October 07, 2014 Agenda • Warm Up • Go over homework • Key Concepts/Notes • Example #1 & #2 • Extra Practice • Homework Warm Up 1.3.2: Creating and Graphing Exponential Equations pg. 79 Read the scenario and answer the questions that follow. One form of the element beryllium, beryllium-11, has a half-life of about 14 seconds and decays to the element boron. A chemist starts out with 128 grams of beryllium-11. She monitors the element for 70 seconds. 1. What is the equation that models the amount of beryllium-11 over time? 2. How many grams of beryllium-11 does the chemist have left after 70 seconds? October 07, 2014 1.3.2: Creating and Graphing Exponential Equations 1. What is the equation that models the amount of beryllium-11 over time? > y = abx, where y is the final value, a is the initial value, b is the rate of growth or decay, and x is the time. y = unknown a = 128 grams b = 0.5 > Time = 70 seconds, but this needs to be converted to time periods before substituting the value for x. • Convert 70 seconds into 14-second time periods. 1 time period = 14 seconds x=5 > Substitute all the variables into the equation. y = abx y = 128(0.5)5 2. How many grams of beryllium-11 does the chemist have left after 70 seconds? Apply the order of operations to the equation from the end of problem 1. y = 128(0.5)5 y = 4 grams 1.3.2: Creating and Graphing Exponential Equations October 07, 2014 Agenda • Warm Up • Go over homework • Key Concepts/Notes • Example #1 & #2 • Extra Practice • Homework pg. 75 #7-10 October 07, 2014 October 07, 2014 1.3.2 Creating & Graphing Exponential Equations I • • • pg. 78 can identify the following: Exponential Equation Exponential Growth & Decay Independent & Dependent Variables I can create and graph an exponential equation of two variables I can interpret and describe parts of a graph pg. 65 I will be able to... October 07, 2014 y = a(b)x or Key Concepts pg. 80 • The general form of an exponential equation is y = a • bx, where a is the initial value, b is the rate of decay or growth, and x is the time. The final output value will be y. • Since the equation has an exponent, the value increases or decreases rapidly. • The base, b, must always be greater than 0 (b > 0). October 07, 2014 Key Concepts, continued pg. 80 • Look for words such as double, triple, half, quarter—such words give the number of the base. For example, if an experiment begins with 1 bacterium that doubles (splits itself in two) every hour, determining how many bacteria will be present after x hours is solved with the following equation: y = (1)2x, where 1 is the starting value, 2 is the rate, x is the number of hours, and y is the final value. pg. 80 y = a(1 r)t October 07, 2014 Guided Practice pg. 81 Example 1 If a pendulum swings to 90% of its height on each swing and starts out at a height of 60 cm, what is the equation that models this scenario? What is its graph? 1. Read the problem statement and then reread the scenario, identifying the know quantities. 2. Substitute the known quantities into the general form of the exponential equation y = abx, where a is the initial value, b is the rate of decay, x is time (in this case swinging), and y is the final value. Guided Practice pg. 81 Example 1, continued If a pendulum swings to 90% of its height on each swing and starts out at a height of 60 cm, what is the equation that models this scenario? What is its graph? 3. Create a table of values. 4. Set up the coordinate plane. (in student textbook) 5. Plot the points on the coordinate plane and connect the points with a line (curve). October 07, 2014 pg. 81 October 07, 2014 October 07, 2014 Guided Practice pg. 82 Example 2 The bacteria Streptococcus lactis doubles every 26 minutes in milk. If a container of milk contains 4 bacteria, write an equation that models this scenario and then graph the equation. Guided Practice: Example 2, continued • Read the problem statement and then reread the scenario, identifying the known quantities. Initial bacteria count = 4 Base = 2 Time period = 26 minutes Guided Practice: Example 2, continued • Substitute the known quantities into the general form of the exponential equation y = abx, where a is the initial value, b is the base, x is time (in this case, 1 time period is 26 minutes), and y is the final value. Since the base is repeating in units other than 1, use the equation , where t = 26. Guided Practice: Example 2, continued a =4 b=2 October 07, 2014 Guided Practice: Example 2, continued • Create a table of values. x y 0 4 26 8 52 16 78 32 104 64 Guided Practice: Example 2, continued • Set up the coordinate plane. Determine the labels by reading the problem again. The independent variable is the number of time periods. The time periods are in number of minutes. Therefore, “Minutes” will be the x-axis label. The y-axis label will be the “Number of bacteria.” The number of bacteria is the dependent variable because it depends on the number of minutes that have passed. Guided Practice: Example 2, continued minutes and the table of values. The table of values showed 4 time periods. One time period = 26 minutes and so 4 time periods = 4(26) = 104 minutes. This means the x-axis scale needs to go from 0 to 104. Use increments of 26 for easy plotting of the points. For the yaxis, start with 0 and go to 65 in increments of 5. This will make plotting numbers like 32 a little easier than if you chose increments of 10. See the graph that follows. The x-axis needs a scale that reflects the time period of 26 October 07, 2014 Guided Practice: Example 2, continued Guided Practice: Example 2, continued • Plot the points on the coordinate plane and connect the points with a line (curve). When the points do not lie on a grid line, use estimation to approximate where the point should be plotted. Add an arrow to the right end of the line to show that the curve continues in that direction toward infinity, as shown in the graph that follows. October 07, 2014 Guided Practice: Example 2, continued Key Concepts pg. 87 Introducing the Compound Interest Formula: • The general form of the compounding interest formula is where A is the initial value, r is the interest rate, n is the number of times the investment is compounded in a year, and t is the number of years the investment is left in the account to grow. October 07, 2014 pg. 82 Guided Practice Example 3 An investment of $500 is compounded monthly at a rate of 3%. What is the equation that models this situation? Graph the equation. Exit Slip Graph the following exponential equation. X Y October 07, 2014 1.3.2 Creating & Graphing Exponential Equations I • • • pg. 78 can identify the following: Exponential Equation Exponential Growth & Decay Independent & Dependent Variables I can create and graph an exponential equation of two variables I can interpret and describe parts of a graph pg. 85 #1-5 October 07, 2014 Agenda • Warm Up • Go over homework • Example #3 • Extra Practice (if necessary) • Problem Based Task • Homework Exit Slip Graph the following exponential equation. X Y October 07, 2014 Hands free pencils down pg. 86 Warm-Up Exponential Equations Determine if the following are exponential growth or decay. 3. If Downers Grove's population is growing at a rate of 3.5% each year. Determine how many people there will be in 7 years if there are currently 42,378 people. Hands free pencils down Warm-Up Exponential Equations pg. 86 Determine if the following are exponential growth or decay. 3. If Downers Grove's population is growing at a rate of 3.5% each year. Determine how many people there will be in 7 years if there are currently 42,378 people. October 07, 2014 Agenda • Warm Up • Go over homework • Example #3 • Extra Practice (if necessary) • Problem Based Task • Homework pg. 85 #1-5 October 07, 2014 pg. 85 #1-5 pg. 85 #1-5 October 07, 2014 1.3.2 Creating & Graphing Exponential Equations I • • • pg. 78 can identify the following: Exponential Equation Exponential Growth & Decay Independent & Dependent Variables I can create and graph an exponential equation of two variables I can interpret and describe parts of a graph Problem Based Task 1.3.2 pg. 83 October 07, 2014 Problem Based Tasks - Sample Response October 07, 2014 Problem Based Tasks - Sample Response Problem Based Tasks - Sample Response October 07, 2014 Exit Slip 1.3.2 Creating & Graphing Exponential Equations I • • • pg. 78 can identify the following: Exponential Equation Exponential Growth & Decay Independent & Dependent Variables I can create and graph an exponential equation of two variables I can interpret and describe parts of a graph October 07, 2014 pg. 85 #6-10 pg. 85 #6-10 October 07, 2014

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