exponential functions & their graphs
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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. - PowerPoint PPT PresentationTRANSCRIPT
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3.13.1
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Algebraic Functions =
polynomial and rational functions.
Transcendental Functions = exponential and logarithmic functions.
Algebraic vs. Transcendental
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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.
Definition of Exponential Function
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Use a calculator to evaluate each function
at the indicated value of x.A)f(x) = 2x
B)f(x) = 2-x
C)f(x) = .6x
Example 1: Evaluating Exponential Functions
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In the same coordinate plane, sketch the
graph of each function by hand.A)f(x) = 2x
B)g(x) = 4x
Example 2: Graphs of y = ax
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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
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1. ax∙ay = ax+y
2. ax / ay = ax-y
3. a-x = 1 / ax
4. a0 = 15. (ab)x = ax ∙bx
6. (ax)y = axy
7. (a / b)x = ax / bx
8. |a2| = |a|2 = a2
Properties of Exponents
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Each of the following graphs is a
transformation of the graph of f(x) = 3x.
f(x) = 3x+1 one unit to the leftf(x) = 3x-1 one unit to the rightf(x) = 3x + 1 one unit upf(x) = 3x -1 one unit downf(x) = -3x reflect about x-axisf(x) = 3-x reflect about y-axis
Transformations of Graphs of Exponential Functions
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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.
The Natural Base e
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Use a calculator to evaluate the function
f(x) = ex at each indicated value of x.
A)x = -2
B)x = .25
C)x = -.4
Example 4: Evaluating the Natural Exponential
Function
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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
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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 = Pert.
Formulas for Compound Interest
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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