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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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polynomials
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

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 functions4

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 functions5
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 functions6
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
Try It Out
  • Find local extrema … absolute extrema
assignment
Assignment
  • Lesson 4.1A
  • Page 256
  • Exercises 1 – 45 odd
even and odd functions
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 functions10
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 functions11
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
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
applications13
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
assignment14
Assignment
  • Lesson 4.1B
  • Page 258
  • Exercises 91 – 97 odd