Nonlinear Functions and their Graphs

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Nonlinear Functions and their Graphs. Lesson 4.1. Polynomials. General formula a 0 , a 1 , … ,a n are constant coefficients n is the degree of the polynomial Standard form is for descending powers of x a n x n is said to be the “leading term”. •. •. Extrema of Nonlinear Functions.

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Nonlinear Functions and their Graphs

Lesson 4.1

Polynomials
• General formula
• a0, a1, … ,an are constant coefficients
• n is the degree of the polynomial
• Standard form is for descending powers of x
• anxn is said to be the “leading term”

Extrema of Nonlinear Functions
• Given the function for the Y= screeny1(x) = 0.1(x3 – 9x2)
• Use window -10 < x < 10 and -20 < y < 20
• Note the "top of the hill" and the"bottom of thevalley"
• These are localextrema

Extrema of Nonlinear Functions
• Local maximum
• f(c) ≥ f(x) whenx is near c
• Local minimum
• f(n) ≤ f(x) whenx is near n

c

n

Extrema of Nonlinear Functions
• Absolute minimum
• f(c) ≤ f(x) for all xin the domain of f
• Absolute maximum
• f(c) ≥ f(x) for all xin the domain of f
• Draw a function with an absolute maximum

Extrema of Nonlinear Functions
• The calculator can find maximums and minimums
• When viewing the graph, use the F5 key pulldown menu
• Choose Maximum or Minimum
• Specify the upper and lower bound for x (the "near")

Note results

Try It Out
• Find local extrema … absolute extrema
Assignment
• Lesson 4.1A
• Page 256
• Exercises 1 – 45 odd
Even and Odd Functions
• If  f(x) = f(-x)  the graph is symmetric across the y-axis
• It is also an even function
Even and Odd Functions
• If f(x) = -f(x) the graph is symmetric across the x-axis
• But ... is it a function ??
Even and Odd Functions
• A graph can be symmetric about a point
• Called point symmetry
• If f(-x) = -f(x) it is symmetric about the origin
• Also an odd function
Applications
• Consider the U.S. birthrate from 1900 to 2005(births per 1000 people)
• Can be modeled by where x = number of years since 1900
• Evaluate f(95)
• What does it mean?
• With domain 1900 ≤ x ≤ 2005
• Identify the absolute minimum and maximum
Applications
• U.S. natural gas consumption from 1965 to 1980 can be modeled by
• x = 6 is 1966 and x = 20 is 1980
• Consumption measured in trillion cubic feet
• Evaluate f(10) …. What does it mean?
• Graph for 6 ≤ x ≤ 20 and 0.4 ≤ y ≤ 0.8
• Determine local extrema, interpret results
Assignment
• Lesson 4.1B
• Page 258
• Exercises 91 – 97 odd