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4.3 GRAPHS OF EXPONENTIAL FUNCTIONS 1 Graphs of the Exponential Family: The Effect of the Parameter a In the formula Q = abt, the value of a tells us where the graph crosses the Q-axis, since a is the value of Q when t = 0. Q Q Q=150 (1.2)t Q=50 (1.4)t Q=50 (1.2)t Q=100 (1.2)t 50 150 100 50 t 0 Q=50 (0.8)t Q=50 (1.2)t 5 10 Q=50 (0.6)t 0 t 5 2 Graphs of the Exponential Family: The Effect of the Parameter b The growth factor, b, is called the base of an exponential function. Provided a is positive, if b > 1, the graph climbs when read from left to right, and if 0 < b < 1, the graph falls when read from left to right. Q Q=50 (1.4)t 50 Q=50 (1.2)t Q=50 (0.8)t Q=50 (0.6)t 0 t 5 3 Horizontal Asymptotes The horizontal line y = k is a horizontal asymptote of a function, f, if the function values get arbitrarily close to k as x gets large (either positively or negatively or both). We describe this behavior using the notation f(x) → k as x → ∞ or f(x) → k as x → −∞. Alternatively, using limit notation, we write lim x f ( x) k or lim f ( x) k x 4 Interpretation of a Horizontal Asymptote Example 1 A capacitor is the part of an electrical circuit that stores electric charge. The quantity of charge stored decreases exponentially with time. If t is the number of seconds after the circuit is switched off, suppose that the quantity of stored charge (in micro-coulombs) is given by Q = 200(0.9)t, t ≥0. The charge stored by a capacitor over one minute. 200 Q, charge (micro-coulombs) 100 0 15 30 t (seconds) 45 60 5 Solving Exponential Equations Graphically Exercise 42 The population of a colony of rabbits grows exponentially. The colony begins with 10 rabbits; five years later there are 340 rabbits. (a) Give a formula for the population of the colony of rabbits as a function of the time. (b) Use a graph to estimate how long it takes for the population of the colony to reach 1000 rabbits. Solution Solving 340 = 10b5 we get b5 = 34, b = 2.0244, R ≈ 10 (2.0244)t Based on the graph, one would estimate that the population of rabbits would reach 1000 in a little more than 6 ½ years. 6 Finding an Exponential Function for Data Example: Population data for the Houston Metro Area Since 1900 Table showing population (in thousands) since 1900 t 0 10 20 30 40 50 N 184 236 332 528 737 1070 t 60 70 80 90 100 110 N 1583 2183 3122 3733 4672 5937 Graph showing population data with an exponential model P (thousands) 6000 P = 190 (1.034)t 5000 4000 3000 2000 1000 t (years since 1900) 0 0 20 40 60 80 100 Using an exponential regression feature on a calculator or computer the exponential function was found to be P = 190 (1.034)t 7