Introduction to Exponential Functions

Transcription

1 MCR3U Unit 4: Exponential Relations Lesson 4 Date: Learning goal: I can graph and identify key properties of exponential functions. I can distinguish between a linear, quadratic, and exponential function when given a graph, table of values, or equation. Introduction to Exponential Functions Exponential functions are curves that increase or decrease through their domains. They have the basic form =, 0. COMPARING GRAPHS Example 1: Create a table of values for each function, and graph them both on the same axis. a) = 2 b) = c) h = h Note: The equation h() = is equivalent to h() = 2. During transformations last unit we saw that when the value of a function is negative it causes a across the -axis. That is exactly what we have done here Property () () () Domain Range -intercept -intercept Horizontal Asymptote Increasing/Decreasing? As

2 COMPARING LINEAR, QUADRATIC, AND EXPOENTIAL FUNCTIONS There are 3 ways we can compare linear, quadratic, and exponential functions: 1. Equations Use the equations of the following relations to determine whether they are linear, quadratic, or exponential. a) = b) = 2 c) = 3(2 ) 2. Tables of Values Use the table of values to confirm whether the relations are linear, quadratic, or exponential. a) = b) = 2 c) = 3(2 ) 1 st x y diff. x y diff. diff. x Graphs 1 st 2 nd y 1 st diff. 2 nd diff. Ratio Use the graphs to recognize whether the relations are linear, quadratic, or exponential (growth or decay). Success Criteria for Determining Between Linear, Quadratic, and Exponential Functions Functions Equations Table of Values Graphs Linear Quadratic Exponential

3 SUMMARY HW: Introduction to Exponentials Worksheet

4 Introduction to Exponentials Worksheet = 2 = 5 2 = 0.8 Key Points:, Key Points:, = 10 = 3 Key Points:, = 1 2 Key Points:, = 2 3 Key Points:, = 3 2 Key Points:, = 1.1 Key Points:, Key Points:, Key Points:,

5 = 4 = 7 = Key Points:, Key Points:, = 1 = 0.3 Key Points:, = 3 4 Key Points:,, (, ) = 1 10 Key Points:, = 1.4 Key Points:, = 1 4 Key Points:, Key Points:, Key Points:,

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7 MCR3U Unit 4: Exponential Relations Lesson 5 Learning goal: I can apply transformations to exponential functions and sketch their graphs. Date: Transformations of Exponential Functions Exponential Functions of the form = can be transformed using the same algorithm as our other functions we saw last unit, where point (. ) on = maps onto the point ( +, " + ) on = () +. = () + Example 1: Each function () is transformed to result in the function (). For each: i) write an equation for () using function notation. ii) write an equation for () given (). a) Transformations apply to = 5 Vertical stretch by a factor of 3 Horizontal stretch by a factor of 4 Horizontal translation 5 unit right Vertical translation 1 unit up b) Transformations apply to = Vertical reflection across the -axis Horizontal stretch by a factor of Vertical translation 7 units down Success Criteria for Graphing Square Root Functions Label your scale Label your equation 5 key points are clearly marked with a dot Label any intercepts Label and sketch any asymptotes

8 Example 2: Sketch the following transformed functions on the grids below (use success criteria). First write the exponential function that is to be transformed. List the transformations in order on the base function = and the new mapping. State the domain and range. a) Base function: = ( ) b) Base function: = (2) = = Description: Description:, (, ), (, ) D = R = HW: Transforming Exponential Functions Worksheet D = R =

9 Transforming Exponential Functions Worksheet 1. Sketch the following transformed functions on a grid (use success criteria). First write the exponential function that is to be transformed. List the transformations in order on the base function = and the new mapping. State the domain and range.**check your answers on Desmos a) = + 5 4, = 2 b) = ( + 3) + 2, = 2 c) = , = ( ) d) = 2, = 3 e) = 2[3 + 6] + 1, = ( ) f) = , = 3 2. Each function () is transformed to result in the function (). For each: iii) write an equation for () using function notation. iv) write an equation for () given (). a) Transformations apply to = 3 Vertical reflection across the x-axis Vertical stretch by a factor of 5 Horizontal translation 6 units left Vertical translation 3 units down b) Transformations apply to = Vertical stretch by a factor of Horizontal stretch by a factor of 3 Horizontal translation 6 units right Vertical translation 2 units down

10 MCR3U Unit 4: Exponential Relations Lesson 6 Learning goal: I can determine more than one equation when given an exponential graph. Date: Similar Exponential Functions Example 1: Identify 3 points and determine the equation of each graph. a) b) c) DIFFERENT BASES Some interesting things happen with transformations of exponential functions. Unlike our other parents, many transformations of base exponential functions are not unique. Different Bases Function Transformations Domain Range = 8 = 2 We know that 2 = 8 so we can also say that = 2 = 2 = 8 = Therefore, we cannot distinguish between a graph of = 2 that has been stretched horizontally by and a graph of = 8.

11 STRETCHS & SHIFTS Function Transformations Domain Range = 3 = 3 3 = 3 We know that 3 3 = so = Therefore we cannot tell the difference between: Example 2: Determine another equation that is the same graph as = 3 REFLECITONS AND RECIPROCALS Function Transformations Domain Range = 3 = 1 3 = 1 = 1 3 = 3 We know that = and = so = = = Therefore we cannot tell the difference between:

12 Example 3: Determine two equations for the graphs below. a) b) c) HW: Similar Exponential Functions Worksheet

13 Similar Exponential Functions Worksheet 1. State the domain & range, and then match each function to its graph. i) = ii) = iii) = 2 1 iv) = 2 1 v) = vi) = 2 1 vii) = viii) = Given =, write an equation for = and h =, then graph f, and h. Write the equation of another exponential function on a different base that is equivalent to or h. Communicate your discoveries articulately and attempt to state as many properties about these functions as you can. 3. Given = 2, write the transformed function and graph each. Write the equation of another exponential function on a different base that is equivalent to h, and. Communicate your discoveries articulately and attempt to state as many properties about these functions as you can. a) = 2 b) h = + 1 c) = d) = 2 4. Match the equation of the functions from the list to the appropriate graph. 1 4 x a) f ( x) = b) y = c) g ( x) = + 3 x x d) h ( x) = x

14 Similar Exponential Functions Worksheet Solutions 1. State the domain & range, and then match each function to its graph. i) = Domain: = ℜ Range: = > 1, ℜ ii) = Domain: = ℜ Range: = < 1, ℜ iii) = 2 1 Domain: = ℜ Range: = < 1, ℜ iv) = 2 1 Domain: = ℜ Range: = > 1, ℜ v) = Domain: = ℜ Range: = > 1, ℜ vi) = 2 1 Domain: = ℜ Range: = > 1, ℜ vii) = Domain: = ℜ Range: = < 1, ℜ viii) = 2 1 Domain: = ℜ Range: = < 1, ℜ vi) vii) viii) iii) ii) iv) v) i)

15 4. a) ii b) iv c)i d)iii

16 MCR3U Unit 4: Exponential Relations Lesson 7 Date: Learning goal: I can create an equation to model exponential growth and decay. I can solve for an unknown in an exponential application. Applications of Growth and Decay Exponential growth or decay occurs when quantities increase or decrease at a rate proportional to the initial quantity present. This growth or decay occurs in savings accounts, the size of populations, appreciation, depreciation, and with radioactive chemicals. Exponential Functions Exponential growth and decay problems can be modelled using the formula: () = (). () is the final amount Where is the initial value Where is the growth/decay factor The base is called the growth factor when > 1 The base is called the decay factor when 0 < < 1 The growth or decay rate is 1 The function neither grows nor decays when = 1 Where is the number of grow/decay periods Example 1: The population of Guelph is expected to grow by 3% per year. The population was in a) Find an equation to model the population. b) What would you expect the population to be in 2018? c) How long would it take for the population to each ?

17 Example 2: A car costs $24,000. A virtual cost associated with the time-value of the car is called depreciation. This car depreciates an average of 18% per year. a) Model this situation with an equation b) What is the approximate value after 31 months? Example 3: A bacteria population doubles every 20 minutes. a) Write an equation for a population that starts with 100 bacteria. b) How many bacteria will you have after 2 hours? Example 4: Ryan has been saving for his college tuition for 4 years. He put $5,550 in a savings account 4 years ago (without adding to it) and now has $ Calculate the annual growth rate as a percent to two decimals.

18 Half-life is the amount of time required for an amount to diminish by half the initial value. Half-life Half-life problems can be modelled using the formula: () = h () is the final mass Where is the initial mass Where is time Where is half-life Example 5: A 200g sample of radioactive polonium-210 has a half-life of 138 days. a) Write an equation for the mass remaining after t days. b) Determine the mass left after 5 years, to the nearest thousandth. c) How long ago was the sample 800g? HW: Pg. 80 #9-13, 19, 20, 23, pg. 95 #12, 14, 17, 20