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Lesson 3 RCSD Geometry Local MATHEMATICS CURRICULUM Name:___________________________________ U4 Period:________ Date:__________GEOMETRY Lesson 3: Linear Versus Exponential Functions Learning Targets: ο· I can distinguish between situations that can be modeled with linear functions and with exponential functions ο· I can state that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors/ ratios over equal intervals. Linear and exponential functions share many characteristics. This is because they are based on two different, but similar, sets of principles. LINEAR VERSUS EXPONENTIAL Linear functions are based on repeatedly adding the same amount The slope (π) β Constant rate of change- Common difference Exponential functions are based on repeatedly multiplying by the same amount The base (π)- Growth or decay factor- Common Ratio Example #1: The two tables below represent a linear function and an exponential function. Part 1 : Which type is each function below? Explain how you arrive at your answer. Type ______________________________________ Type _________________________________ Part 2 : Find equations in standard form for each of the functions from Example #1. Table 1 Equation : __________________________ Part 3 : Sketch the graph of each equation Table 2 Equation : ___________________________ Lesson 3 RCSD Geometry Local MATHEMATICS CURRICULUM U4 Period:________ Date:__________GEOMETRY Name:___________________________________ Example 2: Consider the linear function π¦ = 20π₯ + 5 and the exponential function π¦ = 5(2)2. Make a sketch of their graphs. Which one of these grows faster? Example 3 Example 4: Find the equation of the exponential function, in y = a(b)x form for the function given in the table below. Linear functions grow _________________ while exponential functions grow _________________________ Example 5: Write an equation of the function represented in the table below. x f(x) β1 2 3 Type ____________________ 0 1 2 3 4 2 6 18 54 162 Equation _________________________ Lesson 3 RCSD Geometry Local MATHEMATICS CURRICULUM U4 Period:________ Date:__________GEOMETRY Name:___________________________________ Example 6: Write an equation of the function represented in the table below. x β3 β2 β1 0 1 2 f(x) 5.5 5 4.5 4 3.5 3 Type ____________________ Equation _________________________ REASONING. You can determine the equation of a line or the equation of an exponential given any two points that lie on these curves. In this exercise we will pick two special points. Consider the points (0, 5) and (1,15) a. Write the equation of the line that passes between these two points in y = mx + b form. b. Write the equation of the exponential that passes between these two points in c. Using your calculator, sketch the two graphs on the axes below. Label with their equations d. Is it fair to say that an exponential function always grows faster than a linear? e. What can we say about an increasing exponential function when compared with an increasing linear function? f. What is the difference between the way a linear function increases and the way an exponential function increases? Lesson Summary ο· ο· ο· Linear function of the form πΏ(π₯) = ππ₯ + π grows additively by the same number over equal length intervals. Exponential function of the form π(π₯) = ππ π₯ grows multiplicatively by the same factor π over equal length intervals An increasing exponential function will eventually exceed any linear function. Sometimes this is not apparent in a graph displayed on a graphing calculator because the graphing window does not show enough of the graphs for us to see the sharp rise of the exponential function in contrast with the linear function. Lesson 3 RCSD Geometry Local MATHEMATICS CURRICULUM U4 Period:________ Date:__________GEOMETRY Name:___________________________________ Lesson 3: Linear Versus Exponential Functions Problem Set 1. Look at the tables below. Which tables show a linear function and which show an exponential function? Table 1 Table 2 Table 3 x y x y x y 0 2 0 20 0 81 1 8 1 17 1 27 2 32 2 14 2 9 3 128 3 11 3 3 4 512 4 8 4 1 Type __________________ Type __________________ Type __________________ Equation:_______________ Equation:_______________ Equation:_______________ 2. Determine the function type: linear or exponential that can be used to model the data set and then write an equation. Table 1 Table 2 Table 3 x y x y x y 0 1 0 300 0 -6 1 5 1 150 1 2 2 25 2 75 2 10 3 125 3 37.5 3 18 4 625 4 18.75 4 26 Type __________________ Type __________________ Type __________________ Equation:_______________ Equation:_______________ Equation:_______________ Lesson 3 RCSD Geometry Local MATHEMATICS CURRICULUM U4 Period:________ Date:__________GEOMETRY Name:___________________________________ 3. Consider the points π΄(0,9) and π΅(1,3). a. Write the equation of the line that passes between these two points in y = mx + b form. b. Write the equation of the exponential that passes between these two points in c. Using your calculator, sketch the two graphs on the axes below. Label with their equations 4. Write an equation of the function represented in the table below. x β3 β2 β1 0 1 f(x) 48 24 12 6 3 Type ____________________ 2 3 2 Equation _________________________ 5. A lab researcher records the growth of the population of a yeast colony and finds that the population doubles every hour. a. Complete the researcherβs table of data: Hours into study Yeast colony population (thousands) 0 1 2 3 4 5 b. What is the exponential function that models the growth of the colonyβs population? Lesson 3 RCSD Geometry Local MATHEMATICS CURRICULUM Name:___________________________________ U4 Period:________ Date:__________GEOMETRY 6. Write an equation of each function represented in the table below. x β3 β2 β1 0 1 2 π(π) β7 β3 1 5 9 13 π(π₯) = ____________________ x β1 0 1 2 3 4 π(π) 1.25 5 20 80 320 1280 β(π₯) = ____________________ How does the y-intercept of the function π(π₯) compare to the y-intercept of β(π₯) ? How does the rate of change of the function π(π₯) compare to the rate of change of β(π₯) ? 7. Write an exponential equation for the graph shown below. Regents 01322015 Equation:__________________________________ Explain how you determined the equation.