9-5 Exponential Functions

Slide 1 : Five-Minute Check (over Lesson 9-4) Main Ideas and Vocabulary California Standards Key Concept: Exponential Function Example 1: Graph an Exponential Function with a > 1 Example 2: Graph an Exponential Function with 0 < a < 1 Example 3: Real-World Example: Use Exponential Functions to Solve Problems Example 4: Identify Exponential Behavior

Slide 2 : exponential function Graph exponential functions. Identify data that displays exponential behavior.

Slide 3 : Preparation for Algebra II Standard 12.0 Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. (Key)

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Slide 5 : Graph an Exponential Function with a > 1 A. Graph y = 3x. State the y-intercept. Graph the ordered pairs and connect the points with a smooth curve. Answer: The y-intercept is 1.

Slide 6 : Graph an Exponential Function with a > 1 B. Use the graph to determine the approximate value of 31.5. The graph represents all real values of x and their corresponding values of y for y = 3x. Answer: The value of y is about 5 when x = 1.5. Use a calculator to confirm this value. 31.5 ≈ 5.196

Slide 7 : A B C D A. Graph y = 5x.

Slide 8 : A B C D B. Use the graph to determine the approximate value of 50.25. A. about 2.5 B. about 5 C. about 2 D. about 1.5

Slide 9 : Graph the ordered pairs and connect the points with a smooth curve. Answer: The y-intercept is 1. Graph Exponential Functions with 0 < a < 1

Slide 10 : Use a calculator to confirm this value. Answer: The value of y is about 8 when x = –1.5. Graph Exponential Functions with 0 < a < 1

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Slide 12 : A B C D A. about 1 B. about 3 C. about 2 D. about 0.1

Slide 13 : DEPRECIATION Some people say that the value of a new car decreases as soon as it is driven off the dealer’s lot. The function V = 25,000 ● 0.82t models the depreciation of the value of a new car that originally cost $25,000. V represents the value of the car and t represents the time in years from the time the car was purchased. Graph the function. What values of V and t are meaningful in the function? Use a graphing calculator to graph the function. Use Exponential Functions to Solve Problems

Slide 14 : Only the values of 0 ≤ V ≤ 25,000 and t ≥ 0 are meaningful in the context of the problem. Answer: Use Exponential Functions to Solve Problems

Slide 15 : B. What is the value of the car after one year? V = 25,000 ● 0.82t Original equation V = 25,000 ● 0.821 t = 1 V = 20,500 Use a calculator. Use Exponential Functions to Solve Problems Answer: After one year, the car's value is about $20,500.

Slide 16 : C. What is the value of the car after five years? V = 25,000 ● 0.82t Original equation V = 25,000 ● 0.825 t = 5 V = 9268.50 Use a calculator. Use Exponential Functions to Solve Problems Answer: After five years, the car’s value is about $9,270.

Slide 17 : A B C D A. Depreciation The function V = 22,000 ● 0.82t models the depreciation of the value of a new car that originally cost $22,000. V represents the value of the car and t represents the time in years from the time the car was purchased. Graph the function.

Slide 18 : A B C D A. $21,000 B. $23,600 C. $18,040 D. $20,000 B. What is the value of the car after one year?

Slide 19 : A B C D A. $12,130 B. $25,120 C. $10,000 D. $15,000 C. What is the value of the car after three years?

Slide 20 : Determine whether the set of data displays exponential behavior. Explain why or why not. Method 1 Look for a Pattern The domain values are at regular intervals of 10. Look for a common factor among the range values. 10 25 62.5 156.25 Identify Exponential Behavior

Slide 21 : Method 2 Graph the Data Answer: The graph shows a rapidly increasing value of y as x increases. This is a characteristic of exponential behavior. Identify Exponential Behavior Answer: Since the domain values are at regular intervals and the range values have a common factor, the data are probably exponential. The equation for the data may involve (2.5)x.

Slide 22 : A B C A. no B. yes C. cannot be determined Determine whether the set of data displays exponential behavior.

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Slide 24 : Five-Minute Check (over Lesson 9-5) Main Ideas and Vocabulary California Standards Key Concept: General Equation for Exponential Growth Example 1: Real-World Example: Exponential Growth Example 2: Real-World Example: Compound Interest Key Concept: General Equation for Exponential Decay Example 3: Real-World Example: Exponential Decay

Slide 25 : compound interest exponential decay exponential growth Solve problems involving exponential growth. Solve problems involving exponential decay.

Slide 26 : Preparation for Algebra II Standard 12.0 Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. (Key)

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Slide 28 : A. POPULATION In 2005 the town of Flat Creek had a population of about 280,000 and a growth rate of 0.85% per year. Write an equation to represent the population of Flat Creek since 2005. The rate 0.85% can be written has 0.0085. y = C(1 + r)t General equation for exponential growth y = 280,000(1 + 0.0085)t C = 280,000 and r = 0.0085 y = 280,000(1.0085)t Simplify. Answer: An equation to represent the population of Flat Creek is y = 280,000(1.0085)t, where y is the population and t is the number of years since 2005. Exponential Growth

Slide 29 : B. POPULATION In 2005 the town of Flat Creek had a population of about 280,000 and a growth rate of 0.85% per year. According to the equation, what will be the population of Flat Creek in the year 2015? In 2015, t will equal 2015 – 2005 or 10. y = 280,000(1.0085)t Equation for population of Flat Creek y = 280,000(1.0085)10 t = 10 y ≈ 304,731 Use a calculator. Answer: In 2015, there will be about 304,731 people in Flat Creek. Exponential Growth

Slide 30 : A B C D A. y = 4500(1.0015) B. y = 4500(1.0015)t C. y = 4500(0.0015)t D. y = (1.0015)t A. POPULATION In 2000, Scioto School District had a student population of about 4500 students, and a growth rate of about 0.15% per year. Write an equation to represent the student population of the Scioto School District since the year 2000.

Slide 31 : A B C D A. about 9000 students B. about 4600 students C. about 4540 students D. about 4700 students B. POPULATION In 2000, Scioto School District had a student population of about 4500 students, and a growth rate of about 0.15% per year. According to the equation, what will be the student population of the Scioto School District in the year 2006?

Slide 32 : COLLEGE When Jing May was born, her grandparents invested $1000 in a fixed rate savings account at a rate of 7% compounded annually. The money will go to Jing May when she turns 18 to help with her college expenses. What amount of money will Jing May receive from the investment? Compound interest equation Compound Interest P = 1000, r = 7% or 0.07, n = 1, and t = 18

Slide 33 : A = 1000(1.07)18 Compound interest equation A = 3379.93 Simplify. Answer: She will receive about $3380. Compound Interest

Slide 34 : A B C D A. about $4682 B. about $5000 C. about $4600 D. about $4500 COMPOUND INTEREST When Lucy was 10 years old, her father invested $2500 in a fixed rate savings account at a rate of 8% compounded semiannually. When Lucy turns 18, the money will help to buy her a car. What amount of money will Lucy receive from the investment?

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Slide 36 : A. CHARITY During an economic recession, a charitable organization found that its donations dropped by 1.1% per year. Before the recession, its donations were $390,000. Write an equation to represent the charity’s donations since the beginning of the recession. y = 390,000(0.989)t Simplify. Exponential Decay y = C(1 + r)t General equation for exponential growth y = 390,000(1 – 0.011)t C = 390,000 and r = 1.1% or 0.011 Answer: y = 390,000(0.989)t

Slide 37 : B. CHARITY During an economic recession, a charitable organization found that its donations dropped by 1.1% per year. Before the recession, its donations were $390,000. Estimate the amount of the donations 5 years after the start of the recession. Answer: The amount of donations should be about $369,017. Exponential Decay y = 390,000(0.989)t General equation for exponential growth y = 390,000(0.989)5 t = 5 y = 369,016.74

Slide 38 : A B C D A. y = (0.975)t B. y = 24,000(0.975)t C. y = 24,000(1.975)t D. y = 24,000(0.975) A. CHARITY A charitable organization found that the value of its clothing donations dropped by 2.5% per year. Before this downturn in donations, the organization received clothing valued at $24,000. Write an equation to represent the value of the charity’s clothing donations since the beginning of the downturn.

Slide 39 : A B C D A. about $23,000 B. about $21,000 C. about $22,245 D. about $24,000 B. CHARITY A charitable organization found that the value of its clothing donations dropped by 2.5% per year. Before this downturn in donations, the organization received clothing valued at $24,000. Estimate the value of the clothing donations 3 years after the start of the downturn.

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Slide 41 : Five-Minute Checks Image Bank Math Tools Animation Menu Exploring Quadratic Functions

Slide 42 : Lesson 9-1 (over Chapter 8) Lesson 9-2 (over Lesson 9-1) Lesson 9-3 (over Lesson 9-2) Lesson 9-4 (over Lesson 9-3) Lesson 9-5 (over Lesson 9-4) Lesson 9-6 (over Lesson 9-5)

Slide 43 : To use the images that are on the following three slides in your own presentation: 1. Exit this presentation. 2. Open a chapter presentation using a full installation of Microsoft® PowerPoint® in editing mode and scroll to the Image Bank slides. 3. Select an image, copy it, and paste it into your presentation.

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Slide 47 : 9-1 Families of Quadratic Functions 9-2 Solving Quadratic Equations By Graphing 9-3 Completing the Square

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Slide 51 : A B C D A. (a + 3)(a – 3) B. (a + 3)(a + 3) C. (a – 3)(a – 3) D. prime Factor the polynomial a2 – 5a + 9, if possible. If the polynomial cannot be factored using integers, write prime. (over Chapter 8)

Slide 52 : A B C D A. (2z – 1)(3z – 1) B. (2z – 1)(3z + 1) C. (2z + 1)(3z – 1) D. prime Factor the polynomial 6z2 – z – 1, if possible. If the polynomial cannot be factored using integers, write prime. (over Chapter 8)

Slide 53 : A B C D Solve the equation 5x2 = 125. (over Chapter 8)

Slide 54 : A B C D Solve the equation 2x2 + 11x – 21 = 0. (over Chapter 8)

Slide 55 : A B C D A certain basketball player’s hang time can be described by 4t2 = 1, where t is time in seconds. How long is this player’s hang time? (over Chapter 8)

Slide 56 : A B C D A. 2y + 3 B. y + 3 C. 2y – 3 D. y – 3 The area of a rectangle is given by 6y2 + 5y – 6 and the width is given by 3y – 2. What is the length? (over Chapter 8)

Slide 57 : A B C D Which of the following options shows a graph and the solution of y = x2 + 2x – 1? (over Lesson 9-1)

Slide 58 : A B C D For y = –x2 + 2, identify the equation of axis of symmetry, and the coordinates of the vertex of the graph of the equation. Identify the vertex as a maximum or a minimum. (over Lesson 9-1)

Slide 59 : A B C D A. x = –2.5; (–2.5, 18.75);maximum B. x = –2.5; (–2.5, 18.75);minimum C. x = 2.5; (2.5, –6.25);maximum D. x = 2.5; (2.5, –6.25);minimum For y = x2 – 5x, identify the equation of the axis of symmetry, and the coordinates of the vertex of the graph of the equation. Identify the vertex as a maximum or a minimum. (over Lesson 9-1)

Slide 60 : A B C D A. 66 ft B. 65 ft C. 64 ft D. 63 ft What is the maximum height of a rocket fired straight up if the height in feet is described by h = –16t2 + 64t + 1, where t is time in seconds? (over Lesson 9-1)

Slide 61 : A B C D A. y = x2 – 4x + 2 B. y = 2x2 + 4x + 2 C. y = x2 – 4x + 3 D. y = x2 – 4x – 2 The graph of which of the following equations has a vertex at (2, –1)? (over Lesson 9-1)

Slide 62 : A B C D Which of the following options shows a graph and the solution of m2 – 2m – 3 = 0? (over Lesson 9-2)

Slide 63 : A B C D Which choice shows the graph of w2 + 5w – 1 = 0, and its roots or the consecutive integers between which the roots lie? (over Lesson 9-2)

Slide 64 : A B C D Use a quadratic equation to find two numbers whose difference is 3 and whose product is 10. (over Lesson 9-2)

Slide 65 : A B C D A. 2 and 4 B. 8 and 1 C. –8 and –1 D. –4 and –2 Which of the following are the roots of x2 + 6x + 8 = 0? (over Lesson 9-2)

Slide 66 : A B C D A. –8, 0 B. 8, 0 C. 2, 16 D. –2, 16 Solve the equation x2 + 8x + 16 = 16. (over Lesson 9-3)

Slide 67 : A B C D A. 1, –7 B. –1, –7 C. –1, 7 D. 1, 7 Solve the equation x2 – 6x – 2 = 5. (over Lesson 9-3)

Slide 68 : A B C D Find the value of c that makes z2 + z + c a perfect square. (over Lesson 9-3)

Slide 69 : A B C D A. 2.24 in. B. 3.7 in. C. 10 in. D. 13.7 in. The area of a square can be tripled by increasing the length and width by 10 inches. What is the original length of the square? Round to the nearest tenth of an inch. (over Lesson 9-3)

Slide 70 : A B C D A. The sum of the roots is greater than the product of the roots. B. The product of the roots is greater than the sum of the roots. C. The product of the roots is positive. D. The sum of the roots is negative. Use the roots of x2 – 14x + 49 = 81 to determine which statement is true. (over Lesson 9-3)

Slide 71 : A B C D Solve the equation x2 – 6x – 7 = 0 by using the Quadratic Formula. Round to the nearest tenth if necessary. (over Lesson 9-4) A. 1, 7 B. –1, 7 C. –7, 1 D. Ø

Slide 72 : A B C D Solve the equation y2 – 11y = –30 by using the Quadratic Formula. Round to the nearest tenth if necessary. (over Lesson 9-4) A. 5, 6 B. –5, 6 C. –5, –6 D. Ø

Slide 73 : A B C D Solve the equation 5z2 + 16z + 3 = 0 by using the Quadratic Formula. Round to the nearest tenth if necessary. (over Lesson 9-4)

Slide 74 : A B C D Solve the equation 4n2 = –19n – 25 by using the Quadratic Formula. Round to the nearest tenth if necessary. (over Lesson 9-4)

Slide 75 : A B C D A. 0 B. 1 C. 2 D. 3 Without graphing, determine the number of x-intercepts of the graph of f(x) = 3x2 + x + 7. (over Lesson 9-4)

Slide 76 : A B C D A. 3x2 – 2x + 4 = 0 B. 4x2 – 3x – 4 = 0 C. 2x2 – 3x + 4 = 0 D. 2x2 + 3x – 4 = 0 (over Lesson 9-4)

Slide 77 : A B C D Which choice shows the graph of y = 4x, and also states the y-intercept and the approximate value of 40.6? Use a calculator to confirm the value. (over Lesson 9-5)

Slide 78 : A B C D A. Yes, the domain values are at a regular intervals and the range values have a common difference. B. Yes, the domain values are at regular intervals and the range values have a common ratio of 3. C. No, the domain values are at regular intervals and the range values have a common ratio of 3. D. No, the domain values are at regular intervals and the range values have a common difference. Which choice states and explains whether or not the data in the given table display exponential behavior? (over Lesson 9-5)

Slide 79 : A B C D (over Lesson 9-5)

Slide 80 : A B C D A. –5 B. –4 C. 3 D. 11 What is the y-intercept of the graph of y = 2x – 5? (over Lesson 9-5)

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