Understanding Exponential Functions and Their Graphs
This guide explores exponential functions, distinguishing between algebraic and transcendental functions. Algebraic functions include polynomial and rational functions, while transcendental functions encompass exponential and logarithmic types. The definition of an exponential function is presented, along with various examples of evaluating and graphing these functions. Key properties of exponents and transformations of graph functions are explained, including natural exponential functions using base e. Additionally, the guide covers formulas for compound interest, illustrating practical applications of exponential growth.
Understanding Exponential Functions and Their Graphs
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Presentation Transcript
Algebraic vs. Transcendental Algebraic Functions = polynomial and rational functions. Transcendental Functions = exponential and logarithmic functions.
Definition of Exponential Function The exponential function f with base a is denoted by f(x) = ax where a > 0, a ≠ 1, and where x is any real number. Sometimes you will have irrational exponents.
Example 1: Evaluating Exponential Functions Use a calculator to evaluate each function at the indicated value of x. • f(x) = 2x • f(x) = 2-x • f(x) = .6x
Example 2: Graphs of y = ax In the same coordinate plane, sketch the graph of each function by hand. • f(x) = 2x • g(x) = 4x
Example 3: Graphs of y = a-x In the same coordinate plane, sketch the graph of each function by hand. • f(x) = 2-x • g(x) = 4-x
Properties of Exponents • ax∙ay = ax+y • ax / ay = ax-y • a-x = 1 / ax • a0 = 1 • (ab)x = ax ∙bx • (ax)y = axy • (a / b)x = ax / bx • |a2| = |a|2 = a2
Transformations of Graphs of Exponential Functions Each of the following graphs is a transformation of the graph of f(x) = 3x. f(x) = 3x+1 one unit to the left f(x) = 3x-1 one unit to the right f(x) = 3x + 1 one unit up f(x) = 3x -1 one unit down f(x) = -3x reflect about x-axis f(x) = 3-x reflect about y-axis
The Natural Base e e ≈ 2.718281828 ← natural base The function f(x) = ex is called the natural exponential function and the graph is similar to that of f(x) = ax. The base e is your constant and x is the variable. The number e can be approximated by the expression [1 + 1 / x] x.
Example 4: Evaluating the Natural Exponential Function Use a calculator to evaluate the function f(x) = ex at each indicated value of x. • x = -2 • x = .25 • x = -.4
Example 5: Graphing Natural Exponential Functions Sketch the graph of each natural exponential function. • f(x) = 2e.24x • g(x) = 1 / 2e-.58x
Formulas for Compound Interest 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: • For n compoundings per year: A = P (1 + r / n)nt • For continuous compoundings: a = Pert.
Example 6: Finding the Balance for Compound Interest • 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. • 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.
