Lesson 3.1, page 376 Exponential Functions Objective: To graph exponentials equations and functions, and solve applied problems involving exponential functions and their graphs.

Look at the following… Polynomial Exponential

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.

REVIEW Remember: x 0 = 1 Translation – slides a figure without changing size or shape

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.)

Examples of Exponential Functions, pg. 376

See Example 1, page 377. Check Point 1: Use the function f(x) = (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.

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.

Check Point 2 -- Graph the exponential function y = f(x) = 3 x. (  3,1/27) 1/27 33 (  2, 1/9) 1/9 22 (  1, 1/3) 1/3 11 (3, 27) y = f(x) = 3 x (2, 9)2 (1, 3)1 (0, 1)0 (x, y)x

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 3 (  2, 9) 22 (  1, 3) 11 (0, 1)0 (x, y)x

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.

Observing Relationships

Connecting the Concepts

Example -- Graph y = 3 x + 2. The graph is that of y = 3 x shifted left 2 units /3 y= 3 x+2 11 22 33 x

Example: Graph y = 4  3  x 55  23 y 11 22 33 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.

The number e (page 381) The number e is an irrational number. Value of e  Note: Base e exponential functions are useful for graphing continuous growth or decay. Graphing calculator has a key for e x.

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) b) c) d)

Natural Exponential Function Remember e is a number e lies between 2 and 3

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

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

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.