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Exponential Functions & Their Graphs

Exponential Functions & Their Graphs. 3.1. Algebraic vs. Transcendental. Algebraic Functions = polynomial and rational functions. Transcendental Functions = exponential and logarithmic functions. Definition of Exponential Function.

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Exponential Functions & Their Graphs

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  1. Exponential Functions & Their Graphs 3.1

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

  3. 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.

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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.

  10. 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

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

  12. 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.

  13. 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.

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