3.1

 Algebraic Functions = polynomial and rational functions. Transcendental Functions = exponential and logarithmic functions. Algebraic vs. Transcendental

 The exponential function f with base a is denoted by f(x) = a x where a > 0, a ≠ 1, and where x is any real number. Sometimes you will have irrational exponents. Definition of Exponential Function

 Use a calculator to evaluate each function at the indicated value of x. A)f(x) = 2 x B)f(x) = 2 -x C)f(x) =.6 x Example 1: Evaluating Exponential Functions

 In the same coordinate plane, sketch the graph of each function by hand. A)f(x) = 2 x B)g(x) = 4 x Example 2: Graphs of y = a x

 In the same coordinate plane, sketch the graph of each function by hand. A)f(x) = 2 -x B)g(x) = 4 -x Example 3: Graphs of y = a -x

 1.a x ∙a y = a x+y 2.a x / a y = a x-y 3.a -x = 1 / a x 4.a 0 = 1 5.(ab) x = a x ∙b x 6.(a x ) y = a xy 7.(a / b) x = a x / b x 8.|a 2 | = |a| 2 = a 2 Properties of Exponents

 Each of the following graphs is a transformation of the graph of f(x) = 3 x. f(x) = 3 x+1 one unit to the left f(x) = 3 x-1 one unit to the right f(x) = 3 x + 1one unit up f(x) = 3 x -1one unit down f(x) = -3 x reflect about x-axis f(x) = 3 -x reflect about y-axis Transformations of Graphs of Exponential Functions

 e ≈ ← natural base The function f(x) = e x is called the natural exponential function and the graph is similar to that of f(x) = a x. The base e is your constant and x is the variable. The number e can be approximated by the expression [1 + 1 / x] x. The Natural Base e

 Use a calculator to evaluate the function f(x) = e x at each indicated value of x. A)x = -2 B)x =.25 C)x = -.4 Example 4: Evaluating the Natural Exponential Function

 Sketch the graph of each natural exponential function. A)f(x) = 2e.24x B)g(x) = 1 / 2e -.58x Example 5: Graphing Natural Exponential Functions

 After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: 1.For n compoundings per year: A = P (1 + r / n) nt 2.For continuous compoundings: a = Pe rt. Formulas for Compound Interest

 A)A total of $12,000 is invested at an annual interest rate of 4% compounded annually. Find the balance in the account after 1 year. B)A total of $12,000 is invested at an annual interest rate of 3%. Find the balance after 4 years if the interest is compounded quarterly. Example 6: Finding the Balance for Compound Interest