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Unit 6 – Chapter 8 Unit 6 • • • • Chapter 7 Review and Chap. 8 Skills Section 8.1 – Exponent Properties with products Section 8.2 – Exponent Properties with quotients Section 8.3 – Define and Use Zero and Negative Exponents • Section 8.4 – Use Scientific Notation • Section 8.5 – Write and Graph Exponential Growth Functions • Section 8.6 – Write and Graph Exponential Decay Functions Warm-Up – Ch. 8 Daily Homework Quiz 2. For use after Lesson 7.1 Solve the linear system by graphing. 2x + y = – 3 – 6x + 3y = 3 ANSWER (–1, –1) Daily Homework Quiz 3. For use after Lesson 7.1 A pet store sells angel fish for $6 each and clown loaches for $4 each . If the pet store sold 8 fish for $36, how many of each type of fish did it sell? ANSWER 2 angel fish and 6 clown loaches Daily Homework Quiz For use after Lesson 7.2 Solve the linear system using substitution 1. –5x – y = 12 3x – 5y = 4 (–2, –2) ANSWER 2. 2x + 9y = –4 x – 2y = 11 ANSWER (7, –2 ) Daily Homework Quiz For use after Lesson 7.3 Solve the linear system using elimination. 1. –5x +y = 18 3x – y = –10 ANSWER 2. (–4, –2) 4x + 2y = 14 4x – 3y = –11 ANSWER 3. (1, 5) – 7x – 3y = 11 4x – 2y = 16 ANSWER (1, – 6) Daily Homework Quiz For use after Lesson 7.4 4. A recreation center charges nonmembers $3 to use the pool and $5 to use the basketball courts. A person pays $42 to use the recreation facilities 12 times. How many times did the person use the pool. ANSWER 9 times Daily Homework Quiz For use after Lesson 7.3 3. A group of 12 students and 3 teachers pays $57 for admission to a primate research center. Another group of 14 students and 4 teachers pays $69. Find the cost of one student ticket. ANSWER $3.50 Daily Homework Quiz 1. For use after Lesson 7.6 Write a system of inequalities for the shaded region. ANSWER x < 2, y > x +1 Prerequisite Skills 1. VOCABULARY CHECK Identify the exponent and the base in the expression 138. ANSWER Exponent: 8, base: 13 2. Copy and complete: An expression that represents repeated multiplication of the same factor is called a(n) ? . ANSWER Power Prerequisite Skills SKILLS CHECK Evaluate the expression. 3. x2 when x = 10 4. ANSWER ANSWER 100 5. r2 when r = 5 6 ANSWER 25 36 a3 when a = 3 27 6. z3 when z = 1 2 ANSWER 1 8 Prerequisite Skills SKILLS CHECK Order the numbers from least to greatest. 7. 6.12, 6.2, 6.01 8. ANSWER 0.073, 0.101, 0.0098 ANSWER 6.01, 6.12, 6.2 0.0098, 0.073, 0.101 Write the percent as a decimal. 9. 4% ANSWER 0.04 10. 0.5% ANSWER 0.005 11. 13.8% ANSWER 0.138 12. 145% ANSWER 1.45 Prerequisite Skills 13. SKILLS CHECK Write a rule for the function. ANSWER f (x) = x + 2 Warm-Up – 8.1 Lesson 8.1, For use with pages 488-494 Evaluate the expression. 1. x4 when x = 3 3. ANSWER 2. a2 81 when a = –6 ANSWER -a2 when a = –6 36 ANSWER -36 Lesson 8.1, For use with pages 488-494 Evaluate the expression. 3. m3 when m = –5 ANSWER –125 4. A food storage container is in the shape of a cube. What is the volume of the container if one side is 4 inches long? Use V = s3. ANSWER 64 in.3 Vocabulary – 8.1 • Power • Repeated multiplication • Exponent • How many times to multiply a quantity • Base • Quantity multiplied • Order of Magnitude • The “power of 10” nearest the number Activity • What is a2 * a3? • Expand it out and combine like terms. • Do you notice anything about the exponent? • Try x3 * x4. What do you get? See any patterns? • What is (x2)3? • Expand it and combine like terms. • What do you notice about the exponent? • Try (x3)3 • What do you get? Notice any patterns? Notes – 8.1 – Exponents with Products •Product of Powers Rule • To Multiply Exponents w/ SAME BASE(!) •ADD THE EXPONENTS •am * an = a (m+n) •Power of a Power Rule •To raise powers to a power w/SAME BASE(!) •MULTIPLY THE EXPONENTS •(am)n = a (m * n) •Order of Magnitude •Round number to nearest power of 10 •The exponent of 10 is the “order of magnitude” Examples 8.1 GUIDED PRACTICE for Example 1 Simplify the expression. 1. 32 37 = 32 + 7 = 39 2. 5 59 = 51 59 = 51 + 9 = 510 3. (– 7)2(– 7) = (– 7)2 (– 7)1 = (– 7)2+1 = (–7)3 4. x2 x6 x = x2 x6 x1 = x2 + 6 + 1 = x9 EXAMPLE 2 a. Use the power of a power property (25)3 = 25 3 b. = 215 c. (x2)4 = x2 = x8 [(–6)2]5 = (–6)2 5 = (–6)10 4 d. [(y + 2)6]2 = (y+ 2)6 = (y + 2)12 2 GUIDED PRACTICE for Example 2 Simplify the expression. 5. (42)7 = 42 7 6. = 414 7. (n3)6 = n3 = n18 [(–2)4]5 = (–2)4 5 = (–2)20 6 8. [(m + 1)5]4 = (m + 1)5 4 = (m + 1)20 EXAMPLE 3 b. b. Use the power of a product property a. (24 13)8 = ? a. 248 138 (9xy)2 = (9 x y)2 = 92 x2 y2 = 81x2y2 EXAMPLE 3 c. (–4z)2 = (–4 z)2 (–4)2 z2 = 16z2 c. d. d. d. d. d. d. Use the power of a product property – (4z)2 = – (4 z)2 = – (42 z2) = –16z2 What is the order of magnitude of 90,000? 90,000 is closest to 100,000 = 105, so Order of magnitude is 5. What is the order of magnitude of 99? 99 is closest to 100 = 102, so Order of magnitude is 2. EXAMPLE 4 Use all three properties Simplify (2x3)2 x4 (2x3)2 x4 = 22 (x3)2 x4 Power of a product property = 4 x6 x4 Power of a power property = 4x10 Product of powers property for Examples 3, 4 and 5 GUIDED PRACTICE Simplify the expression. 9. 10. 11. (42 12)2 = 422 122 (–3n)2 = (–3 n)2 = (–3)2 n2 = 9n2 (9m3n)4 = 94 (m3) 4 n 4 = 6561 m12 n4 = 6561 m12n4 Power of a product property Product of powers property Power of a power property GUIDED PRACTICE 12. 5 (5x2)4 for Examples 3, 4 and 5 =5 54 (x2)4 Power of a product property = 5 625 x8 Power of a power property = 3125x8 Product of powers property EXAMPLE 5 Solve a real-world problem Bees In 2003 the U.S. Department of Agriculture (USDA) collected data on about 103 honeybee colonies. There are about 104 bees in an average colony during honey production season. About how many bees were in the USDA study? EXAMPLE 5 Solve a real-world problem SOLUTION To find the total number of bees, find the product of the number of colonies, 103, and the number of bees per colony, 104. 103•104 = 103+4 = 107 ANSWER The USDA studied about 107, or 10,000,000, bees. Warm-Up – 8.2 Lesson 8.2, For use with pages 495-501 Evaluate the expression. 1 1. q3 when q = 4 ANSWER 1 64 2. c2 when c = ANSWER 9 25 3 5 Lesson 8.2, For use with pages 495-501 Evaluate the expression. 3. A magazine had a circulation of 9364 in 2001. The circulation was about 125 times greater in 2006. Use order of magnitude to estimate the circulation in 2006. ANSWER 10. about 106 or 1,000,000 (–5n2)2 = (–5)2 (n2)2 = 25n4 Vocabulary – 8.2 • No new vocab words!! Activity • What is a3 / a2? • Expand it out and cancel like terms. • Do you notice anything about the exponents? • Try x5/x2. What do you get? See any patterns? • What is (1/2)3? • Expand it and combine like terms. • What do you notice about the exponent? • Try (2/3)3 • What do you get? Notice any patterns? Notes – 8.2 –Exponents with quotients •Quotient of Powers Rule • To Divide Powers w/ SAME BASE(!) •SUBTRACT THE EXPONENTS •am / an = a (m-n) •Power of a Quotient Rule •To raise fractions to a power: •Raise numerator and denominator to the power •(a/b)m = am/bm Examples 8.2 EXAMPLE 1 a. Use the quotient of powers property 810 = 810 – 4 84 = 86 b. (– 3)9 = (– 3)9 – 3 (– 3)3 = (– 3)6 c. 12 54 58 = 5 57 57 = 512 – 7 = 55 EXAMPLE 1 d. Use the quotient of powers property x6 1 6 x = 4 x x4 = x6 – 4 = x2 GUIDED PRACTICE for Example 1 Simplify the expression. 1. 611 11 – 5 =6 65 = 66 2. (– 4)9 9–2 = (– 4) 2 (– 4) = (– 4)7 3. 94 93 = 97 92 92 = 97 – 2 = 95 GUIDED PRACTICE 4. y8 1 8 y = 5 y y5 = y8 – 5 = y3 for Example 1 EXAMPLE 2 a. b. Use the power of quotient property x 3 x3 y = y3 –7 x 2 = –7 x 2 (– 7)2 49 = = x2 x2 EXAMPLE 3 4x2 5y a. b. 5 2 a b 3 Use properties of exponents (4x2)3 = (5y)3 43 (x2)3 = 3 3 5y 64x6 = 125y3 (a2)5 1 2a2 = b5 a10 = 5 b a10 = 2a2b5 8 = a 2b5 1 2a2 1 2a2 Power of a quotient property Power of a product property Power of a power property Power of a quotient property Power of a power property Multiply fractions. Quotient of powers property GUIDED PRACTICE for Examples 2 and 3 Simplify the expression. 5. a 2 a2 = 2 b b 5 6. = – y 7. x2 4y 3 –5 = y 2 3 2)2 ( x = (4y)2 x4 = 2 2 4y x4 = 16y2 (– 5)3 125 = 3 =– y y3 Power of a quotient property Power of a product property Power of a power property GUIDED PRACTICE 8. 2s 3t 3 t5 16 23 s3 = 3 3 3 t for Examples 2 and 3 t5 16 8 s3 t 5 = 27 t3 16 8 s3 t5 = 27 16t3 s3 t 2 = 54 Power of a quotient property Power of a power property Multiply fractions. EXAMPLE 4 Solve a multi-step problem Fractal Tree To construct what is known as a fractal tree, begin with a single segment (the trunk) that is 1 unit long, as in Step 0. Add three shorter segments that are 1 unit 2 long to form the first set of branches, as in Step 1. Then continue adding sets of successively shorter branches so that each new set of branches is half the length of the previous set, as in Steps 2 and 3. EXAMPLE 4 Solve a multi-step problem a. Make a table showing the number of new branches at each step for Steps 1 - 4. Write the number of new branches as a power of 3. b. How many times greater is the number of new branches added at Step 5 than the number of new branches added at Step 2? EXAMPLE 4 Solve a multi-step problem SOLUTION a. Step Number of new branches 1 3 = 31 b. 2 9 = 32 3 27 = 33 4 81 = 34 The number of new branches added at Step 5 is 35. The number of new branches added at Step 2 is 32. So, the number of new branches added at 5 Step 5 is 3 = 33 = 27 times the number of new 32 branches added at Step 2. GUIDED PRACTICE for Example 4 9. FRACTAL TREE In Example 4, add a column to the table for the length of the new branches at each step. Write the length of the new branches as power of 1 . What is the length of a new branch 2 added at Step 9? SOLUTION 1 1 9 ( 9 ) = 512 units EXAMPLE 5 Solve a real-world problem ASTRONOMY The luminosity (in watts) of a star is the total amount of energy emitted from the star per unit of time. The order of magnitude of the luminosity of the sun is 1026 watts. The star Canopus is one of the brightest stars in the sky. The order of magnitude of the luminosity of Canopus is 1030 watts. How many times more luminous is Canopus than the sun? EXAMPLE 5 Solve a real-world problem SOLUTION Luminosity of Canopus (watts) 1030 30 - 26 4 10 10 = = = Luminosity of the sun (watts) 1026 ANSWER Canopus is about 104 times as luminous as the sun. GUIDED PRACTICE for Example 5 9. WHAT IF? Sirius is considered the brightest star in the sky. Sirius is less luminous than Canopus, but Sirius 1 appears to be brighter because it is much closer to the 2 Earth. The order of magnitude of the luminosity of Sirius is 1028 watts. How many times more luminous is Canopus than Sirius? SOLUTION Luminosity of Canopus (watts) 1030 30 - 28 2 10 10 = = = Luminosity of the sun (watts) 1028 ANSWER Canopus is about 102 times as luminous as Sirius Warm-Up – 8.3 Lesson 8.3, For use with pages 502-508 1. Simplify (– 3x)2. ANSWER 9x2 a3 2. Simplify 2b ANSWER 5 . a15 32b5 Lesson 8.3, For use with pages 502-508 3. The order of magnitude of Earth’s mass is about 1027 grams. The order of magnitude of the sun’s mass is about 1033 grams. About how many times as great is the sun’s mass as Earth’s mass? ANSWER about 106 Vocabulary – 8.3 • No New Vocabulary in this section either!! Activity • What was our rule for Quotients of Exponents? • What is a2 / a3? Expand it and cancel like terms. • Apply the quotient exponent rule. What do you get? • Try x2/x5. What do you get when you apply the quotient exponent rule? See any patterns? • Try x3/x3. What do you get? See any patterns? • What is the square root of 64? • What do you get when you raise 64^(1/2) on your calculator? Notes – 8.3 – Zero and Negative Exponents • Zero Exponents •Any number to the zero power is 1 EXCEPT 0! •Negative Exponents •an is the reciprocal of a-n (an = 1 / a-n) •a-n is the reciprocal of an (a–n = 1 / an) •TO EVALUATE NEGATIVE EXPONENTS: •Take the reciprocal and change the sign of the exponent. •Think “Change the line and flip the sign” •Fractional Exponents •Raising a number to a fractional exponent looks Notes – 8.3 – Zero and Negative Exponents Summary of Exponent Rules – on page 504 of your textbook Examples 8.3 EXAMPLE 1 3 –2= 1 32 1 = 9 b. (–7 )0 = 1 a. Use definition of zero and negative exponents Definition of negative exponents Evaluate exponent. Definition of zero exponent EXAMPLE 1 1 5 c. –2 1 1 2 5 1 = 1 25 = = 25 d. Use definition of zero and negative exponents Definition of negative exponents Evaluate exponent. Simplify by multiplying numerator and denominator by 25. 0 – 5 = 1 (Undefined) a – n is defined only for a nonzero 05 number a. GUIDED PRACTICE for Example 1 Evaluate the expression. 1. 2 3 0 =1 Definition of zero exponents Evaluate the expression. 2. (–8) –2 1 = (–8) 2 1 = 64 Definition of negative exponents Evaluate exponent. GUIDED PRACTICE for Example 1 Evaluate the expression. 3. 1 13 = –3 1 2 2 Definition of negative exponents 1 = 1 8 =8 Simplify by multiplying numerator and denominator by 8. GUIDED PRACTICE for Example 1 Evaluate the expression. 4. (–1 )0 = 1 Definition of zero exponent EXAMPLE 2 a. Evaluate exponential expressions 6– 4 64 = 6– 4 + 4 Product of a power property = 60 Add exponents. =1 Definition of zero exponent EXAMPLE 2 b. Evaluate exponential expressions (4– 2)2 = 4– 2 Power of a product property = 4– 4 Multiply exponents. 1 = 4 4 Definition of negative exponents 1 = 256 c. 2 1 4 = 3 3– 4 = 81 Evaluate power. Definition of zero exponent Evaluate power. EXAMPLE 2 d. Evaluate exponential expressions 5– 1 –1 –2 = 5 52 Quotient of powers property = 5– 3 Subtract exponents. 1 = 3 5 Definition of negative exponents = 1 125 Evaluate power. GUIDED PRACTICE for Example 2 Evaluate the expression. 5. 1 3 = 4 4– 3 Definition of negative exponent = 64 Evaluate power. Evaluate the expression. 6. (5– 3) – 1 = 5– 3 –1 Power of a product property = 53 Multiply exponents. = 125 Evaluate power. GUIDED PRACTICE for Example 2 Evaluate the expression. 7. (– 3 ) 5 (– 3 ) – 5 = (– 3 ) 5 + ( – 5) Product of a power property = (– 3)0 Add exponents. =1 Definition of zero exponent GUIDED PRACTICE for Example 2 Evaluate the expression. 8. 6– 2 –2 –2 = 6 62 = 6– 4 = 1 64 1 = 1296 Quotient of powers property Subtract exponents. Definition of negative exponents Evaluate power. EXAMPLE 3 Use properties of exponents Simplify the expression. Write your answer using only positive exponents. a. (2xy – 5)3 = 23 x3 (y – 5)3 Power of a product property = 8 x3 y – 15 Power of a power property 8x3 = 15 y Definition of negative exponents EXAMPLE 3 b. Use properties of exponents y5 (2x)– 2y5 = (2x)2(– 4x2y2) – 4x2y2 Definition of negative exponents y5 = (4x)2(– 4x2y2) Power of a product property y5 = –16x4y2 Product of powers property = – y3 16x4 Quotient of powers property EXAMPLE 1 Fractional Exponents Warm-Up – 8.4 Lesson 8.4, For use with pages 512-519 1. Simplify using only positive exponents: x2y-3 ANSWER x2/y3 1. Simplify using only positive exponents: (6x-2y3)-3 ANSWER x6 / (216y9) 2. Simplify using positive exponents 2 8x-2y-6 ANSWER x2y6 4 Lesson 8.4, For use with pages 512-519 1. Order the numbers 0.014, 0.1, 0.01 from least to greatest. ANSWER 0.01, 0.014, 0.1 2. Find the ratio of the mass of the Milky Way galaxy, which is about 1044 grams, to the mass of the universe, which is about 1055 grams. ANSWER about 1 1011 Vocabulary – 8.4 • Scientific Notation • Used to represent very large or very small numbers Notes – 8.4 – Using Scientific Notation. 1. To Write Scientific Notation •Of the form C * 10n where 1<= C < 10 and n is an integer •n is the number of places to move the decimal 2. Comparing using Scientific Notation •I can only compare #’s in math that ?????? •Convert all to standard form or Sci. Not. •Order the exponents (powers of ten) first •Order the C’s second Notes – 8.4 – Using Sci. Not. – cont 1. To Add or Subtract using Scientific Notation •Convert numbers so that n is the same •Line up decimals •Add the C’s and leave the n’s •Convert back to sci. not. •Example: Add 1.23*102 + 4.56*103 •Answer: 4.683*103 2. To Multiply or Divide using Scientific Not. •Multiply/Divide the C’s •Multiply/Divide the 10n •Convert to proper sci. not. Examples 8.4 EXAMPLE 1 Write numbers in scientific notation a. 42,590,000 = 4.259 107 Move decimal point 7 places to b. 0.0000574 = 5.74 10-5 the left. Exponent is 7. Move decimal point 5 places to the right. Exponent is – 5. EXAMPLE 2 a. Write numbers in standard form 2.0075 106 = 2,007,500 Exponent is 6. Move decimal point 6 places to the right. b. 1.685 10-4 = 0.0001685 Exponent is – 4. Move decimal point 4 places to the left. GUIDED PRACTICE for Examples 1 and 2 Write the number 539,000 in scientific notation. Then write the number 4.5 3 10 – 4 in standard form. 1. 539,000 = 5.39 105 4.5 10 – 4 = 0.00045 Move decimal point 5 places to the left. Exponent is 5. Exponent is – 4. Move decimal point 4 places to the left. EXAMPLE 3 Order numbers in scientific notation Order 103,400,000 , 7.8 to greatest. 8 10 , and 80,760,000 from least SOLUTION STEP 1 Write each number in scientific notation, if necessary. 103,400,000 = 1.034 108 80,760,000 = 8.076 107 EXAMPLE 3 Order numbers in scientific notation STEP 2 Order the numbers. First order the numbers with different powers of 10. Then order the numbers with the same power of 10. Because 107 < 108, you know that 8.076 107 is less than both 1.034 10 8 and 7.8 108. Because 1.034 < 7.8, you know that 1.034 108 is less than 7.8 108. So, 8.076 107 < 1.034 108 < 7.8 108. EXAMPLE 3 Order numbers in scientific notation STEP 3 Write the original numbers in order from least to greatest. 80,760,000; 103,400,000; 7.8 108 EXAMPLE 4 Compute with numbers in scientific notation Evaluate the expression. Write your answer in scientific notation. a. (8.5 102)(1.7 106) = (8.5 • 1.7) (102•106) Commutative property and = 14.45 = 1.445 108 109 associative property Product of powers property Product of powers property EXAMPLE 4 b. (1.5 Compute with numbers in scientific notation –3 2 10 ) = 1.52 –3 2 (10 ) Power of a product property = 2.25 (10 – 6) c. (1.2 10 4) 1.2 = 1.6 1.6 (10 – 3) Power of a power property 10 4 10 – 3 = 0.75 (10 7) Product rule for fractions Quotient of powers property = (7.5 10 – 1) 10 7 Write 0.75 in scientific notation. = 7.5 (10 – 1 10 7) Associative property = 7.5 (10 6) Product of powers property GUIDED PRACTICE 2. for Examples 3 and 4 Order 2.7 × 10 5, 3.401 × 10 4, and 27,500 from least to greatest. SOLUTION STEP 1 Write each number in scientific notation, if necessary. 27,500 = 2.75 × 104 for Examples 3 andnotation 4 Order numbers in scientific GUIDED PRACTICE STEP 2 Order the numbers. First order the numbers with different powers of 10. Then order the numbers with the same power of 10. Because 104 < 105, you know that 3.401 104, 0.7 104 is less than both 2.7 105. Because 2.7 < 3.401, you know that 2.7 104 is less than 3.401 104 So, 2.7 104 < 2.7 105 < 3.401 104 EXAMPLE 3 Order numbers in scientific notation STEP 3 Write the original numbers in order from least to greatest. 27,500; 3.401 × 104, and 2.7 × 105 for Examples 3 and 4 GUIDED PRACTICE Evaluate the expression. Write your answer in scientific notation. 3. (1.3 –5 2 10 ) = 1.32 –5 2 (10 ) = 1.69 (10 –10) 4. 4.5 10 5 4.5 = –2 1.5 10 1.5 =3 10 7 10 5 10 – 2 Power of a product property Power of a power property Product rule for fractions Quotient of powers property for Examples 3 and 4 GUIDED PRACTICE Evaluate the expression. Write your answer in scientific notation. 5. (1.1 107) (1.7 = (1.1 = 4.62 102) 1.7) (102 107) Commutative property and 109 associative property Product of powers property Warm-Up – 8.5 Lesson 8.5, For use with pages 520-527 1. Simplify – (– 4x2)3. ANSWER 64x6 2. Simplify ANSWER 2. Simplify ANSWER 2a3 3b 2 4a6 9b2 2a-3b5 3b-1 2b6 3a3 Lesson 8.5, For use with pages 520-527 3. The table shows the cost of tickets for a matinee. Write a rule for the function. Tickets, t 2 5 Cost, c ANSWER c = 2t + 1 4 6 8 9 13 17 Vocabulary – 8.5 • Exponential Function • Function with a variable as an exponent y = abx • Graph is NOT linear!! • Exponential Growth • Base must be b > 1 • Graph curves UP • Exponential Decay • Base must be 0 < b < 1 • Graph curves DOWN • Compound Interest • Interest earned on the initial investment and interest already earned Notes – 8.5 – Exponential Growth Functions. • Exponential Functions are in the form of: • Y = Abx •A = Initial Value •b = Growth Factor •If A > 0 and b > 1, it represents “Exponential Growth” Notes – 8.5 – Exponential Growth Functions. • Exponential Growth is frequently used to calculate population growth and compound interest. •Uses the following model Examples 8.5 EXAMPLE 1 Write a function rule Write a rule for the function. x –2 –1 0 1 2 y 2 4 8 16 32 SOLUTION STEP 1 Tell whether the function is exponential. EXAMPLE 1 Write a function rule +1 +1 +1 +1 X –2 –1 0 1 2 y 2 4 8 16 32 2 2 2 2 Here, the y-values are multiplied by 2 for each increase of 1 in x, so the table represents an exponential function of the form y = a b x where b = 2. EXAMPLE 1 Write a function rule STEP 2 Find the value of a by finding the value of y when x = 0. When x = 0, y = ab0 = a 1 = a. The value of y when x = 0 is 8, so a = 8. STEP 3 Write the function rule. A rule for the function is y = 8 2x. GUIDED PRACTICE 1. for Example 1 Write a rule for the function. x –2 –1 0 1 2 y 3 9 27 81 243 SOLUTION STEP 1 Tell whether the function is exponential. for Example 1 GUIDED PRACTICE +1 +1 +1 +1 X –2 –1 0 1 2 y 3 9 27 81 343 3 3 3 3 Here, the y-values are multiplied by 3 for each increase of 1 in x, so the table represents an exponential function of the form y = a b x where b = 3. GUIDED PRACTICE for Example 1 STEP 2 Find the value of a by finding the value of y when x = 0. When x = 0, y = ab0 = a 1 = a. The value of y when x = 0 is 27, so a = 27. STEP 3 Write the function rule. A rule for the function is y = 27 3x. EXAMPLE 2 Graph an exponential function Graph the function y = 2x. Identify its domain and range. SOLUTION STEP 1 Make a table by choosing a few values for x and finding the values of y. The domain is all real numbers. EXAMPLE 2 Graph an exponential function STEP 2 Plot the points. STEP 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is all positive real numbers. EXAMPLE 3 Compare graphs of exponential functions Graph the functions y = 3 2x and y = –3 2x. Compare each graph with the graph of y = 2x. SOLUTION To graph each function, make a table of values, plot the points, and draw a smooth curve through the points. EXAMPLE 3 Compare graphs of exponential functions Because the y-values for y = 3 2x are 3 times the corresponding y-values for y = 2x, the graph of y = 3 2x is a vertical stretch of the graph of y = 2x. Because the y-values for y = –3 2x are – 3 times the corresponding y-values for y = 2x, the graph of y = – 3 2x is a vertical stretch with a reflection in the x-axis of the graph of y = 2x. GUIDED PRACTICE 2. for Examples 2 and 3 Graph y = 5x and identify its domain and range. SOLUTION Domain: all real numbers, range: all positive real numbers GUIDED PRACTICE for Examples 2 and 3 1 3. Graph y = 3 2x. Compare the graph with the graph of y = 2x. SOLUTION The graph is a vertical shrink of the graph of y = 2x. GUIDED PRACTICE 4. for Examples 2 and 3 Graph y = – 13 2x. Compare the graph with the graph of y = 2x. SOLUTION The graph is a vertical shrink with a reflection in the x–axis of the graph of y = 2x. EXAMPLE 4 Solve a multi-step problem Collector Car The owner of a 1953 Hudson Hornet convertible sold the car at an auction. The owner bought it in 1984 when its value was $11,000. The value of the car increased at a rate of 6.9% per year. EXAMPLE 4 Solve a multi-step problem a. Write a function that models the value of the car over time. b. The auction took place in 2004. What was the approximate value of the car at the time of the auction? Round your answer to the nearest dollar. SOLUTION a. Let C be the value of the car (in dollars), and let t be the time (in years) since 1984. The initial value a is $11,000, and the growth rate r is 0.069. EXAMPLE 4 Solve a multi-step problem C = a(1 + r)t Write exponential growth model. = 11,000(1 + 0.069)t Substitute 11,000 for a and 0.069 fo = 11,000(1.069)t Simplify. b. To find the value of the car in 2004, 20 years after 1984, substitute 20 for t. C = 11,000(1.069)20 ≈ 41,778 Substitute 20 for t. Use a calculator. EXAMPLE 4 Solve a multi-step problem ANSWER In 2004 the value of the car was about $41,778. EXAMPLE 5 Standardized Test Practice You put $250 in a savings account that earns 4% annual interest compounded yearly. You do not make any deposits or withdrawals. How much will your investment be worth in 5 years? A $300 B $304.16 C $1344.56 D $781,250 SOLUTION y = a(1 + r)t = 250(1 + 0.04)5 = 250(1.04)5 304.16 Write exponential growth model. Substitute 250 for a, 0.04 for r, and 5 fo Simplify. Use a calculator. You will have $304.16 in 5 years. EXAMPLE 5 Standardized Test Practice ANSWER The correct answer is B. A B C D GUIDED PRACTICE 5. for Examples 4 and 5 What if ?In example 4, suppose the owner of the car Sold in 1994. Find the value of the car to the nearest dollar. SOLUTION Let C be the value of the car (in dollars), and let t be the time (in years) since 1984. The initial value a is $11,000, and the growth rate r is 0.069. GUIDED PRACTICE C = a(1 + r)t for Examples 4 and 5 Write exponential growth model. = 11,000(1 + 0.069)t Substitute 11,000 for a and 0.069 fo = 11,000(1.069)t Simplify. To find the value of the car in 1994, 30 years after 1984, substitute 30 for t. C = 11,000(1.069)30 ≈ 21,437 ANSWER Substitute 30 for t. Use a calculator. In 2004 the value of the car was about $21,437. GUIDED PRACTICE 6. for Examples 4 and 5 What if ?In example 5, suppose the annual interest rate is 3.5%. How much will your investment be worth in 5 year. SOLUTION y = a(1 + r)t Write exponential growth model. = 250(1 + 0.035) Substitute 250 for a, 0.035 for r, and 5 = 250(1.035)5 Simplify. 296.92 ANSWER Use a calculator. You will have $296.92 in 5 years. Warm-Up – 8.6 Lesson 8.6, For use with pages 530-538 1. Evaluate ANSWER 2. Evaluate ANSWER 1 3 . 2 1 8 1 –2. 4 16 2. Evaluate x . –2 2y ANSWER 4y2/x2 Lesson 8.6, For use with pages 530-538 3. The table shows how much money Tess owes after w weeks. Write a rule for the function. Week, w Owes, m ANSWER 0 1 2 3 50 45 40 35 m = 50 – 5w 3. Given the following points, write the linear equation in Slope intercept form: (1,4) and (2,6) ANSWER y = 2x + 2 EXAMPLE 5 Standardized Test Practice You put $250 in a savings account that earns 5% annual interest compounded yearly. You do not make any deposits or withdrawals. What exponential function will model this growth? How much will your investment be worth in 5 years? In 20 years? SOLUTION y = a(1 + r)t y = 250(1 + .05)t = 250(1.05)5 319.07 = 250(1.05)20 663.32 You will have $319.07 in 5 years and 663.32 in 20 years. EXAMPLE 5 Standardized Test Practice If you buy a house for $100,000 and get a 4.5% loan for 30 years compounded annually, how much will you pay for the house at the end of the loan? SOLUTION y = a(1 + r)t y = 100000(1 + .045)t = 100000(1.045)30 374,532 You will pay almost $375,000 for your $100,000 house! EXAMPLE 5 Standardized Test Practice Some credit card companies charge 18% interest ANNUALLY compounded MONTHLY(!!). If you max out your $5000 credit limit and don’t make any payments for a year, how much will you owe? How about if you don’t make any payments for 2 years? SOLUTION y = a(1 + r)t y = 5000(1 + 0.18/12)12t = 5000(1.015)12 5978.09 You will pay $5978.09 after 1 year and $7,147.51 after two years. Vocabulary – 8.6 • Exponential Decay • Y = abx • a>0 • Base must be 0 < b < 1 • Graph curves DOWN from left to right Notes – 8.6 – Exponential Decay • Exponential Functions are in the form of: • Y = Abx •A = Initial Value •b = Growth Factor •If A > 0 and 0 < b < 1, it represents “Exponential Decay” •To Find the Exponential Rule •Use points for x = 0 and x = 1 •Plug in what you know and ….. •Solve for both “a” and “b” Notes – 8.6 – Exponential Decay Functions. • Exponential Decay is frequently used to calculate population decline and investments that lose money •Uses the following model Examples 8.6 EXAMPLE 2 Graph an exponential function 1 Graph the function y = 2 range. x and identify its domain and SOLUTION STEP 1 Make a table of values. The domain is all real numbers. x –2 –1 0 1 2 y 4 2 1 1 2 1 4 EXAMPLE 2 Graph an exponential function STEP 2 Plot the points. STEP 3 Draw a smooth curve through the points. From either the table or the graph, you can see the range is all positive real numbers. EXAMPLE 3 Compare graphs of exponential functions Graph the functions y = 3 1 1x – and y = 3 2 1x 2 x 1 Compare each graph with the graph of y =. 2 Compare graphs of exponential functions EXAMPLE 3 SOLUTION x 1 y= 2 –2 4 12 –1 2 6 0 1 3 1 1 2 1 4 3 2 3 4 2 x y =3 1 2 x 1 1 y= – 3 2 –4 3 –2 3 –1 3 –1 6 – 1 12 x EXAMPLE 3 Compare graphs of exponential functions 1 x Because the y-values for y = 3 2 are 3 times the 1 x corresponding y-values for y = , the graph of 2 1 x 1 x y=3 is a vertical stretch of the graph of y = . 2 2 1 1 x – 1 times the Because the y-values for y = – are 3 2 3 1 x corresponding y-values for y = 2 , the graph of 1 1 x y = – 3 2 is a vertical shrink with reflection in the 1 x x-axis of the graph of y = 2 . GUIDED PRACTICE for Examples 2 and 3 2. Graph the function y =(0.4) and range. x and identify its domain SOLUTION STEP 1 Make a table of values. The domain is all real numbers. x –2 –1 y 0.16 –0.4 0 1 2 1 0.4 0.16 GUIDED PRACTICE for Examples 2 and 3 STEP 2 Plot the points. STEP 3 Draw a smooth curve through the points. From either the table or the graph, you can see the range is all positive real numbers. EXAMPLE 4 Classify and write rules for functions Tell whether the graph represents exponential growth or exponential decay.Then write a rule for the function. SOLUTION The graph represents exponential growth (y = abx where b > 1). The yintercept is 10, so a = 10. Find the value of b by using the point (1, 12) and a = 10. y = abx Write function. 12 = 10 . b1 Substitute. 1.2 = b Solve. A function rule is y = 10 (1.2)x. EXAMPLE 4 Classify and write rules for functions Tell whether the graph represents exponential growth or exponential decay.Then write a rule for the function. The graph represents exponential decay (y = abx where 0 < b < 1).The y-intercept is 8, so a = 8. Find the value of b by using the point (1, 4) and a = 8. y = abx Write function. 4 = 8 . b1 Substitute. 0.5 = b Solve. A function rule is y = 8 (0.5)x. GUIDED PRACTICE for Example 4 4. The graph of an exponential function passes through the points (0, 10) and (1, 8). Graph the function. Tell whether the graph represents exponential growth or exponential decay. Write a rule for the function. GUIDED PRACTICE for Example 4 SOLUTION The graph represents exponential decay (y = abx where 0 < b < 1).The y-intercept is 10, so a = 10. Find the value of b by using the point (1, 8) and a = 10. y = abx Write function. 8 = 10 . b1 Substitute. 0.8 = b Solve. A function rule is y = 10 (0.8)x. EXAMPLE 5 Solve a multi-step problem Forestry The number of acres of Ponderosa pine forests decreased in the western United States from 1963 to 2002 by 0.5% annually. In 1963 there were about 41 million acres of Ponderosa pine forests. a. Write a function that models the number of acres of Ponderosa pine forests in the western United States over time. EXAMPLE 5 b. Solve a multi-step problem To the nearest tenth, about how many million acres of Ponderosa pine forests were there in 2002? SOLUTION a. Let P be the number of acres (in millions), and let t be the time (in years) since 1963. The initial value is 41, and the decay rate is 0.005. P = a(1 – r)t Write exponential decay model. = 41(1 –0.005)t Substitute 41 for a and 0.005 for r. = 41(0.995)t Simplify. EXAMPLE 5 b. Solve a multi-step problem To the nearest tenth, about how many million acres of Ponderosa pine forests were therein 2002? Substitute 39 for t. Use a calculator. P = 41(0.995)39 33.7 ANSWER There were about 33.7 million acres of Ponderosa pine forests in 2002. GUIDED PRACTICE 5. for Example 5 WHAT IF? In Example 5, suppose the decay rate of the forests remains the same beyond 2002. About how many acres will be left in 2010? SOLUTION a. Let P be the number of acres (in millions), and let t be the time (in years) since 1963. The initial value is 41, and the decay rate is 0.005. P = a(1 – r)t Write exponential decay model. = 41(1 –0.005)t Substitute 41 for a and 0.005 for r. = 41(0.995)t Simplify. GUIDED PRACTICE for Example 5 To find the number of acres will be left in 2010, 47 years after 1968,substitute 47 for t P = 41(0.995)47 Substitute 47 for t. = 32.4 ANSWER There will be about 32.4 million acres of Ponderosa pine forest in 2010. Review – Ch. 8 – PUT HW QUIZZES HERE Daily Homework Quiz For use after Lesson 8.1 Simplify the expression. Write your answer using exponents. 1. 145 142 ANSWER 2. 147 [(–8)4]3 ANSWER (–8)12 Simplify the expression. 3. [(m –3)6]4 ANSWER (m – 3)24 Daily Homework Quiz 4. –(2s)3 ANSWER 5. For use after Lesson 8.1 -8s3 A website had about 102 hits after a week. After a year,it had about 103 times the number of hits of the first week. About how many hits did it have at the end of the year ? ANSWER About 10,000 hits Daily Homework Quiz 3 65 6 1. Simplify . 64 ANSWER 64 2. Simplify 107 ANSWER 103 –1 4. 10 8 3. s 3. Simplify 3r ANSWER s24 27r3 For use after Lesson 8.2 Daily Homework Quiz For use after Lesson 8.2 4. The order of magnitude of the power output of a nuclear-powered aircraft carrier is about 106 watts. The order of magnitude of peak power at Hoover Dam is about 109 watts. How many times as great is the power output of Hoover Dam as the power output of a nuclear-powered aircraft carrier? ANSWER 103 Daily Homework Quiz 1. Evaluate 5 2 ANSWER 2. 3. 8 125 Evaluate 4–7 ANSWER –3 43 1 256 Simplify 6a –4 b 0. ANSWER 6 a4 For use after Lesson 8.3 Daily Homework Quiz For use after Lesson 8.3 8x3y –4 4. Simplify . 12x2y –3 ANSWER 2x 3y 5. A human cell uses on average about 10–12 watts of power. The laser in a CD-R drive uses 109 times as many watts. About how many watts of power does the laser in a CD-R drive use? ANSWER About 10–3 Daily Homework Quiz For use after Lesson 8.4 Write the number in scientific notation. 1. 100,500 ANSWER 2. 1.005 105 0.0203 ANSWER 2.03 3. Write 3.06 ANSWER 10–2 107 in standard form. 30,600,000 Daily Homework Quiz For use after Lesson 8.4 4. The diameter of Mercury is about 4.9 103 kilometers. The diameter of Venus is about 1.2 104 kilometers. Find the ratio of the diameter of Venus to that of Mercury. Round to the nearest hundredth. ANSWER about 2.5 Daily Homework Quiz For use after Lesson 8.4 Write the number in scientific notation. 1. 100,500 ANSWER 2. 1.005 105 0.0203 ANSWER 2.03 3. Write 3.06 ANSWER 10–2 107 in standard form. 30,600,000 Daily Homework Quiz For use after Lesson 8.4 4. The diameter of Mercury is about 4.9 103 kilometers. The diameter of Venus is about 1.2 104 kilometers. Find the ratio of the diameter of Venus to that of Mercury. Round to the nearest hundredth. ANSWER about 2.5 Daily Homework Quiz 1. For use after Lesson 8.5 Graph y = 1.4x. ANSWER 2. Your family bought a house for $150,000 in 2000. The value of the house increases at an annual rate of 8%. What is the value of the house after 5 years? ANSWER About $220,399 Daily Homework Quiz For use after Lesson 8.6 x 1 1. Graph y = ( ) 3 ANSWER 2. The population in a town has been declining at a rate of 2% per year since 2001. The population was 84,223 in 2001.What was the population in 2006? ANSWER about 76,130 Warm-Up – X.X Vocabulary – X.X • Holder • Holder 2 • Holder 3 • Holder 4 Notes – X.X – LESSON TITLE. • Holder •Holder •Holder •Holder •Holder Examples X.X