1. Slide 4.1- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Exponential Functions Learn the definition of exponential function. Learn to graph exponential functions. Learn to solve exponential equations. Learn to use transformations on exponential functions. SECTION 4.1 1 2 3 4

3. Slide 4.1- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXPONENTIAL FUNCTION A function f of the form is called an exponential function with base a. The domain of the exponential function is (–∞, ∞).

4. Slide 4.1- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Evaluating Exponential Functions Solution

5. Slide 4.1- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley RULES OF EXPONENTS Let a, b, x, and y be real numbers with a > 0 and b > 0. Then

6. Slide 4.1- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Graphing an Exponential Function with Base a > 1 Graph the exponential function Solution Make a table of values. x–3–2–10123 y = 3 x 1/271/91/313927 Plot the points and draw a smooth curve.

7. Slide 4.1- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Graphing an Exponential Function with Base a > 1 Solution continued This graph is typical for exponential functions when a > 1.

8. Slide 4.1- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Graphing an Exponential Function with Base 0 < a < 1 Sketch the graph of Solution Make a table of values. x–3–2–10123 y = (1/2) x 84211/21/41/8 Plot the points and draw a smooth curve.

9. Slide 4.1- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Graphing an Exponential Function with Base 0 < a < 1 Solution continued As x increases in the positive direction, y decreases towards 0.

10. Slide 4.1- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROPERTIES OF EXPONENTAIL FUNCTIONS Let f (x) = a x, a > 0, a ≠ 1. Then 1.The domain of f (x) = a x is (–∞, ∞). 2.The range of f (x) = a x is (0, ∞); thus, the entire graph lies above the x-axis. 3.For a > 1, i. f is an increasing function; thus, the graph is rising as we move from left to right.

11. Slide 4.1- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ii.As x  ∞, y = a x increases indefinitely and very rapidly. 4.For 0 < a < 1, i. f is a decreasing function; thus, the graph is falling as we scan from left to right. iii.As x  –∞, the values of y = a x get closer and closer to 0. ii.As x  –∞, y = a x increases indefinitely and very rapidly. iii.As x  ∞, the values of y = a x get closer and closer to 0.

12. Slide 4.1- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.Each exponential function f is one-to- one. Thus, ii.f has an inverse 6.The graph of f (x) = a x has no x-intercepts. In other words, the graph of f (x) = a x never crosses the x-axis. Put another way, there is no value of x that will cause f (x) = a x to equal 0. i. if x 1 = x 2 ;

13. Slide 4.1- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.The graph of f (x) = a x has y-intercept 1. If we substitute x = 0 in the equation y = a x, we obtain y = a 0 = 1, which yields 1 as the y-intercept. 8.The x-axis is a horizontal asymptote for every exponential function of the form f (x) = a x.

14. Slide 4.1- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding a Base Value for an Exponential Function Find a if the graph of the exponential function f (x) = a x contains the point (2, 49). Solution Write y = f (x) so that we have y = a x. Solution set is {–7, 7}. The base for an exponential function must be positive, so a = 7 and the exponential function is f (x) = 7 x. Since the point (2, 49) is on the graph, we have,

15. Slide 4.1- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving an Exponential Equation Solve for x: Solution The solution set is

16. Slide 4.1- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Finding the First Coordinate, Given the Second a. Let Find x so that b. Let Find x so that So the point is on the graph of f. Solution

17. Slide 4.1- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Finding the First Coordinate, Given the Second Solution continued So there are two points on the graph of g. and

18. Slide 4.1- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRANSFORMATIONS ON EXPONENTIAL FUNCTION f (x) = a x TransformationEquationEffect on Equation Horizontal Shift y = a x+b = f (x + b) Shift the graph of y = a x, b units (i) left if b > 0. (ii) right if b < 0. Vertical Shift y = a x + b = f (x) + b Shift the graph of y = a x, b units (i) up if b > 0. (ii) down if b < 0.

19. Slide 4.1- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRANSFORMATIONS ON EXPONENTIAL FUNCTION f (x) = a x TransformationEquationEffect on Equation Stretching or Compressing (Vertically) y = ca x = c f (x) Multiply the y coordinates by c. The graph of y = a x is vertically (i) stretched if c > 1. (ii) compressed if 0 < c < 1.

20. Slide 4.1- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRANSFORMATIONS ON EXPONENTIAL FUNCTION f (x) = a x TransformationEquationEffect on Equation Reflectiony = –a x = – f (x)The graph of y = a x is reflected in the x-axis. The graph of y = a x is reflected in the y-axis. y = a –x = f (–x)

21. Slide 4.1- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Sketching Graphs Use transformations to sketch the graph of each function. State the domain and range of each function and the horizontal asymptote of its graph.

22. Slide 4.1- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Sketching Graphs Solution a. Domain: (–∞, ∞) Range: (–4, ∞) Horizontal Asymptote: y = –4

23. Slide 4.1- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Sketching Graphs Solution b. Domain: (–∞, ∞) Range: (0, ∞) Horizontal Asymptote: y = 0

24. Slide 4.1- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Sketching Graphs Solution c. Domain: (–∞, ∞) Range: (–∞, 0) Horizontal Asymptote: y = 0

25. Slide 4.1- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Sketching Graphs Solution d. Domain: (–∞, ∞) Range: (–∞, 2) Horizontal Asymptote: y = 2

26. Slide 4.1- 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Comparing Exponential and Power Functions Compare the graphs of The graph of g(x) is higher than f (x) in the interval [0, 2). Thus, 2 x > x 2 for 0 < x < 2. for x ≥ 0. and Solution

27. Slide 4.1- 27 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Comparing Exponential and Power Functions The graph of f intersects the graph of g at x = 2. Thus, 2 x = x 2 at x = 2; both = 4, (2, 4) is a point on each graph Solution continued

28. Slide 4.1- 28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Comparing Exponential and Power Functions The graph of f is higher than the graph of g in the interval (2, 4). Thus, x 2 > 2 x for 2 < x < 4. Solution continued

29. Slide 4.1- 29 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Comparing Exponential and Power Functions The graph of f intersects the graph of g at x = 4. Thus, 2 x = x 2 at x = 4; both = 16, (2, 16) is a point on each graph Solution continued

30. Slide 4.1- 30 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Comparing Exponential and Power Functions The graph of g is higher than the graph of f for x > 4. Thus, 2 x > x 2 for x > 4. Solution continued

31. Slide 4.1- 31 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Bacterial Growth A technician to the French microbiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubles every hour. If the bacteria count B(t) is modeled by the equation a.the initial number of bacteria, b.the number of bacteria after 10 hours; and c.the time when the number of bacteria will be 32,000. with t in hours, find

32. Slide 4.1- 32 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Bacterial Growth a. Initial size c. Find t when B(t) = 32,000 4 hours after the starting time, the number of bacteria will be 32,000. Solution