1. 2.2.1 MATHPOWER TM 12, WESTERN EDITION 2.2 Chapter 2 Exponents and Logarithms

2. An exponential function is a function of the form f(x) = Ab x, where A is a constant and b > 0. Many real phenomena can be related to exponents; for example, population growth, the growth or decay of substances, and the value of investments earning compound interest. Example: A person sends a letter to each of 2 people. They, in turn, send a letter to 2 other people, and so on... 1 2 4 8 The number of letters grows exponentially and can be represented by the equation y = 2 x. 2.2.2 Exponential Functions

3. y = 2 x We can use the graph of y = 2 x to interpolate information. For example, how many letters have been written after the 6th round of writing letters? 2.2.3 Using the Graph of an Exponential Function: Round of Letter Writing Number of Letters (6, 64) Exponential Functions [cont’d]

4. y = 2 x General Observations of the Graph of y = b x y = 2 x y = 3 x y = 4 x The y-intercept is 1. There is no x-intercept. The domain is {x| x  R}. The range is {y|y > 0}. There is a horizontal asymptote at y = 0. 2.2.4

5. The y-intercept is 1. There is no x-intercept. The domain is {x| x  R}. The range is {y|y > 0}. 2.2.5 y = 2 x General Observations of the Graph of y = b x [cont’d]

6. You invest $400 in an account paying 8% per annum compounded semi-annually. A(t) = P(1 + i) n n = 2t i = 0.04 A(t) = 400(1 + 0.04) 2t A(t) = 400(1.04) 2t Estimate when you would expect your investment to double. From the graph, it would double in approximately 18 years. 2.2.6 Using the Graph of y = b x Number of Years Investment ($) (17.69, 800.46)

7. For every metre that a diver descends below the surface of the water, the light intensity is reduced by 3.5%. A)Write an exponential function relating the light intensity to the depth of the diver. B)About what percent of the original intensity remains at 10 m below the surface? C)Use a graph to estimate how far below the surface the diver would be when the light intensity is 40% of the surface intensity. Since the light is being reduced, there is 96.5% (100% - 3.5%) of the light intensity remaining for each metre descended. I d = I 0 (0.965) d IdId intensity at a depth of d metres I0I0 original intensity 2.2.7 The Exponential Function

8. B) When d = 10 m: I d = I 0 (0.965) d I d = I 0 (0.965) 10 I d = 0.7003 At 10 m below the surface, about 70% of the light remains. I d = I 0 (0.965) d C) Estimate the depth when the light intensity is 40%. From the graph, the diver would be at a depth of about 26 m. 2.2.8 The Exponential Function [cont’d] Depth (m) Light Intensity (%) (25.69, 0.4) (10, 0.7)

9. Suggested Questions: Pages 81 and 82 1-23, 25, 26 abc, 29 2.2.9