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Chapter 4

Chapter 4. 4-1 exponential functions, growth and decay. Objectives. Write and evaluate exponential expressions to model growth and decay situations. Example.

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Chapter 4

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  1. Chapter 4 4-1 exponential functions, growth and decay

  2. Objectives • Write and evaluate exponential expressions to model growth and decay situations.

  3. Example • Moore’s law, a rule used in the computer industry, states that the number of transistors per integrated circuit (the processing power) doubles every year. Beginning in the early days of integrated circuits, the growth in capacity may be approximated by this table.

  4. Exponential function • Growth that doubles every year can be modeled by using a function with a variable as an exponent. This function is known as an exponential function. The parent exponential function is f(x) = bx, where the baseb is a constant and the exponent x is the independent variable.

  5. Example • The graph of the parent function f(x) = 2xis shown. The domain is all real numbers and the range is {y|y> 0}.

  6. Example • Notice as the x-values decrease, the graph of the function gets closer and closer to the x-axis. The function never reaches the x-axis because the value of 2xcannot be zero. In this case, the x-axis is an asymptote. An asymptote is a line that a graphed function approaches as the value of x gets very large or very small.

  7. Exponential growth and decay • A function of the form f(x) = abx, with a > 0 and b> 1, is an exponential growth function, which increases as x increases. When 0 < b < 1, the function is called an exponential decay function, which decreases as x increases.

  8. Example #1 • Tell whether the function shows growth or decay. Then graph. • Solution: • Get base ¾ and compare it to 1 • The base , 3/4 ,is less than 1. This is an exponential decay function.

  9. Example#1 continue

  10. Example#2 • Tell whether the function shows growth or decay. Then graph. • g(x) = 100(1.05)x • Solution: • Get base 1.05 and compare it to 1 • The base, 1.05, is greater than 1. This is an exponential growth function.

  11. Student guided practice • Do problems 2-4 in your book page 237

  12. Exponential growth and decay • You can model growth or decay by a constant percent increase or decrease with the following formula: • In the formula, the base of the exponential expression, 1 + r,is called the growth factor. Similarly, 1 – ris the decay factor.

  13. Example#3 • Clara invests $5000 in an account that pays 6.25% interest per year. After how many years will her investment be worth $10,000?

  14. Example#4 • A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below 8000.

  15. Example#5 • A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100.

  16. Student guided practice • Do problems 1-3 in your worksheet

  17. Exponential activity • Do PowerPoint presentation

  18. Homework!! • Do problems 7-9,12-16 in your book page 238

  19. Closure • Today we learned about exponential growth and decay • Next class we are going to learn about inverse functions

  20. Have a great day

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