Unit 4. Exponential Function

Transcription

1 Unit 4. Exponential Function In mathematics, an exponential function is a function of the form, f(x) = a(b) x + c + d, where b is a base, c and d are the constants, x is the independent variable, and f(x) is the dependent variable. Lesson Essential Question 1: How can we apply exponent properties? Let s revisit the rules of exponents!

2 Lesson Essential Question 1: How can we solve Exponential Equations in One Variable? Remember to use algebraic properties! Steps: 1. Examine the information given 2. Define a variable and write an expression for the exponential term 3. Write an exponential equation for the situation 4. Solve the equation Example 1: Solve for n. 256( 1 2 )n = 2 Example 2: Solve for m. 6(3) n = 162

3 Example 3: Solve for k. -3(2) k + 1 = - 47 Example 4: Solve for k. 5(4) k = 81 Essential Question 2: Why is the concept of a function important? Essential Question 3: How do I build an exponential function that models a relationship between two quantities? MGSE9 12.A.F.IF.1 Understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range, i.e. each input value maps to exactly one output value. If f is a function, x is the input (an element of the domain), and f(x) is the output (an element of the range). Graphically, the graph is y = f(x). MGSE9 12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MGSE9 12.F.BF.1 Write a function that describes a relationship between two quantities.

4 MGSE9 12.A.CED.2 Create exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase in two or more variables refers to formulas like the compound interest formula, in which A = P (1 + r n )nt has multiple variables, which is a Literal Equation.) Let s revisit the meaning of functions! Example 1: The set of ordered pairs shown represents a function f. { (-5, 6), (2, 7), (3, 0), (0, 6)} Select three ordered pairs that could be added to the set so that f remains a function. (-5, 10) (-4, 8) (0, -1) (-3, 10) (10, 10) (2, -1)

5 Example 2: Brenda is a salon owner. Yesterday, she did 1 haircut and colored the hair of 2 clients, charging a total of $211. Today, she did 1 haircut and colored the hair of 5 clients, charging a total of $466. Part1. Write the system modeled by the situation. Part 2. How much does the haircut cost and the colored the hair? What does an exponential function look like? Example 1

6 Examples of Exponential Growth and Decay

7 We can see the main characteristics of Exponential Functions! Domain: input values - x values Range: output values y values X-intercepts: x value when y = 0 Y-intercepts: y value when x = 0 Asymptotes: A Straight line approached by a given curve as one of the variables in the equation of the curve approaches infinity.

8 Intervals of Increase: A function is increasing, if as x increases (reading from left to right), y increases. Using interval notation, it is described as increasing on the interval (1,3). Intervals of Decrease: A function is decreasing, if as x increases (reading from left to right), y decreases. Using interval notation, it is described as decreasing on the interval (3, 5). In plain English, as you look at the graph, from left to right, the graph goes down-hill. The graph has a negative slope.

9 February 8 Lesson 3 Essential Question 1: How do I use function notation to show a variety of situations modeled by exponential functions? MGSE9 12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from exponential functions (integer inputs only). MGSE9 12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Generally this subset is the set of natural numbers 1,2,3,4...) By graphing or calculating terms, students should be able to show how the recursive sequence a1=7, an=an-1 +2; the sequence sn = 2(n-1) + 7; and the function f(x) = 2x + 5 (when x is a natural number) all define the same sequence.

10 Example 1. Y = 4 (2) x a = b = Growth or Decay y Intercept = Example 2. Y = 30 (1.5) x a = b = Growth or Decay y Intercept = Example 3. Y = 5 (0.8) x a = b = Growth or Decay y Intercept = Example 4. Y = 9 (0.4) x a = b = Growth or Decay y Intercept =

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13 February 12 Lesson 4 Essential Question 1: Why are geometric sequences functions? Essential Question 2: How do I interpret exponential expressions for functions in terms of the situation they model? MGSE9 12.F.BF.1a Determine an explicit expression and the recursive process (steps for calculation) from context. For example, if Jimmy starts out with $15 and earns $2 a day, the explicit expression 2x+15 can be described recursively (either in writing or verbally) as to find out how much money Jimmy will have tomorrow, you add $2 to his total today. Jn = Jn 1 + 2, J0 = 15 MGSE9 12.F.BF.2 Write arithmetic and geometric sequences recursively and explicitly, use them to model situations, and translate between the two forms. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions. MGSE9 12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Geometric Sequence In Mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The general form of a geometric sequence is where r 0 is the common ratio and a is a scale factor, equal to the sequence's start value. a(r) n

14 Example 1. The sequence 2, 6, 18, 54,... is a geometric progression with common ratio 3. Example 2. The sequence 10, 5, 2.5, 1.25,... is a geometric sequence with common ratio 1/2. Geometric and Arithmetic Sequences

15 Examples 1) 3, 8, 13, 18, 23,... 2) 1, 2, 4, 8, 16,... 3) 24, 12, 6, 3, 3/2, 3/4,... 4) 55, 51, 47, 43, 39, 35,... Exit Ticket Express the exponential functions by using the below tables. 1. x y x y -3 72

16 y-intercepts 1, -5, 25, -125,... 3, 0, -3, -6, -9,...

17 February 13 Lesson 5 Essential Question 1: How do I interpret exponential functions that arise in applications in terms of context? Essential Question 2: How do I use different representations to analyze exponential functions? Essential Question 3: How can we use real-world situations to construct and compare exponential models and solve problems? MGSE9 12.F.IF.4 Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. MGSE9 12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one function and an algebraic expression for another, say which has the larger maximum. MGSE9 12.F.IF.7 Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.

18 Examples of the applications of the Arithmetic and Geometric sequences Example 1. A bag of Cat food weighs 18 pounds. Each day, the cats are feed 0.5 pound of food. How much does the bag of cat food weigh after 30days? Step 1. Determine whether the situation appears to be arithmetic. The sequence for the situation is arithmetic because the cat food decreased by 0.5 pound each day. Step 2. Find d, a 1, and n. Since the bag weighs 18 pounds to start, a 1 = 18. Since the cat food decreased by 0.5 pound each day, d = 0.5. Since you want to find the weight of the bag after 30 days, you will need to find the 31 st term of the sequence so, n = 3

19 Example 2. A Ferrari rental company charges a base rate to rent a Ferrari plus an additional fee based on how many miles you drive the car. The cart below represents the total cost to rent the Ferrari. How much will you have to pay if you drive the car 143 miles? Number of Miles Cost of Car Rental ( $) Is this Arithmetic or Geometric Sequence? a n = a 1 + (n-1)d a n = a 1(r) n-1 a 64 = 452 Growth and Decay Key Words Exponential Growth Key Words Earns/ Pays interest Appreciates/ Appreciation Increases/Grows/Accrues A = P(1 + r) t Exponential Decay Key Words Depreciates/ Depreciation Decays Decreases A = P(1 - r) t A : P: r: t:

20 Example 3. A rabbit population grew in the following pattern: 2, 4, 8, 16, If all the rabbits live and the pattern continues, how many rabbits will be in the 8 th generation? Example 4. Dr. Franklin begins an experiment with 100 bacteria in a container. She finds that the number of bacteria present at any given time is modeled by the following recursive formula: A 0 = 100, a n = 2a n-1, where n is the number of hours after the beginning of the experiment. How many bacteria are present 5 hours after the beginning of the experiment?

21 Example 5. In 1991, the average cell phone bill was $ The average cell phone bill decreased 4% each year. What is the average cost of a cell phone bill in the year 2002? a 1 = $87.60 r = 0.96 a n = (1-0.04) n-1 Example 6. Suppose that you invest $2000 in an account that pays 8% interest annually, what is the balance after 5 years?

22 Lesson 6 February 14 Warming - up Find the sixth term of the sequence 2, 10, 50,... Amy owns a graphic design store. She purchases a new printer to use in her store. The printer depreciates by a fixed rate of 15% per year. The function A = 3,400(1 0.15) t can be used to model the value of the printer A dollars after t years. 1. Explain what the parameter 3,400 represents in the equation of the function 2. What is the factor by which the printer depreciates each year?

23 3. Amy also considered purchasing a printer that costs $4,000 and depreciates by 25% each year. Which printer will have more value in 5 years? Essential Question 1: How do I build new functions from existing exponential functions? Essential Question 2: What are the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative)? Essential Question 3: How do I find the value of k given the graphs? MGSE9 12.F.IF.7e Graph exponential and logarithmic functions showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

24 MGSE9 12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. MGSE9 12.F.IF.7 Graph functions expressed algebraically and show key features of the graph both by hand and by using technology. MGSE9-12.A.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y intercept.)

25 Horizontal Translation Which function shows the function f(x) = 3 x being translated 5 units left? f(x) = 3 x 5 f(x) = 3 x + 5 f(x) = 3 (x+5) f(x) = 3 (x-5)

26 Vertical Stretching or Shrinking Which function shows the function f(x) = 3 x being 4 times stretched? f(x) = -4(3) x f(x) = 4(3) x + 5 f(x) = 4(3) x f(x) = 3 (x-4)

27 Reflection Which function shows the function f(x) = 3 x being reflected about the x-axis? f(x) = -3 x 5 f(x) = -3 x + 5 f(x) = -3 x f(x) = 3 (x-5)

28 Vertical Transformation Which function shows the function f(x) = 3 x being translated 5 units down? f(x) = 3 x 5 f(x) = 3 x + 5 f(x) = 3 (x+5) f(x) = 3 (x-5)

29 Multi-Transformation Which function shows the function f(x) = 3 x being translated 5 units down and 2 units right? f(x) = 3 x 5 f(x) = 3 x + 5 f(x) = 3 (x+5) - 5 f(x) = 3 (x-2) - 5

30 Lesson 7 February 15 Unit4. Exponential Function Study Guide The graph of the exponential function f(x) = 4(0.5) x + 2 is shown below: Part A: Which function has the same end behavior for large, positive values of x? g(x) = 0.5(0.5) x + 2 g(x) = 7(1.5) x + 3

31 g(x) = 0.5(1.5) x + 2 g(x) = 4(0.7) x + 3 Part B: Select TWO functions whose graphs have a y-intercept of 1. h(x) = (2) x h(x) = 0.5(2) x h(x) = (0.5) x + 1 h(x) = 5(2) x h(x) = 5(0.5) x Sandy decided to raise rabbits. She started out with 3 rabbits. At the end of the seventh year, she determined that the population increases at a rate of 23%. Find the exponential function which describes how many rabbits she will have at the end of x years.

32 The table shows the number of squirrels in a particular forest t years after a forest fire. Create an exponential function to model the above table. What is the y-intercept of the equation g(x) = 3 (5) x? What is the recursive rule and explicit rule for the following sequence? 81, 54, 36, 24,... Recursive rule is A 1 = 81; A n= 2 3 (An-1) Explicit rule is Write the explicit formula for the geometric sequence, where a 1 = 5 and r = 4. Then, find the fourth term in the sequence.

33 Which function has a y-intercept of (0, -4)? Which function has a x-intercept of (-1, 0)? Which function decreases over the domain? Which function has the greatest rate of change over the interval [0, 3]?

34 What is the Average rate of Change of the exponential function from x = -3 to x = 3? Solve for k. 5(4) k = 81 Solve for n. 256( 1 2 )n = 2

35 The set of ordered pairs shown represents a function f. { (-5, 6), (2, 7), (3, 0), (0, 6)} Select three ordered pairs that could be added to the set so that f remains a function. (-5, 10) (-4, 8) (0, -1) (-3, 10) (10, 10) (2, -1) Write the explicit formula for the Arithmetic sequence, where a 1 = 5 and d = 4. Then, find the fourth term in the sequence. What is the x-intercept and y-intercept of the equation g(x) = 3 (5) x + 3 5?