Maximum, Minimum, Even, and Odd
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Maximum, Minimum, Even, and Odd NYOS Charter School Precalculus
description
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. - PowerPoint PPT Presentation
Transcript of Maximum, Minimum, Even, and Odd
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
