Algebra 2 Unit 4 Exponential Functions PDF

Unit 4 ALGEBRA 2 Exponential Functions Expressions and and Equations Equations Lesson 13 Exponential Functions with Base e e on a Calculator Warm-up The other day, you learned that e is a mathematical constant whose value is approximately...

Unit 4 ALGEBRA 2 Exponential Functions Expressions and and Equations Equations Lesson 13 Exponential Functions with Base e e on a Calculator Warm-up The other day, you learned that e is a mathematical constant whose value is approximately 2.718. When working on problems that involve e, we often rely on calculators to estimate values. 1. Find the e button on your calculator. Experiment with it to understand how it works. (For example, see how the value of 2e or e2 can be calculated.) 2. Evaluate each expression. Make sure your calculator gives the indicated value. If it doesn’t, check in with your partner to compare how you entered the expression. a. 10 ⦁ e(1.1) should give approximately 30.04166 b. 5 ⦁ e(1.1)(7) should give approximately 11,041.73996 c. e9/23 + 7 should give approximately 8.47891 Unit 4 Lesson 13 Activity 1 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Same Situation, Different Equations The population of a colony of insects is 9 thousand when it was first being studied. Here are two functions that could be used to model the growth of the colony months after the study began. P(t) = 9 ⦁ (1.02)t Q(t) = 9 ⦁ e(0.02t) 1. Use technology to find the population of the colony at different times after the beginning of the study and complete the table. 2. What do you notice about the populations in the two models? Unit 4 Lesson 13 Activity 2 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Same Situation, Different Equations 3. Here are pairs of equations representing the populations, in thousands, of four other insect colonies in a research lab. The initial population of each colony is 10 thousand and they are growing exponentially. is time, in months, since the study began. Colony 1 Colony 2 f(t) = 10 ⦁ (1.05)t k(t) = 10 ⦁ (1.03)t g(t) = 10 ⦁ e(0.05t) l(t) = 10 ⦁ e(0.03t) Colony 3 Colony 4 p(t) = 10 ⦁ (1.01)t v(t) = 10 ⦁ (1.005)t q(t) = 10 ⦁ e(0.01t) w(t) = 10 ⦁ e(0.005t) a. Graph each pair of functions on the same coordinate plane. Adjust the graphing window to the following boundaries to start: 0 < x < 50 and 0 < y < 80. b. What do you notice about the graph of the equation written using and the counterpart written without ? Make a couple of observations. Unit 4 Lesson 13 Activity 2 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. e in Exponential Models Exponential models that use e often use the format shown in this example: Here are some situations in which a percent change is considered to be happening continuously. For each function, identify the missing information and the missing growth rate (expressed as a percentage). 1. At time t = 0, measured in hours, a scientist puts 50 bacteria into a gel on a dish. The bacteria are growing and the population is expected to show exponential growth. function: b(t) = 50 ⦁ e(0.25t) continuous growth rate per hour: Unit 4 Lesson 13 Activity 3 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. e in Exponential Models 2. In 1964, the population of the United States was growing at a rate of 1.4% annually. That year, the population was approximately 192 million. The model predicts the population, in millions, t years after 1964. function: p(t) = ________ ⦁ e_____t continuous growth rate per year: 1.4% 3. In 1955, the world population was about 2.5 billion and growing. The model predicts the population, in billions, t years after 1955. function: q(t) = ________ ⦁ e(0.0168t) continuous growth rate per year: Unit 4 Lesson 13 Activity 3 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Graphing Exponential Functions with Base e 1. Use graphing technology to graph the function defined by f(t) = 50 ⦁ e(0.25t). Adjust the graphing window as needed to answer these questions: a. The function f models the population of bacteria in t hours after it was initially measured. About how many bacteria were in the dish 10 hours after the scientist put the initial 50 bacteria in the dish? b. About how many hours did it take for the number of bacteria in the dish to double? Explain or show your reasoning. 2. Use graphing technology to graph the function defined by p(t) = 192 ⦁ e(0.014t). Adjust the graphing window as needed to answer these questions: a. The equation models the population, in millions, in the U.S. t years after 1964. What does the model predict for the population of the U.S. in 1974? b. In which year does the model predict the population will reach 300 million? Unit 4 Lesson 12 Activity 4 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Exponential Functions with Base e Lesson Synthesis v(t) = 10 ⦁ (1.005)t w(t) = 10 ⦁ e(0.005t) The first form can be used to represent both discrete and exponential functions. The second form (using base e) is used only to represent situations where change happens “continuously” or at every moment. Unit 4 Lesson 13 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Two Population Predictions LessonCool-down Synthesis The functions a(t) = 8.75 ⦁ (1 + 0.01)t and b(t) = 8.75 ⦁ e(0.01t) each model the population of a city, in thousands, t years after 2010. Describe how each model predicts that the population in the town will grow. Unit 4 Lesson 13 Activity 5 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.  Unit 4 Lesson 14 Learning Let’s solve equations using Goal logarithms. Algebra 2 A Valid Solution? Warm-up To solve the equation 5 ⦁ e3a = 90, Lin wrote the following: 5 ⦁ e3a = 90 e3a = 18 3a = loge 18 Is her solution valid? Be prepared to explain what she did in each step to support your answer. Unit 4 Lesson 14 Activity 1 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Natural Logarithm 1. Complete the table with equivalent equations. The first row is completed for you. 2. Solve each equation by expressing the solution using notation. Then, find the approximate value of the solution using the “ln” button on a calculator. a. em = 20 b. en = 30 c. ep = 7.5 Unit 4 Lesson 14 Activity 2 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Solving Exponential Equations Without using a calculator, solve each equation. It is expected that some solutions will be expressed using log notation. Be prepared to explain your reasoning. 1. 10x = 10,000 2. 5 ⦁ 10x = 500 3. 10(x+3) = 10,000 4. 102x = 10,000 5. 10x = 315 6. 2 ⦁ 10x = 800 7. 10(1.2x) = 4,000 8. 7 ⦁ 10(0.5x) = 70 9. 2 ⦁ ex = 16 10. 10 ⦁ e3x = 250 Unit 4 Lesson 14 Activity 3 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Solve Some Equations LessonCool-down Synthesis 1. Solve each equation. Show or explain your reasoning. a. 10(x-2) = 1,000 b. 4 ⦁ 10x = 88 2. Solve the equation et = 120. Then find the approximate value of the solution using a calculator. Unit 4 Lesson 14 Activity 5 Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics.