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Lesson 3.1, page 376 Exponential Functions Objective: To graph exponentials equations and functions, and solve applied problems involving exponential functions and their graphs.
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Look at the following… Polynomial Exponential
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Real World Connection Exponential functions are used to model numerous real-world applications such as population growth and decay, compound interest, economics (exponential growth and decay) and more.
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REVIEW Remember: x 0 = 1 Translation – slides a figure without changing size or shape
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Exponential Function The function f(x) = b x, where x is a real number, b > 0 and b 1, is called the exponential function, base b. (The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers.)
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Examples of Exponential Functions, pg. 376
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See Example 1, page 377. Check Point 1: Use the function f(x) = 13.49 (0.967) x – 1 to find the number of О-rings expected to fail at a temperature of 60° F. Round to the nearest whole number.
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Graphing Exponential Functions 1. Compute function values and list the results in a table. 2. Plot the points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises.
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Check Point 2 -- Graph the exponential function y = f(x) = 3 x. ( 3,1/27) 1/27 33 ( 2, 1/9) 1/9 22 ( 1, 1/3) 1/3 11 (3, 27)273 9 3 1 y = f(x) = 3 x (2, 9)2 (1, 3)1 (0, 1)0 (x, y)x
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Check Point 3: Graph the exponential function (3,1/27)1/273 (2, 1/9)1/92 (1, 1/3)1/31 ( 3, 27) 27 33 9 3 1 ( 2, 9) 22 ( 1, 3) 11 (0, 1)0 (x, y)x
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Characteristics of Exponential Functions, f(x) = b x, pg. 379 Domain = (-∞,∞) Range = (0, ∞) Passes through the point (0,1) If b>1, then graph goes up to the right and is increasing. If 0<b<1, then graph goes down to the right and is decreasing. Graph is one-to-one and has an inverse. Graph approaches but does not touch x-axis.
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Observing Relationships
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Connecting the Concepts
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Example -- Graph y = 3 x + 2. The graph is that of y = 3 x shifted left 2 units. 2433 812 271 90 3 1 1/3 y= 3 x+2 11 22 33 x
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Example: Graph y = 4 3 x 3.963 3.882 3.671 30 1 55 23 y 11 22 33 x The graph is a reflection of the graph of y = 3x across the y-axis, followed by a reflection across the x-axis and then a shift up of 4 units.
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The number e (page 381) The number e is an irrational number. Value of e 2.71828 Note: Base e exponential functions are useful for graphing continuous growth or decay. Graphing calculator has a key for e x.
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Practice with the Number e Find each value of e x, to four decimal places, using the e x key on a calculator. a) e 4 b) e 0.25 c) e 2 d) e 1 Answers: a) 54.5982b) 0.7788 c) 7.3891d) 0.3679
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Natural Exponential Function Remember e is a number e lies between 2 and 3
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Compound Interest Formula A = amount in account after t years P = principal amount of money invested R = interest rate (decimal form) N = number of times per year interest is compounded T = time in years
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Compound Interest Formula for Continuous Compounding A = amount in account after t years P = principal amount of money invested R = interest rate (decimal form) T = time in years
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See Example 7, page 384. Compound Interest Example Check Point 7: A sum of $10,000 is invested at an annual rate of 8%. Find the balance in that account after 5 years subject to a) quarterly compounding and b) continuous compounding.
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