1. Chapter 2: Functions and Models Lesson 4: Exponential Functions Mrs. Parziale

2. Example 1: Make a table of values and graph f(x) = 2 x and g(x) =.5 x x2x2x 0.5 x -3 -2 0 1 2 3 What transformation maps f onto g?

3. Important Terms exponential function with base b: a function of the form Growth factor – ratio of change Growth when b > 1 Decay when 0 < b <1 Initial value

4. More terms exponential growth function: a > 0, b > 0 exponential growth curve: graph of an exponential growth function where a > 0, b > 1 exponential decay curve: graph of an exponential growth function where a > 0 and 0 < b < 1

5. More Terms strictly increasing: as x-values increase, the corresponding y-values increase. strictly decreasing: as x-values increase, the corresponding y-values decrease. asymptote: a line the graph gets very close to but never touches. The x-axis is the asymptote for exponential functions.

6. Example 2: Use a graph to estimate the solution to

8. Example 3: The population of the US in 1995 was estimated at 264,000,000 and was expected to grow at 0.9% per year. (a) Write an equation for the population (x) years after 1995 ____________________ (b) Estimate the US population in 2010: _________________________

9. Example 4: Will the function with the given equation be an exponential function? (a)k(m) = 4 m (b) s(t) = 6 (c) j(z) = z 2 (d) p(x) = 0.6 x

10. Example 5: The most populous country in the world is the People’s Republic of China. In 1995, its population was estimated at 1,198,000,000 people, and the average annual growth rate was about 1.01%. Suppose this rate remains unchanged. (a) Estimate the population in 1996. (b) Estimate the population in 1997. (c) Express the population P as a function of n, the number of years after 1995. (d) Use your answer to part c to predict the population of China in 2005. (e) Use your answer to part c to predict the population in 2995. Is this possible?

11. Characteristics of the graph of the exponential function f(x) = ab x : General Properties 1. Domain is the set of all real numbers 2. Range is the set of POSITIVE real numbers (not 0) 3. Because the range is the set of positive reals, every positive real can be expressed as a multiple of the power of b. 4. Graph crosses the point (0, a) 5. The graph does NOT cross or touch the x-axis

12. 6.The function is strictly increasing (if b > 1) or strictly decreasing (if 0 < b < 1) 7. Growth: As x gets larger, f(x) increases without bound. Decay: As x gets smaller, f(x) increases without bound. 8. There is a horizontal asymptote at the x-axis (y = 0). Characteristics of the graph of the exponential function f(x) = ab x :

13. Closure What is the general form of an exponential growth function? What makes the graph show growth? What makes the graph show decay? What is the asymptote of exponential growth functions?