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Pre-Calculus Section 4.1: Exponential Functions Objective: SWBAT evaluate and graph exponential functions and calculate compound interest. Homework: Page 336 Day 1 (1-13 odd,19-29 odd) Day 2 (31-37 odd, 51, 59, 69, 75, 77) Daily Warm Up: 1. Solve the system by graphing. 2y x 2 x 2y 8 Exponential Functions The exponential function with base a is defined for all real numbers x by f ( x) a x where a > 0 and a 1 . *We assume a 1 because the function f ( x) 1x 1 is just a constant function. Example 1: Evaluating Exponential Functions (calculator) Let f ( x) 3x and evaluate the following: (a) f(2) 2 (b) f 3 (c) f ( ) (d) f ( 2) Example 2: Graphing Exponential Functions by Plotting Points Draw the graph of each function. (a) f ( x) 3x 1 (b) g ( x) 3 x x f ( x ) 3x 1 g ( x) 3 x -3 -2 -1 0 1 2 3 y Note that the family of exponential functions f ( x) a x have graphs that all pass through the point (0, 1) because a 0 1 for a 0 . If 0 a 1 , the exponential function decreases rapidly. If a > 1, the function increases rapidly. Graphs of Exponential Functions The exponential function f ( x) a x (a 0, a 1) has domain R and range (0, ). The line y = 0 (the x-axis) is a horizontal asymptote of f. Example 3: Identifying Graphs of Exponential Functions Find the exponential function f ( x) a x whose graph is given. . Example 4: Transformations of Exponential Functions Use the graph of f ( x) 2 x to sketch the graph of each function. (a) g ( x) 1 2 x (b) h( x) 2 x (c) k ( x) 2 x 1 Example 5: Comparing Exponential and Power Functions Compare the rates of growth of the exponential function f ( x) 2 x and the power function g ( x) x 2 by drawing the graphs of both functions in the following viewing rectangles. (a) [0, 3] by [0, 8] (b) [0, 6] by [0, 25] (c) [0, 20] by [0, 1000] The Natural Exponential Function The natural exponential function is the exponential function f ( x) e x with base e. It is often referred to as the exponential function. e 2.71828 … Since 2 e 3 , the graph of the natural exponential function lies between the graphs of y 2x and y 3x . Example 6: Evaluating the Exponential Function Evaluate each expression correct to five decimal places. (a) e3 (b) 2e0.53 (c) e 4.8 Example 7: Transformations of the Exponential Function Sketch the graph of each function. (a) f ( x) e x (b) g ( x) 3e0.5 x Example 8: An Exponential Model for the Spread of a Virus An infectious disease begins to spread in a small city of population 10,000. After t days, the number of persons who have succumbed to the virus is modeled by the function 10, 000 v(t ) 5 1245e 0.97 t (a) How many infected people are there initially (at time t = 0) (b) Find the number of infected people after one day, two days, and five days. (c) Use your calculator to graph the function v and describe its behavior. Compound Interest Compound Interest is calculated by the formula nt r A(t ) P 1 n where A(t) = amount after t years P = principal r = interest rate per year n = number of times interest is compounded per year t = number of years Example 9: Calculating Compound Interest A sum of $1000 is invested at an interest rate of 12% per year. Find the amounts in the account after 3 years if interest is compounded annually, semiannually, quarterly, monthly, and daily. Continuously Compounded Interest Continuously compounded interest is calculated by the formula A(t ) Pert where A(t) = amount after t years P = principal r = interest rate per year t = number of years Example 10: Calculating Continuously Compound Interest Find the amount after 3 years if $1000 is invested at an interest rate of 12% per year, compounded continuously.