Name Learning Objective We will solve problems using exponential equations in one variable. What are we going to do? CFU Activate Prior Knowledge An exponential equation is an equation which contains a constant factor raised to a variable exponent. Exponential Equation 2x = 32 constant factor variable exponent Solve the exponential equations. 1. 3x = 27 2. 2x = 16 Students, you already know how to write numbers as exponential expressions. Now, we will write numbers as exponential expressions to solve exponential equations in one variable. Make Connection

Concept Development Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity A growth describes a situation in which a quantity increases. In an equation, an exponential growth has a constant factor greater than 1. A decay describes a situation in which a quantity decreases. In an equation, an exponential decay has a constant factor less than 1. Exponential Growth Exponential Decay A population of bacteria doubles every day. If the population of bacteria starts at 2,000, how many days would it take for it to reach 32,000? The half-life1 of a radioactive substance is one hour. If the initial amount of the substance is 12,000 atoms, how many hours would it take for the substance to decay to 1,500 atoms? Days 1 2 3 4 5 6 8,000 16,000 24,000 32,000 40,000 48,000 Bacteria Population 2,000 Hours 1 2 3 4 5 6 2,000 4,000 6,000 8,000 10,000 12,000 Atoms 1,500 2,000(2)x = 32,000 2,000 (2)x = 16 (2)x = (2)4 x = 4 12,000( )x = 1,500 12,000 ( )x = ( )x = ( )3 x = 3 1 2 8 It would take 4 days for the population of bacteria to reach 32,000. It would take 3 hours for the substance to decay to 1,500 atoms. Which equation below represents an exponential growth? How do you know? A 4,000(3)x = 108,000 B 22,400( )x = 1,400 What is the difference between an exponential growth and an exponential decay? CFU 1 4 1 (half-life) time it takes for an amount to be reduced to half Vocabulary

Skill Development/Guided Practice Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity A growth describes a situation in which a quantity increases. In an equation, an exponential growth has a constant factor greater than 1. A decay describes a situation in which a quantity decreases. In an equation, an exponential decay has a constant factor less than 1. Solve problems using exponential equations in one variable. 1 Write an exponential equation to represent the problem, if necessary. Divide each side of the equation by the initial amount. Solve for the variable. Interpret2 the solution. 2 3 4 1. Mia is creating a new social networking website with three of her classmates. She knows that if each of the members recruits 2 more members each month, the number of members will triple every month. How long will it take for her to reach 972 members? 2. Mia’s social networking website has reached 3,000 members. She estimates the membership will now begin to double every month. How long will it take for the website to increase to 24,000 members? Months 1 2 3 4 5 6 200 400 600 800 1,000 1,200 Members 4(3)x = 972 Months 1 2 3 4 5 6 4,000 8,000 12,000 16,000 20,000 24,000 Members 3,000 3,000(2)x = 24,000 How did I/you write an exponential equation? How did I/you solve for the variable? How did I/you interpret the solution? CFU 1 3 4 2 explain (synonym) Vocabulary

Skill Development/Guided Practice (continued) Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity A growth describes a situation in which a quantity increases. In an equation, an exponential growth has a constant factor greater than 1. A decay describes a situation in which a quantity decreases. In an equation, an exponential decay has a constant factor less than 1. Solve problems using exponential equations in one variable. 1 Write an exponential equation to represent the problem, if necessary. Divide each side of the equation by the initial amount. Solve for the variable. Interpret the solution. 2 3 4 3. In a given environment, scientists estimate that removing snakes in the area would allow the population of mice to double every year. If there are currently 2,500 mice, how long would it take for the population of mice to reach 80,000 if snakes were removed? 4. In a given region, it is hypothesized that extinction of a population of owls would result in annual tripling of the opossum population. If there are currently 3,000 opossums, how long would it take for the opossum population to reach 81,000 if the owls became extinct? Years 1 2 3 4 5 6 20,000 40,000 60,000 80,000 100,000 120,000 Rat Population 2,500 Years 1 2 3 4 5 6 20,000 40,000 60,000 80,000 100,000 120,000 Opossum Population 3,000 How did I/you write an exponential equation? How did I/you solve for the variable? How did I/you interpret the solution? CFU 1 3 4 3 yearly Vocabulary

Grasshopper Population Skill Development/Guided Practice (continued) Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity A growth describes a situation in which a quantity increases. In an equation, an exponential growth has a constant factor greater than 1. A decay describes a situation in which a quantity decreases. In an equation, an exponential decay has a constant factor less than 1. Solve problems using exponential equations in one variable. 1 Write an exponential equation to represent the problem, if necessary. Divide each side of the equation by the initial amount. Solve for the variable. Interpret the solution. 2 3 4 5. After the rapid growth in the rat population, it is predicted that every year the grasshopper population will be reduced by one-fourth as a direct result of the increase of rats. How long will it take for the grasshopper population to drop to 1,875 in the region, given that it is now 120,000? 6. A rapid growth in the opossum population would cause competition over insects with the frog population. As a result, it is expected that the frog population would be reduced by one-fifth every year. How long would it take for the frog population in the region to drop to 200, given that it is now 25,000? Years 1 2 3 4 5 6 20,000 40,000 60,000 80,000 100,000 120,000 Grasshopper Population Years 1 2 3 4 5 6 5,000 10,000 15,000 20,000 25,000 30,000 Frog Population How did I/you write an exponential equation? How did I/you solve for the variable? How did I/you interpret the solution? CFU 1 3 4

2,000(2)x = 32,000 Exponential Equation Skill Development/Guided Practice (continued) Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity 7. This week, Thomas bought a new car for $16,000. Every year, the car depreciates4 by 20%. How long will it take for the car to be worth $8,192? 8. For graduation, Karen receives $2,500 as a gift from her grandfather. She invests all the money, earning annual interest at a rate of 10%. How long will it take for Karen to have $3,025 invested? Years 1 2 3 4 5 6 4,000 8,000 12,000 16,000 20,000 24,000 Car Value Years 1 2 3 4 5 6 2,500 2,600 2,700 2,800 2,900 3,000 Money Invested How did I/you write an exponential equation? How did I/you solve for the variable? How did I/you interpret the solution? CFU 1 3 4 4 decreases in value Vocabulary

Relevance Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity A growth describes a situation in which a quantity increases. In an equation, an exponential growth has a constant factor greater than 1. A decay describes a situation in which a quantity decreases. In an equation, an exponential decay has a constant factor less than 1. 1 Solving problems using exponential equations will help you solve real-world problems. This week, Ronald bought a new car for $20,000. Every year, the car depreciates by 30%. How long will it take for the car to be worth $4,802? Years 1 2 3 4 5 6 $4,000 $8,000 $12,000 $16,000 $20,000 $24,000 Car Value 20,000(0.7)x = 4,802 2 Solving problems using exponential equations will help you do well on tests. Sample Test Question: 22 The functions f(x)=500(1.015)x and g(x)=500(1.021)x give the total amounts in two different savings accounts after x years. How do the graphs of f(x) and g(x) compare? They have the same y-intercept, but the graph of f(x) rises more quickly over time. They have the same y-intercept by the graph of g(x) rises more quickly over time. The function f(x) has a greater y-intercept and rises more quickly over time. The function g(x) has a greater y-intercept and rises more quickly over time. A B C D Does anyone else have another reason why it is relevant to solve problems using exponential equations in one variable? (Pair-Share) Why is it relevant to solve problems using exponential equations in one variable? You may give one of my reasons or one of your own. Which reason is more relevant to you? Why? CFU

Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity A growth describes a situation in which a quantity increases. In an equation, an exponential growth has a constant factor greater than 1. A decay describes a situation in which a quantity decreases. In an equation, an exponential decay has a constant factor less than 1. Skill Closure Solve problems using exponential equations in one variable. 1 Write an exponential equation to represent the problem, if necessary. Divide each side of the equation by the initial amount. Solve for the variable. Interpret the solution. 2 3 4 1. A population of bacteria quadruples every day. If the population of bacteria starts at 5,000, how many days will it take for it to reach 1,280,000? Days 1 2 3 4 5 6 300,000 600,000 900,000 1,200,000 1,500,000 1,800,000 Bacteria Population 5,000 5,000(4)x = 1,280,000 Word Bank exponential equation growth decay Access Common Core Explain the difference between an exponential growth and an exponential decay. ________________________________________________________ Summary Closure What did you learn today about solving problems using exponential equations in one variable? (Pair-Share) Use words from the word bank. Day 1 ____________________________________________________________ _________________________________________________________________ Day 2 ____________________________________________________________

Name Independent Practice Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity A growth describes a situation in which a quantity increases. In an equation, an exponential growth has a constant factor greater than 1. A decay describes a situation in which a quantity decreases. In an equation, an exponential decay has a constant factor less than 1. Solve problems using exponential equations in one variable. 1 Write an exponential equation to represent the problem, if necessary. Divide each side of the equation by the initial amount. Solve for the variable. Interpret the solution. 2 3 4 1. Nancy is creating a new club at school with four of her friends. She knows that if each of the members recruits 1 more member each week, the number of members will double every week. How long will it take for her club to reach 40 members? 2. Nancy’s new club has reached 50 members. She estimates the membership will now begin to double every semester. How long will it take for the club to reach 200 members? Weeks 1 2 3 4 5 6 10 20 30 40 50 60 Members 5(2)x = 40 1 2 3 4 5 6 100 200 300 400 500 600 Members 50 Semesters

2,000(2)x = 32,000 Exponential Equation Independent Practice (continued) Exponential Equation 2,000(2)x = 32,000 initial quantity constant factor # of growths/decays final quantity 3. The half-life of a radioactive substance is one hour. If the initial amount of the substance is 36,000 atoms, how many hours would it take for the substance to decay to 1,125 atoms? 4. Holly earned $3,000 working a summer job. She invested $1,000 of her earnings at a 10% annual interest rate. If Holly does not withdraw any of the money, how long will it take for her to have $1,331 invested? 1 2 3 4 5 6 6,000 12,000 18,000 24,000 30,000 36,000 Atoms Hours Years 1 2 3 4 5 6 500 1,000 1,500 2,000 2,500 3,000 Money Invested

Name Periodic Review 1 1. A population of bacteria triples every day. If the population of bacteria starts at 15,000, how many days will it take for it to reach 405,000? 2. This week, Ronald bought a new car for $20,000. Every year, the car depreciates by 30%. How long will it take for the car to be worth $4,802? Days 1 2 3 4 5 6 100,000 200,000 300,000 400,000 500,000 600,000 Bacteria Population 15,000 Years 1 2 3 4 5 6 $4,000 $8,000 $12,000 $16,000 $20,000 $24,000 Car Value Access Common Core For problems 1 and 2 above: Explain how you set up the equation to solve the problem. Explain each step for solving the equation. 1. ________________________________________________________________ ________________________________________________________________ 2. ________________________________________________________________ ________________________________________________________________

Name Periodic Review 2 1. In a given environment, scientists estimate that removing owls in the area would allow the population of rodents to double every year. If there are currently 3,700 rodents, how long would it take for the population of rodents to reach 118,400 if owls were removed? 2. After the rapid growth in the rodent population, it is predicted that every year the insect population will be reduced by one-fifth as a direct result of the increase of rodents. How long would it take for the insect population to drop to 1,600 in the region, given that it is now 200,000? Years 1 2 3 4 5 6 20,000 40,000 60,000 80,000 100,000 120,000 Rodent Population 3,700 Years 1 2 3 4 5 6 40,000 80,000 120,000 160,000 200,000 240,000 Insect Population Access Common Core 1. Write a word problem for a situation that could be modeled with an exponential growth. ________________________________________________________________ 2. Write a word problem for a situation that could be modeled with an exponential decay. ________________________________________________________________

A The problem is an exponential growth. Name Periodic Review 3 1. The half-life of a radioactive substance is two hours. If the initial amount of the substance is 50,000 atoms, how many hours would it take for the substance to decay to 12,500 atoms? Hours 1 2 3 4 5 6 10,000 20,000 30,000 40,000 50,000 60,000 Atoms Access Common Core 1. Choose Yes or No to indicate whether each statement is true about the problem below. O Yes O No A The problem is an exponential growth. B The initial amount of bacteria is 10,000. C The equation that represents the problem is 10,000(4)x = 640,000. D It will take 4 days for the bacteria to reach 640,000. A population of bacteria quadruples every day. If the population of bacteria starts at 10,000, how many days will it take for it to reach 640,000? 2. Choose Yes or No to indicate whether each statement is true about the problem below. O Yes O No A The problem is an exponential decay. B The final amount of atoms is 100,000. C The equation that represents the problem is 100,000(2)x = 6,250. D It will take 4 hours for substance to drop to 6,250 atoms. The half-life of a radioactive substance is one hour. If the initial amount of the substance is 100,000 atoms, how many hours would it take for the substance to drop to 6,250 atoms?

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