Exponential Growth and Exponential Decay Section 8.1 and 8.2.

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Exponential Growth and Exponential Decay Section 8.1 and 8.2

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Transcript of Exponential Growth and Exponential Decay Section 8.1 and 8.2.

  • Slide 1
  • Exponential Growth and Exponential Decay Section 8.1 and 8.2
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  • WHAT YOU WILL LEARN: 1.How to graph exponential growth functions. 2.How to graph exponential decay functions.
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  • Exponential Growth This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week.
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  • Exponential Growth W01234567891011121314151617 1.01.02.04.08.16.32.641.282.565.1210.2 4 20.4 8 40. 96 81. 92 163.84 327.68 655.36 131 0.7 2 2$1$2$3$4$5$6$7$8$9$10$11$12$13$14$15$16$17$18 This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week. Although the second option, growing at a constant rate of $1/week, pays more in the short run, the first option eventually grows much larger:
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  • Why! Exponential Growth! The equation for option 1 is: y = 2 n where n is the number of weeks. The equation for option 2 is y = 1 + n where n is the number of weeks.
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  • Oh Boy! Vocabulary An exponential function involves the expression b x where the base b is a positive number other than 1. The variable is going to be in the position of the exponent.
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  • Let s Graph an Example Question: Will the graph ever pass below y of 0?
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  • Let s Graph an Example Question: Will the graph ever pass below y of 0? We say that there is an asymptote at y = 0.
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  • Let s Graph an Example Question: Will the graph ever pass below y of 0? We say that there is an asymptote at y = 0. An asymptote is a line that a graph approaches as you move away from the origin.
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  • Try the following on your graphing calculator Group 1:Group 2: How does a in the function affect the graph?
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  • A Definition y = ab x is an exponential growth function. When a is greater than 0 and b is greater than 1.
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  • Graphing Examples Graph
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  • Another Example Graph
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  • Graphing by Translation The generic form of an exponential function is: y = ab x-h + k Where h is movement along the x axis and k is movement along the y axis.
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  • An Example of Graphing by Translation Graph
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  • You Try Graph
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  • Exponential Growth Model We will use the formula: y = a(1 + r) t a is the initial amount, r is the percent increase expressed as a decimal and t is the number of years. The term 1 + r is called the growth factor.
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  • An Example Problem In January 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year. Write a model. How many hosts were there in 1996? Graph the model. When will there be 30 million hosts?
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  • Section 8.2 Exponential Decay These functions will have the form y = ab x where a is greater than zero and b is between 0 and 1. 19
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  • Example 1 20 State whether the function is an exponential growth or exponential decay function.
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  • You Try State whether the function is an exponential decay or growth function.
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  • A Basic Graph A graph of
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  • Graphing Exponential Functions again Graph:
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  • Another Example Graph:
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  • Graphing by Translation The generic form of an exponential function is: y = ab x-h + k Where h is movement along the x axis and k is movement along the y axis.
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  • Graphing by Translation Graph:
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  • An Exponential Decay Word Problem We will use the formula: y = a(1 - r) t (1-r) is called the decay factor.
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  • The Word Problem You buy a new car for $24,000. The value y of the car decreases by 16% each year. 1. Write an exponential decay model for the value of the car. 2. Use the model to estimate the value after 2 years. 3. Graph the model. 4. When will the car have a value of $12,000.
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  • Homework : Page 469, 14-18 even, 19-24 all, 34, 36, 38, 43-45 all Page 477, 12, 16, 18, 19-24 all, 36, 40, 42, 47-49 all