maximum, minimum, even, and odd
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Maximum, Minimum, Even, and Odd
NYOS Charter School
Precalculus
Nonlinear Functions and Their Graphs
An Absolute (Global) Maximum is the maximum y-value across the entire domain of a function.
An Absolute (Global) Minimum is the minimum y-value across the entire domain of a function.
Nonlinear Functions and Their Graphs
An Local (Relative) Maximum is the maximum y-value within a restricted domain of a function.
An Local (Relative) Minimum is the minimum y-value within a restricted domain of a function.
Nonlinear Functions and Their Graphs
Example: What are the local and global extrema?
Local Maximum: 10 Local Minimum: -10
Global Maximum: none Global Minimum: none
Nonlinear Functions and Their Graphs
Example: What are the local and global extrema?
Local Maximum: Local Minimum:
Global Maximum: Global Minimum:
Nonlinear Functions and Their Graphs
Example: What are the local and global extrema?
Local Maximum: 1, 3 Local Minimum: -2, 0
Global Maximum: 3 Global Minimum: none
Nonlinear Functions and Their Graphs
Assignment
Draw a graph that has at least three local maximums and three local minimums.
Nonlinear Functions and Their Graphs
A function is an even function if f(-x) = f(x) for every x in the domain.
The graph is symmetric with respect to the y-axis.
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = x2 – 10 even?
f(-x) = (-x)2 – 10
= x2 – 10
which is f(x). Thus, f(x) is even.
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = 2x4 + 21 even?
f(-x) =
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = 2x4 + 21 even?
f(-x) = 2(-x)4 + 21
= 2x4 + 21
which is f(x). Thus, f(x) is even.
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = x3 – 5 even?
f(-x) =
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = x3 – 5 even?
f(-x) = (-x)3 – 5
= -x3 – 5
which is not f(x). Thus, f(x) is not even.
Nonlinear Functions and Their Graphs
Example: Is the function represented in the graph even?
Nonlinear Functions and Their Graphs
Example: Is the function represented in the graph even?
The graph is symmetric with respect
to the y-axis. Thus, f(x) is even.
Nonlinear Functions and Their Graphs
A function is an odd function if f(-x) = -f(x) for every x in the domain.
The graph is symmetric with respect to the origin.
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = 5x odd?
f(-x) = 5(-x)
= -5x
which is –f(x). Thus, f(x) is odd.
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = -x5 odd?
f(-x) =
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = -x5 odd?
f(-x) = -(-x)5
= x5
which is -f(x). Thus, f(x) is odd.
Nonlinear Functions and Their GraphsExample: Is the function represented by the graph odd?
which is f(x).
Nonlinear Functions and Their GraphsExample: Is the function represented by the graph odd?
which is f(x). The graph is symmetric with
respect to the origin. Thus, f(x) is odd.
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = -x6 + 5x2 even or odd?
f(-x) =
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = -x6 + 5x2 even or odd?
f(-x) = -(-x)6 + 5(-x)2
= -x6 + 5x2
which is f(x). Thus, f(x) is even.
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = x + 3 even, odd, or neither?
f(-x) =
Nonlinear Functions and Their Graphs
Example: Is the function f(x) = x + 3 even, odd, or neither?
f(-x) = -x + 3
which is neither f(x) nor -f(x). Thus, f(x) is neither.
Nonlinear Functions and Their Graphs
Assignment:
Draw a graph that has at least three local maximums and three local minimums.
Create three functions One of each: even, odd, neither Must contain at least three terms each.
Nonlinear Functions and Their Graphs
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Complete the “Even and Odd Functions” exercise
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