average, about. 0, or,0, pounds of potatoes were produced for each acre harvested. 6. t = d = r 0 = = =.6 0 So, it takes.6 0, or 600, seconds for sunlight to reach Jupiter. 7. a. You should use the Power of a Product Propert because for each cube, ou must raise a product of two factors to the third power to find the volume. b. Use the Power of a Quotient Propert to express (6x) (x) as ( 6x x ). Simplif the expression inside the parentheses to produce (), so the volume is 7 times greater. 8. a. kilobte 0 btes 0 btes terabtes = 0 kilobtes 0 terabtes = 0 0 kilobtes per terabte = 0 kilobtes per terabte So, there are 0 kilobtes in terabte. b. megabte 0 btes 0 btes gigabte 6 gigabtes = 0 0 megabtes = 0+ = 0 megabtes 0 megabtes = 0 megabtes = megabtes So, there are, or 6,8, megabtes in 6 gigabtes. Copright Big Ideas Learning, LLC Algebra 09

6 c. In order to convert the number of btes in each unit of measure to bits, multipl each number in the table b 8. Because 8 can be expressed as, multipl each number in the table b. Because the values have a common base of, the can be simplified using the Product of Powers Propert. So, ou can simpl add to each of the exponents in the table. 9. 8a b = a b = (ab) 60. 6r s = r s = (rs) 6. 6w 8 z = 6 w 6 z 6 = (w z ) x 8 = x = (x ) or 8x 8 = 9 x = (9x ) 6. a. ( 6. a. 6) n = n 6 n = 6 n b. 6 = n 6 = 96 c. ; The probabilit of flipping heads once is, and the probabilit of flipping heads five times in a row is ( =. ) = Figure The shaded part is of the original. Figure The shaded part is of the original. b. Figure : = Figure : = = Figure : 8 = = Figure : 6 = = Figure The shaded part is 8 of the original. 6. Using the Quotient of Powers Propert and the first Figure The shaded part is 6 of the original. equation bx b = b9, ou can conclude that x = 9. Using the Product of Powers Propert, the Quotient of Powers Propert, and the second equation bx b = b, ou can conclude that x + =. Use these equations to solve a sstem of linear equations b substitution. Step : x = 9 x + = 9 + x = 9 + b Step : x + = (9 + ) + = Step : + = x = 9 x ( ) = 9 x + = 9 x = 8 = = = So, x = 8 and =. 66. Sample answer: Let r = 9x. V = πr h 7πx 8 = π(9x ) h 7πx 8 = π 9 x h 7πx 8 = π 8 x h 7πx 8 = 7πx h x 7πx 8 7πx = 7πx h 7πx x = h So, one possibilit is r = 9x and h = x (0 6 ) = 0 0+( 6) = 0 0 g kg 0 g = 0 kg 0 = 0 kg = 0 kg So, the mass of the seed from the double coconut palm is 0 kilograms, which means our friend is incorrect. 68. a. Each of the questions has choices. So, represents the number of different was that a student can answer all the questions in Part. b. Each of the questions has choices. So, represents the number of different was that a student can answer all the questions on the entire surve. c. Because each question now has choices, the answer for part (a) becomes, and the answer for part (b) becomes. 0 Algebra Copright Big Ideas Learning, LLC

7 69. a. When a > and n < 0, a n < a n because a n will be less than and a n will be greater than. When a > and n = 0, a n = a n =, because an number to the zero power is. When a > and n > 0, a n > a n because a n will be greater than and a n will be less than. b. When 0 < a < and n < 0, a n > a n, because a n will be greater than and a n will be less than. When 0 < a < and n = 0, a n = a n =, because an number to the zero power is. When 0 < a < and n > 0, a n < a n because a n will be less than and a n will be greater than. Maintaining Mathematical Proficienc 70. Because =, = = represents the negative square root. Because 0 = 00, 00 = 0 = ± represents the positive or negative square root. 6 Because ( 8) = 6, ± 6 = ± ( 8) = ± is a natural number, whole number, integer, rational number, and real number is a rational number and a real number π is an irrational number and a real number. 6. Explorations (p. 99). a. 7 ft = ft Check: =? 7 =? 7 9 =? 7 7 = 7 b. cm = cm Check: =? =? =? = c. 7 in. = in. Check: =? 7 =? 7 =? 7 7 = 7 d..7 m =. m Check:. =?.7... =?.7.. =?.7.7 =.7 e. d = d Check: =? =? =? = f. 8 mm = mm, or. mm Check: ( ) =? 8 =? 8 =? 8 8 = 8 Of the sides measured in metric units, the longest length is. meters. Of the sides measured in standard units, the longest length is ard. Because meter is approximatel the same length as ard, the cube in part (d) has the largest side length of. meters. The cubes in parts (a) and (e) have equal side lengths because feet = ard.. a. Sample answer:., which is represented b point C. is between = 6 and = 8, and C is the onl point on the graph between and. b. Sample answer: , which is represented b point A. 0. is between 0 = 0 and =, and A is the onl point on the graph between 0 and. Also, 0. is close to 0.9, which equals 0.7 because (0.7) =.9. c. Sample answer:.., which is represented b point B.. is between = and =, and B is the onl point on the graph between and. d. Sample answer: 6.0, which is represented b point E. 6 is between = 6 and =, and E is the onl point on the graph between and. e. Sample answer:.8, which is represented b point D. is between = 7 and = 6, and D is the onl point on the graph between and. f. Sample answer: 6 0,000., which is represented b point F. 0,000 is between 6 =,6 and 6 6 = 6,66, and F is the onl point on the graph between and 6.. Find what real number multiplied b itself n times gives ou that number. If that is not possible, determine which nth powers the number is between and estimate the decimal part.. m = (0.0006)C = (0.0006)C = (0.0006)C ,000,000 = C = C.7 Use guess and check with a calculator: So, the circumference of its femur was about.7 millimeters. Copright Big Ideas Learning, LLC Algebra

8 6. Monitoring Progress (pp. 00 0). The index n = is odd, so has one real cube root. Because ( ) =, the cube root of is 7 =, or ( ) / =.. The index n = 6 is even, and a > 0. So, 6 has two real sixth roots. Because 6 = 6 and ( ) 6 = 6, the sixth roots of 6 are ± 6 6 = ±, or ±6 /6 = ±.. = ( ) ( ) ( ) =. ( 6) / = ( 6 / ) =( ) =6. 9 / = (9 / ) = = 6. 6 / = (6 / ) 7. r = ( V = = 6 π ) / = ( (7,000) (.) ) / = (,000.6 ) / 6 So, the radius of the beach ball is about 6 inches. 8. r = ( F P) /n = (, ) /8.88 / The annual inflation rate is about 6.0%. 6. Exercises (pp. 0 0) Vocabular and Core Concept Check. Find the fourth root of 8, or what real number multiplied b itself four times produces 8.. The expression that does not belong is ( 7 ) because it is the onl one that is not equivalent to 9. Monitoring Progress and Modeling with Mathematics. 0 = 0 /. = /. / = 6. 0 /8 = The index n = is even and a > 0. So, 6 has two real square roots. Because 6 = 6 and ( 6) = 6, the square roots of 6 are ± 6 = ±6, or ±6 / = ±6. 8. The index n = is even and a > 0. So, 8 has two real fourth roots. Because = 8 and ( ) = 8, the fourth roots of 8 are ± 8 = ±, or ±8 / = ±. 9. The index n = is odd, so 000 has one real cube root. Because 0 = 000, the cube root of 000 is 000 = 0, or (000) / = The index n = 9 is odd, so has one real ninth root. Because ( ) 9 =, the ninth root of is 9 =, or ( ) /9 =.. 6 in. = in. Check: =? 6 =? 6 So, each side of the cube is inches. 6 =? 6 6 = 6. 6 cm = 6 cm Check: 6 =? =? =? 6 6 = 6 So, each side of the cube is 6 centimeters.. 6 = =. 6 = ( 6) ( 6) ( 6) = 6. = ( 7) ( 7) ( 7) = = ( ) = () = 7. 8 /7 = 7 8 = 7 = 8. ( 6) / is not a real number because there is no real number that can be multiplied b itself two times to produce ( 8 ) = (8 / ) = 8 / 0. ( ) 6 = [( ) / ] 6 = ( ) 6/. ( ) /7 = [( ) /7 ]. 9 / = (9 / ) = ( 7 ) = ( 9 ). / = ( / ) = = 8 Algebra Copright Big Ideas Learning, LLC

9 . / = ( / ) = =. ( 6) / is not a real number because ( 6) / = [( 6) / ], and there is no real number that can be multiplied b itself two times to produce ( ) / = [( ) / ] = ( ) = 9 8. V = 6 π S/ = 6 π (60)/ 0.67 (6.78).7 The volume of the sphere is about cubic meters. 9. Write the radicand, a, as the base and write the exponent as a fraction with the power, m, as the numerator and the index, n, as the denominator. 7. ( 8) /7 = [( 8) /7 ] = ( ) = 8. / = ( / ) = 7 = 0 9. The numerator and denominator are reversed. ( ) = ( / ) = / 0. ( 8) / is not a real number because there is no real number that can be multiplied b itself four times to produce 8.. ( 000) / / = 000 / = 000. ( 6) /6 /6 6 = 6 /6 = = 6 6. (7) / = 7 / =. (9) / = 9 / =. A = w ( 7 ) = ( 9 ) = = 0 = 9 = = ( 6 79 ) ( / ) = ( 6 ) ( ) = () ( ) = ()() = 6 The area of the bake sale sign is 6 square feet = 7 / = (7 / ) = = One side of the box is millimeters. 7. r = ( V πh ) / = ( () (.)()) / = (.6) / (.9) / The radius of the paper cup is about inch. 0. A = s Copright Big Ideas Learning, LLC Algebra x = s x = s x = s x / = s An expression that represents the side length of the square is x / inches.. r = ( F P) /n = (,00, ,000 ) /6 = (.7) /6 0.0 The annual inflation rate is about.%.. r = ( F P) /n = (..6) /0 (.78) / The annual inflation rate is about 9.%.. = / =, 0 = 0 / = 0, = ( ) / = So, x = x / is true for x =, x = 0, and x =.. no; The value of n a is not alwas positive, and the value of n a is not alwas negative. If n is odd and a is negative, then n a will be negative and n a will be positive.. ( /6 ) x = (/6) x / = / x / ( 6 = 6 = 6, or = (x) / The simplified expression is (x) /, or x. )

10 6. ( / ) / = ( +/ ) / ) = ( / ) / ( + = + = = (/)(/) = ( The simplified expression is. = 6 ), or 7. x 6 + x = x 6/ + x / = x + x = (x ) = x The simplified expression is x. 8. (x / / ) 9 = x (/) 9 (/) 9 / = x 9/ 9/ / = x (9/)+(/) = x 0/ = x The simplified expression is x. 9. V = = = The edge length of the dodecahedron is approximatel.8 feet. 0. Sample answer: The formula for the period of a pendulum is T = π g (, or T = π g ) /.. The statement (x / ) = x is alwas true because (x / ) = x (/) = x / = x = x b the Power of a Power Propert.. The statement x / = x is sometimes true. If x =, then x / = / = = and x = = = =. So, the statement is true if x =. Otherwise, it is false.. The statement x / = x is alwas true because of the definition of a rational exponent.. The statement x / = x is sometimes true because it is onl true for x = 0 and x =.. 0 / =? 0 / =? 0 =? =? 0 0 =? =? 0 = 0 = 8 / =? 8 8 =? =? 6 8 x / x / =? x x (/) (/) =? x / x / = x / The statement x/ x / = x is sometimes true. It is true b the Quotient of Powers Propert and the definition of rational exponents except when x = 0 because division b 0 is undefined. 6. x = x / x Let x =. x = x (/)+ x = x 0/ x = x (/)+(9/) =? ( ) 0/ x = x 0/ =? ( ) 0 =? ( ) 0 Let x = 0. Let x =. x = x 0/ x = x 0/ 0 =? 0 0/ =? 0/ 0 =? ( 0 ) 0 =? ( ) 0 0 =? 0 0 =? 0 0 = 0 = Let x = 8. So, the statement x = x / x is x = x 0/ sometimes true. If x = 0 or x =, 8 =? 8 0/ the statement is true. Otherwise 8 =? ( 8 ) 0 it is false. 8 =? Algebra Copright Big Ideas Learning, LLC

11 Chapter 6 Maintaining Mathematical Proficienc 7. f (x) = x 0 f (x) = x 0 f ( ) = ( ) 0 f (0) = (0) 0 = 6 0 = 0 0 = 6 = 0 f (x) = x 0 f (8) = (8) 0 = 6 0 = 6 So, f ( ) = 6, f (0) = 0, and f (8) = w(x) = x w(x) = x w( ) = ( ) w(0) = (0) = = 0 = = w(x) = x w(8) = (8) = 0 = So, w( ) =, w(0) =, and w(8) =. 9. h(x) = x h(x) = x h(x) = x h( ) = ( ) h(0) = 0 h(8) = 8 = + = = = 6 So, h( ) = 6, h(0) =, and h(8) =. 6. Explorations (p. 0). x 6() x 0 6() () + 6() 6 + 6() 8 + 6() 6 + 6() x 6() x 0 6() () 6 + 6() () () () 0 6,8 Each value of x increases b the same amount, while each value of is multiplied b the same factor. 60. g(x) = 8x + 6 g(x) = 8x + 6 g(x) = 8x + 6 g( ) = 8( ) + 6 g(0) = 8(0) + 6 g(8) = 8(8) + 6 = + 6 = = = 8 = 6 = 80 So, g( ) = 8, g(0) = 6, and g(8) = 80. Copright Big Ideas Learning, LLC Algebra

12 Chapter 6. x 6 ( x ) 0 6 ( ) ( ) ( ) + 6 ( ) + 6 ( ) + 6 ( x 6 ( x ) ) 0 6 ( ) ( ) + 6 ( ) ( ) ( ) ( 6 ) 0 The statement seems to be true because as the exponent increases b a constant amount, the base is multiplied b itself the same number of additional times.. a. x 0 6 x = x x b. x 0 6 () x () () 0 () () () = () x x c. x 0 6 (.) x (.) (.) 0 (.) (.) (.) ,000, = 6( ) x 0 0 = (.) x 000 = 6() x x 8 x 6 x Both are curved and do not intersect the x-axis; The graph from Exploration is increasing, the graph from Exploration is decreasing.. Sample answer: All graphs of an exponential function seem to have a similar curved shape, and the do not intersect the x-axis. 6 Algebra Copright Big Ideas Learning, LLC

13 d. x 6 ( ( ) x ) 6 = 6 ( ) = 6 6 f. x 6 ( ) x ) 6 = ( 6 ) 6 ( ) = ( (. 6. ) x 0 ( ( ) x ) = ( ) 0 = ( ) = x 0 ) x ( ) = ( ) ( ) 0 = () ( ) = ( (.6. ) = ( ) x 6 80 = ( ) x x e. 6 x x 6 ( x ) ( ) 6 = (6) ( ) = (6) 9 8 x 0 ) x ( ) = () ( ) 0 = () ( ) = ( ( 0.7 ) These graphs have the same characteristics as the graphs from Exploration. The have the same general curved shape, and the do not intersect the x-axis. 6. Monitoring Progress (pp ) x 0 8 As x increases b, is multiplied b. So, the function is exponential. 6 = ( ) x x x ( ) +( ) +( ) As x increases b, decreases b. The rate of change is constant. So, the function is linear.. = (9) x = (9) x = (9) x = (9) = (9) 0 = (9) = ( 8) = () = ( 9 ) Copright Big Ideas Learning, LLC Algebra 7 = 8 = = (). =.() x =.() x =.() x =.() =.() 0 =.() =.(0.) =.() =. = 0.7 =..(.).

14 . x 0 () x () () () 0 () () f (x) x f(x) = () x 8 7. x () x+ () () 9 x 0 () x+ () 0 () () x The parent function is g(x) = x. The graph of f is a vertical stretch b a factor of and a reflection in the x-axis of the graph of g. The -intercept of the graph of f,, is below the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is < x 0 ( ) x ( ) ( ) ) 0 ( ) ( ) ( f (x) = () x From the graph, ou can see that the domain is all real numbers and the range is <. x (0.) x + (0.) + (0.) + f (x) 9 7 x 0 (0.) x + (0.) 0 + (0.) + (0.) + f (x) f(x) = ( ) x 8 0 x 0 The parent function is g(x) = ( ) x. The graph of f is a vertical stretch b a factor of of the graph of g. The -intercept of the graph of f,, is above the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is > = (0.) x + x From the graph, ou can see that the domain is all real numbers and the range is >. x 0 g(x) x Both functions have the same value when x = 0, but the value of f is less than the value of g over the rest of the interval. 8 Algebra Copright Big Ideas Learning, LLC

15 Chapter 6 0. a. Because the graph crosses the -axis at (0, 00), the -intercept is 00. Also, the -values increase b a factor of 00 = as x increases b. 00 = ab x = 00() x So, the population can be modeled b = 00() x. b. = 00() x = 00() 6 = 00(6) = 600 So, there are 600 bacteria after 6 das. c. The bacteria population in Example 7 is growing b a factor of, and the bacteria population in this problem is onl growing b a factor of. So, this bacteria population does not grow faster. 6. Exercises (pp. 0 ) Vocabular and Core Concept Check. Sample answer: 8 6 x. The -intercept occurs when x = 0. So, when x = 0, the value of the function is = ab 0 = a = a.. The graph of = x is the parent function for the graph of = () x. The graph of = () x is a vertical stretch b a factor of of the graph of = x. The -intercept of = () x,, is above the -intercept of = x,. The both have a domain of all real numbers and a range of > 0.. The equation that does not belong is f (x) = ( ) x. Because the base of the power is a negative number, this is not an exponential function. The other three equations represent exponential functions. Monitoring Progress and Modeling with Mathematics. The equation = (7) x represents an exponential function because it fits the pattern = ab x, where a = and b = The equation = 6x does not represent an exponential function because it fits the pattern = mx + b, and therefore represents a linear function. 7. The equation = x does not represent an exponential function because the exponent is a constant. 8. The equation = x represents an exponential function because it fits the pattern = ab x, where a = and b =. 9. The equation = 9( ) x does not represent an exponential function. Although it fits the pattern = ab x, the definition of an exponential function states that b cannot be negative. 0. The equation = ()x does not represent an exponential function. Although it fits the pattern = ab x, the definition of an exponential function states that b cannot be x As x increases b, increases b. The rate of change is constant. So, the function is linear x 6 8 As x increases b, is multiplied b. So, the function is exponential x As x increases b, is multiplied b. So, the function is exponential x ( 9) +( 9) +( 9) +( 9) As x increases b, decreases b 9. The rate of change, 9, or, is constant. So, the function is linear.. = x = = 9 6. f (x) = () x f ( ) = () ) = ( = Copright Big Ideas Learning, LLC Algebra 9

16 7. = () x = () = () = f (x) = 0. x f ( ) = 0. = ( = = 8 9. f (x) = (6)x ) f () = (6) = (6) = 7 0. = ()x = () = ( ) = ( ) = () = (8) =. C; The parent function of f (x) = (0.) x is g(x) = (0.) x. The graph of the parent function, g, decreases as x increases because 0 < b <. The graph of f is a vertical stretch of the graph of g, and the -intercept of f is because a =. So, the function f matches graph C.. B; The parent function of f (x) = (0.) x is g(x) = (0.) x. The graph of the parent function, g, decreases as x increases because 0 < b <. The graph of f is a vertical stretch and a reflection in the x-axis of the graph of g, and the -intercept of f is because a =. So, the function f matches graph B.. x 0 (0.) x (0.) (0.) (0.) 0 (0.) (0.) f (x) f(x) = (0.) x x The parent function is g(x) = (0.) x. The graph of f is a vertical stretch b a factor of of the graph of g. The -intercept of the graph of f,, is above the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is > x 0 x 0 f (x) x f(x) = x 6 The parent function is g(x) = x. The graph of f is a reflection in the x-axis of the graph of g. The -intercept of the graph of f,, is below the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is < 0.. A; The parent function of f (x) = () x is g(x) = () x. The graph of the parent function, g, increases as x increases because b >. The graph of f is a vertical stretch of the graph of g, and the -intercept of f is because a =. So, the function f matches graph A.. D; The parent function of f (x) = () x is g(x) = () x. The graph of the parent function, g, increases as x increases because b >. The graph of f is a vertical stretch and a reflection in the x-axis of the graph of g, and the -intercept of f is because a =. So, the function f matches graph D. 0 Algebra Copright Big Ideas Learning, LLC

17 7. x 0 (7) x (7) (7) (7) 0 (7) (7) f (x) x f(x) = (7) x x 0 (8)x (8) (8) (8)0 (8) (8) f (x) f(x) = (8)x 60 x The parent function is g(x) = 7 x. The graph of f is a vertical stretch b a factor of and a reflection in the x-axis of the graph of g. The -intercept of the graph of f,, is below the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is < x 0 6 ( ) x 6 ( ) 6 ( ) ) 0 6 ( ) 6 ( ) 6 ( f (x) 8 6 The parent function is g(x) = 8 x. The graph of f is a vertical shrink of the graph of g b a factor of. The -intercept of the graph of f,, is below the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is > x 0 (0.)x (0.) (0.) (0.)0 (0.) (0.) f (x) f(x) = 6( ) x 6 8 f(x) = (0.)x x The parent function is g(x) = ( x ). The graph of f is a vertical stretch b a factor of 6 of the graph of g. The -intercept of the graph of f, 6, is above the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is > 0. x The parent function is g(x) = (0.) x. The graph of f is a vertical stretch of the graph of g b a factor of. The -intercept of the graph of f,, or., is above the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is > 0. Copright Big Ideas Learning, LLC Algebra

18 . x 0 x 0 f (x) x ( ) x+ ( ) ( ) 7 8 x 0 f(x) = x x ( ) x+ ( ) 0 ( ) ( ) From the graph, ou can see that the domain is all real numbers and the range is >.. x x+ 0 f (x) x = ( ) x+ 8 6 f(x) = x From the graph, ou can see that the domain is all real numbers and the range is <. 6 x x From the graph, ou can see that the domain is all real numbers and the range is > 0.. x 0 x = x + 7 8(0.7) x+ 8(0.7) 8(0.7) x 0 8(0.7) x+ 8(0.7) 0 8(0.7) 8(0.7) x = 8(0.7) x+ 6 x From the graph, ou can see that the domain is all real numbers and the range is > 7. 6 From the graph, ou can see that the domain is all real numbers and the range is <. Algebra Copright Big Ideas Learning, LLC

19 6. x 0 (6) x (6) (6) f (x).97.. x 0 g(x) f (x) x (6) x (6) 0 (6) (6) f (x) x The value of f is less than the value of g over the entire interval. x 6 f(x) = (6) x From the graph, ou can see that the domain is all real numbers and the range is >. 7. The graph of g is a vertical shrink b a factor of of the graph of f. So, a =. 8. The graph of g is a vertical translation units up of the graph of f. So, k =. 9. The graph of g is a horizontal translation units right of the graph of f. So, h =. 0. The graph of g is a horizontal translation units left of the graph of f. So, h =.. According to the order of operations, the power should be simplified before multipling b 6. g(x) = 6(0.) x ; x = g( ) = 6(0.) = 6() =. The graph approaches the line =, not = 0. The domain is all real numbers and the range is <.. x 0 h(x) 6 8 f (x) x Both functions have the same value when x =, but the value of h is greater than the value of f over the rest of the interval.. a. x 0 0. x Portion of screen displa Zoom Displa = 0. x x Zooms Based on the context of the problem, because ou cannot zoom a negative number of times, the domain is x 0. From the graph, ou can see that the range is 0 <. Copright Big Ideas Learning, LLC Algebra

20 b. The -intercept is. This means that when ou do not zoom in, 00%, or all, of the original screen displa is seen. c. = 0. x = 0. = 0.06 So, ou see 6.% of the original screen if ou zoom in twice. 6. a. x 0 () x () 0 () () () () 0 Population Coote Population = () x 0 0 x Twent-ear periods Based on the context of the problem, because ou cannot have a negative number of 0-ear periods, the domain is x 0. From the graph, ou can see that the range is. b. The -intercept is. This means that the coote population was at the beginning of the first interval. c. = () x = () = (9) = In 0 ears, twent-ear periods have passed. So, there will be cootes in 0 ears x x When x = 0, the value of is 0. So, the -intercept is 0. Each -value is multiplied b a factor of as x increases b. Using a = 0 and b =, an exponential function of the form = ab x that is represented b the table is = 0 ( ) x x 0 0. The graph crosses the -axis at (0, 0.). So, the -intercept is 0.. The -values increase b a factor of as x increases b. Using a = 0. and b =, an exponential function of the form = ab x that is represented b the table is = 0.() x x 0 8 The graph crosses the -axis at (0, 8). So, the -intercept is 8. Each -value is multiplied b a factor of as x increases b. Using a = 8 and b =, an exponential function of the form = ab x that is represented b the graph is = 8 ( ) x. When x = 0, the value of is. So, the -intercept is. The -values increase b a factor of 7 as x increases b. Using a = and b = 7, an exponential function of the form = ab x that is represented b the table is = (7) x. Algebra Copright Big Ideas Learning, LLC

21 . a x When x = 0, the value of is 0. So, the -intercept is 0. The -values increase b a factor of as x increases b. Using a = 0 and b =, an exponential function of the form = ab x that is represented b the graph is = 0 ( x. ) x ) = 0 ( b. = 0 ( ) = 0(7.97) = 0.7 So, after months, the number of visitors is about 0.. The -intercept is 00 because that is the initial value. The -values increase b a factor of + 6% =.06 as the number of ears x increases b. Using a = 00 and b =.06, an exponential function of the form = ab x that represents this situation is = 00(.06) x. = 00(.06) x = 00(.06) 6 00(.8) 68 The store expects to sell about 68 grills in Year 6.. x 0 f (x) = x g(x) = x g(x) = x 8 6 x f(x) = x 7. Your friend is incorrect. This is not an exponential function because even though the values of are being multiplied b a common factor, the values of x are not increasing at a constant rate.. When a is positive, it causes a vertical stretch (for a > ) or shrink (for 0 > a > ) of the graph of the parent function. If a is negative, it causes a vertical stretch (for a > ) or shrink (for 0 > a > ) and a reflection in the x-axis of the graph of the parent function. 6. Sample answer: The equation f(x) = x represents a horizontal translation units right of the graph of h(x) = x. 7. Using the form = ab x h + k, where h = and k =, an equation is g(x) = x a. The point on the graph with a -value of 0 has an x-value of. So, the stock will be worth $0 after weeks. b. The stock price in Week is $0, and the stock price in Week is $0. So, the stock price drops $0 $0 = $0 from Week to Week. 9. The graph crosses the -axis at (0,.). So, the -intercept is.. Each -value is multiplied b a factor of. = 6 = as x increases b. Using a =. and b =, an exponential function of the form = ab x that is represented b the graph is f(x) =.() x. f (x) =.() x f (7) =.() 7 =.(8) = 9 So, f (7) = Sample answer: If ou were to do something nice for people and then ask each of those people to do something nice for more people who will each do something nice for more people and so on, this situation can be modeled b the exponential function = () x. The -intercept is a = because initiall people had someone do something nice for them, and b = because the number of people who have someone do something nice for them is increasing b a common factor of. f(x + k) 6. f(x) = abx+k ab x = bx+k b x = b(x+k) x = b k 6. Using a = and b = =, an exponential function of the form = ab x that represents this situation is f(x) = () x. The -intercept of g is units below the -intercept of f. The domain of both functions is all real numbers. The range of g is < and the range of f is < 0. Copright Big Ideas Learning, LLC Algebra

22 6. Sample answer: Let f(0) = 8. So, a = 8. Use the equation for slope. Solve for f(). f() f(0) m = 0 = f() 8 = f() 8 = f() = f() Use the exponential form of a function. Solve for b. f(x) = a(b) x = 8(b) = 8b 8 8 = b = b = b Using a = 8 and b =, an exponential function for this situation is f(x) = 8() x. Maintaining Mathematical Proficienc 6. % = 00 = % = 00 = % = 8 00 = % = 0 00 =. 6. Explorations (p. ) x As x increases b, is multiplied b about.. So, this situation can be described as exponential growth. Using a = 88 and b =., an exponential function of the form = ab t that can approximatel model this situation is = 88(.) t, where t represents the number of -ear intervals since 98. = 88(.) t 00,000 = 88(.) t 00, = 88(.)t (.) t Use a table of values, or Guess, Check, and Revise to find that if t =, then , which is close to 8.7. So, the population will return to about 00,000 nesting pairs after intervals of -ears, or about ears after 98, which is the ear 06.. a = = = , or about 9.8% b. Time (h) Temperature difference ( F) Bod temperature ( F) = ( 0.098) = 9.8.8( 0.098) = 9..( 0.098) = ( 0.098) = 8..( 0.098) = ( 0.098) = 80.7 So, the time of death was about 6 hours prior to midnight, or about 6 p.m.. As the independent variable changes b a constant amount, the dependent variable is multiplied b a constant factor.. a. Sample answer: the value of a CD each ear that earns.% interest compounded annuall b. Sample answer: the worth of a farm tractor that depreciates at a rate of 0% per ear 6. Monitoring Progress (pp. 8). a. The initial amount is 00,000 and the rate of growth is %, or 0.. = a( + r) t = 00,000( + 0.) t = 00,000(.) t The website membership can be represented b = 00,000(.) t. 6 Algebra Copright Big Ideas Learning, LLC

23 b. The value t = 6 represents 06 because t = 0 represents 00. = 00,000(.) t = 00,000(.) 6 00,000(.),6,0 So, in 06, the website will have about,60,000 members x As x increases b, is multiplied b. So, the table represents an exponential deca function x As x increases b, increases b 7. The function has a constant rate of change. So, it is a linear function and therefore neither an exponential growth nor an exponential deca function.. The function is of the form = a( r) t, where r <. So, it represents exponential deca. Use the deca factor r to find the rate of deca. r = 0.9 r = 0.08 r = 0.08 r = 0.08 So, the rate of deca is 8%.. The function is of the form = a( + r) t, where + r >. So, it represents exponential growth. Use the growth factor + r to find the rate of growth. + r =. r = 0. So, the rate of growth is 0%. 7. = (0.9) t+ = (0.9) t (0.9) (0.9) t (0.9) t So, the function represents exponential deca. 8. = P ( + r n) nt = 00 ( ) t = 00(.007) t Balance (dollars) t Savings Account = 00(.007) t t Year 9. a. The initial value is $,00, and the rate of deca is 9%, or = a( r) t =,00( 0.09) t =,00(0.9) t The value of the car can be represented b =,00(0.9) t. b. =,00(0.9) t =,00(0.9) (/)(t) =,00(0.9 / ) t,00(0.99) t So, r 0.99 r r r The monthl percent decrease is about 0.8%. 6. f(t) = (.0) 0t = (.0 0 ) t (.) t So, the function represents exponential growth. Copright Big Ideas Learning, LLC Algebra 7

24 c. t 0 8 6,00,7 0, Value (dollars),000,000 8,000,000, Value of a Car =,00(0.9) t t Year From the graph, ou can see that the -value is about 7000 when t =. So, the value of the car is about $7000 after ears. 6. Exercises (pp. 9 ) Vocabular and Core Concept Check. In the exponential growth function = a( + r) t, the quantit r is called the rate of growth.. The deca factor is r.. Exponential growth occurs when a quantit increases b the same factor over equal intervals of time. Exponential deca occurs when a quantit decreases b the same factor over equal intervals of time.. The function = ab x represents exponential growth when b > and x represents time. The function = ab x represents exponential deca when 0 < b < and x represents time. Monitoring Progress and Modeling with Mathematics. The initial amount is a = 0, and the rate of growth is r = 0.7, or 7%. = 0( + 0.7) t = 0(.7) 0(6.) 7.6 So, the value of is about 7.6 when t =. 6. The initial amount is a = 0, and the rate of growth is r = 0., or 0%. = 0( + 0.) t = 0(.) 0(.78).8 So, the value of is about.8 when t = r =. r = 0. The initial amount is a =, and the rate of growth is r = 0., or 0%. = (.) t = (.) (.88) 6. So, the value of is about 6. when t = r =.0 r = 0.0 The initial amount is a =, and the rate of growth is r = 0.0, or %. = (.0) t = (.0) (.76). So, the value of is about. when t = r =.07 r = 0.07 The initial amount is a = 00, and the rate of growth is r = 0.07, or 7.%. f (t) = 00(.07) t f () = 00(.07) 00(.9). So, the value of f(t) is about. when t = r =.08 r = 0.08 The initial amount is a = 7, and the rate of growth is r = 0.08, or.8%. h(t) = 7(.08) t h() = 7(.08) 7(.8) 00.9 So, the value of h(t) is about 00.9 when t =. 8 Algebra Copright Big Ideas Learning, LLC

25 . + r = r = The initial amount is a = 6.7, and the rate of growth is r =, or 00%. g(t) = 6.7() t g() = 6.7() = 6.7() =.0 So, the value of g(t) is about.0 when t =.. + r =.8 r = 0.8 The initial amount is a =, and the rate of growth is r = 0.8, or 80%. p(t) =.8 t p() = So, the value of p(t) is about 8.9 when t =.. The initial amount is 0,000, and the rate of growth is 6%, or 0.6. = a( + r) t = 0,000( + 0.6) t = 0,000(.6) t The sales can be represented b = 0,000(.6) t.. The initial amount is,000, and the rate of growth is %, or 0.0. = a( + r) t =,000( + 0.0) t =,000(.0) t Your salar can be represented b =,000(.0) t.. The initial amount is 0,000, and the rate of growth is.%, or 0.. = a( + r) t = 0,000( + 0.) t = 0,000(.) t The population can be represented b = 0,000(.) t. 6. The initial amount is., and the rate of growth is.%, or 0.0. = a( + r) t =.( + 0.0) t =.(.0) t The cost of the item can be represented b =.(.0) t. 7. a. The initial amount is,000, and the rate of growth is %, or 0.0. = a( + r) t =,000( + 0.0) t =,000(.0) t The population of Brookfield can be represented b =,000(.0) t. b. The value t = 0 represents 00 because t = 0 represents 000. =,000(.0) t =,000(.0) 0,000(.89) 68,07 So, in 00, the population will be about 68, a. The initial amount is 0., and the rate of growth is %, or 0.. = a( + r) t = 0.( + 0.) t = 0.(.) t The weight of the catfish during the 8-week period can be represented b = 0.(.) t. b. = 0.(.) t = 0.(.) 0.(.89) 0.9 So, after weeks, the catfish will weigh about 0.9 pound. 9. r = 0.6 r = 0.6 r = 0.6 r = 0.6 The initial amount is a = 7, and the rate of deca is r = 0.6, or 60%. = 7( 0.6) t = 7(0.) = 7(0.06) = 6.8 So, the value of is 6.8 when t =. Copright Big Ideas Learning, LLC Algebra 9

26 0. r = 0. r = 0. r = 0. r = 0. The initial amount is a = 8, and the rate of deca is r = 0., or %. = 8( 0.) t = 8(0.8) 8(0.6).9 So, the value of is about.9 when t =.. r = 0.7 r = 0. r = 0. r = 0. The initial amount is a = 0, and the rate of deca is r = 0., or %. g(t) = 0(0.7) t g() = 0(0.7) 0(0.) 0. So, the value of g(t) is about 0. when t =.. r = 0..0 r = 0. r = 0. r = 0. The initial amount is a = 7, and the rate of deca is r = 0., or 0%. f (t) = 7(0.) t f() = 7(0.) = 7(0.) 9. So, the value of f(t) is about 9. when t =.. r = r = 0.00 r = 0.00 r = 0.00 The initial amount is a = 700, and the rate of deca is r = 0.00, or 0.%. w(t) = 700(0.99) t w() = 700(0.99) 700(0.98) So, the value of w(t) is about when t =.. r = r = 0. r = 0. r = 0. The initial amount is a = 0, and the rate of deca is r = 0., or.%. h(t) = 0(0.86) t h() = 0(0.86) 0(0.67) So, the value of h(t) is about when t =.. r = 7 8 r = 8 ( 7 8 = = 8) ( r) = ( r =, or 0. 8 The initial amount is a =, and the rate of deca is r = 0., or.%. = ( 7 t 8) = ( 7 8) 0.7 So, the value of is about 0.7 when t =. 8) 0 Algebra Copright Big Ideas Learning, LLC

27 6. r = r = ( = = ) ( r) = ( r =, or 0. The initial amount is a = 0., and the rate of deca is r = 0., or %. = 0. ( ) t = 0. ( ) 0.(0.) 0. So, the value of is about 0. when t =. 7. The initial amount is 00,000, and the rate of deca is %, or 0.0. = a( r) t = 00,000( 0.0) t = 00,000(0.98) t The population can be represented b = 00,000(0.98) t. 8. The initial amount is 900, and the rate of deca is 9%, or = a( r) t = 900( 0.09) t = 900(0.9) t The cost of the sound sstem can be represented b = 900(0.9) t. 9. The initial amount is 00, and the rate of deca is 9.%, or = a( r) t = 00( 0.09) t = 00(0.90) t The value of the stock can be represented b = 00(0.90) t. 0. The initial amount is 0,000, and the rate of deca is.%, or 0.. = a( r) t = 0,000( 0.) t = 0,000(0.866) t The compan s profit can be represented b = 0,000(0.866) t. ). Because the rate of growth is 0%, or., the growth factor is +. =., not just.. = a( + r) t b(t) = 0( +.) t b(8) = 0(.) 8 0(.879),9 After 8 hours, there are about,9 bacteria in the culture.. Because % is the rate of deca, the deca factor is 0. not = a( r) t v(t) =,000( 0.) t v() =,000(0.86),000(0.70),76 The value of the car in 0 is about $, x As x increases b, is multiplied b. So, the table represents an exponential deca function x ( ) +( ) +( ) As x increases b, decreases b. The function has a constant rate of change. So, it is a linear function and therefore neither an exponential growth nor an exponential deca function x ( 6) +( 6) +( 6) As x increases b, decreases b 6. The function has a constant rate of change. So, it is a linear function and therefore neither an exponential growth nor an exponential deca function x 7 9 As x increases b, is multiplied b. So, the table represents an exponential growth function. Copright Big Ideas Learning, LLC Algebra

28 x As x increases b, is multiplied b. So, the table represents an exponential growth function.. The function is of the form = a( + r) t, where + r >. So, it represents exponential growth. Use the growth factor + r to find the rate of growth. + r =. r = 0. So, the rate of growth is 0% a. x As x increases b, is multiplied b 6 represents an exponential deca function t Value $7,000 $9,600 $,680 $8, So, the table As t increases b, the value is multiplied b 0.8. So, the table represents an exponential deca function. b. (8,9)(0.8) =,. So, after ears, the value of the camper is about $,. 0. a t Visitors,000,00,0,6... As t increases b, the number of visitors is multiplied b.. So, the table represents an exponential growth function. b. + + t 6 7 Visitors,000,00,0,6 6,0 7,76 So, after the website is online 7 das, about 7,76 people will have visited it.... The function is of the form = a( r) t, where r <. So, it represents exponential deca. Use the deca factor r to find the rate of deca. r = 0.8 r = 0. r = 0. r = 0. So, the rate of deca is 0%.. The function is of the form = a( r) t, where r <. So, it represents exponential deca. Use the deca factor r to find the rate of deca. r = 0.9 r = 0.0 r = 0.0 r = 0.0 So, the rate of deca is %.. The function is of the form = a( + r) t, where + r >. So, it represents exponential growth. Use the growth factor + r to find the rate of growth. + r =.08 r = 0.08 So, the rate of growth is 8%.. The function is of the form = a( + r) t, where + r >. So, it represents exponential growth. Use the growth factor + r to find the rate of growth. + r =.06 r = 0.06 So, the rate of growth is 6%. 6. The function is of the form = a( r) t, where r <. So, it represents exponential deca. Use the deca factor r to find the rate of deca. r = 0.8 r = 0. r = 0. r = 0. So, the rate of deca is %. 7. The function is of the form = a( + r) t, where + r >. So, it represents exponential growth. Use the growth factor + r to find the rate of growth. + r = r = (, or 0. = = So, the rate of growth is %. Algebra Copright Big Ideas Learning, LLC )

29 8. The function is of the form = a( r) t, where r <. So, it represents exponential deca. Use the deca factor r to find the rate of deca. r = r = (, or 0. = = So, the rate of deca is 0%. 9. = (0.9) t = (0.9)t (0.9) = (0.9)t 0.66 = 0.66 (0.9)t.(0.9) t The function is of the form = a( r) t, where r <. So, it represents exponential deca. 0. = (.) t+8 = (.) t (.) 8 (.) t (.8).8(.) t The function is of the form = a( + r) t, where + r >. So, it represents exponential growth.. = (.06) 9t = (.06 9 ) t (.69) t The function is of the form = a( + r) t, where + r >. So, it represents exponential growth.. = (0.8) t = (0.8) ( ) t = (0.8 ) t = ( 0.8 ) t (0.96) t The function is of the form = a( r) t, where r <. So, it represents exponential deca.. x(t) = (.) t = (.) ( ) t = (. ) t = (. ) t (.0) t The function is of the form = a( + r) t, where + r >. So, it represents exponential growth. ). f(t) = 0.(.6) t = 0. (.6)t (.6) = 0..6 (.6)t 0.(.6) t The function is of the form = a( + r) t, where + r >. So, it represents exponential growth.. b(t) = (0.) t+ = (0.) t (0.) = (0.) (0.) t = (0.667) (0.) t 0.67(0.) t The function is of the form = a( r) t, where r <. So, it represents exponential deca. 6. r(t) = (0.88) t = (0.88 ) t (0.60) t The function is of the form = a( r) t, where r <. So, it represents exponential deca. 7. = P ( + n) r nt = 000 ( ) t = 000( + 0.0) t = 000(.0) t 8. = P ( + n) r nt = 00 ( ) t = 00( + 0.0) t = 00(.0) t 9. = P ( + n) r nt = 600 ( ) t = 600( ) t = 600(.007) t 60. = P ( + n) r nt = 00 ( ) t = 00( + 0.0) t = 00(.0) t Copright Big Ideas Learning, LLC Algebra

30 6. a. Tree A Year, t 0 Basal area, A From the table, ou know the initial basal area of Tree A is 0 square inches, and it is multiplied b a growth factor of. each ear. So, P = 0 and + r =.. = P( + r) t = 0(.) t The initial basal area of Tree B is square inches, and the rate of growth is 6%, or = P( + r) t = ( ) t = (.06) t So, the function that represents the basal area of Tree A after t ears is = 0(.) t and the basal area of Tree B after t ears is = (.06) t. b. Tree B Year, t 0 Basal area, A Basal area (in. ) Tree Basal Area A A A = 0(.) t t Year A B = (.06) t The basal area of Tree B is larger than the basal area of Tree A, but the difference between the basal areas is decreasing. 6. a. The principal of the investment account is $00, the annual interest rate is 6%, or 0.06, and because the interest is compounded quarterl, n =. = P ( + n) r nt = 00 ( + 0. ) t = 00( + 0.0) t = 00(.0) t The graph crosses the -axis at (0, 00). So, the principal of the savings account is $00. The points on the graph are approximatel (, ), (, 0), and (, 7). Because ,.08, and.07, the 00 balance of the savings account has a growth factor of about.08. So, P = 00 and + r =.08. = P( + r) t = 00(.08) t So, the function that represents the balance of the investment account is = 00(.0) t, and the function that represents the balance of the savings account is = 00(.08) t. b. Investment account: Year, t 0 Balance (dollars), Balance (dollars) Accounts = 00(.0) t = 00(.08) t t Year Both accounts start with the same balance. The investment account balance is increasing at a faster rate, so it is greater than the savings account balance after the start. 6. a. The initial value is,000, and the rate of growth is.%, or 0.0. = P( + r) t =,000( + 0.0) t =,000(.0) t A function that represents the cit s population is =,000(.0) t. b. Use the fact that t = (t) and the properties of exponents to rewrite the function in a form that reveals the monthl rate of growth. =,000(.0) t =,000(.0) (/)(t) =,000(.0 (/) ) (t),000(.007) (t) + r.007 r So, the monthl percent increase is about 0.%. Algebra Copright Big Ideas Learning, LLC