Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Nonlinear Functions and Their Graphs ♦ Learn terminology about polynomial functions ♦ Identify intervals where a function is increasing or decreasing ♦ Find extrema of a function ♦ Identify symmetry in a graph of a function ♦ Determine if a function is odd, even, or neither 4.1

Slide 4- 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polynomial Functions Polynomial functions are frequently used to approximate data.

Formulas, degrees, and leading coefficients of some polynomial functions include the following Slide 4- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley FormulaDegreeLeading Coefficient f(x) = 10 g(x) = h(x) = k(x) =

Slide 4- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Increasing or Decreasing Functions The concept of increasing and decreasing relate to whether the graph of a function rises or falls. Moving from left to right along a graph of an increasing function would be uphill. Moving from left to right along a graph of a decreasing function would be downhill. We speak of a function f increasing or decreasing over an interval of its domain.

Slide 4- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Increasing or Decreasing Functions continued

Slide 4- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use the graph of shown below and interval notation to identify where f is increasing or decreasing. Solution

Slide 4- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Extrema of Nonlinear Functions Graphs of polynomial functions often have “hills” or “valleys”.

Slide 4- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Extrema of Nonlinear Functions continued Maximum and minimum values that are either absolute or local are called extrema.

Slide 4- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute and Local Extrema

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Group Work The monthly average ocean temperature in degrees Fahrenheit at Bermuda can be modeled by where x = 1 corresponds to January and x = 12 to December. The domain of f is D = {x|1 }. (Source: J. Williams, The Weather Almanac 1995.) a)Graph f in [1, 12, 1] by [50, 90, 10]. b)Estimate the absolute extrema. Interpret the results. Solution

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Symmetry If a graph was folded along the y-axis, and the right and left sides would match, the graph would be symmetric with respect to the y-axis. A function whose graph satisfies this characteristic is called an even function.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Symmetry continued Another type of of symmetry occurs in respect to the origin. If the graph could rotate, the original graph would reappear after half a turn. This represents an odd function.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Identify whether the function is even or odd. Solution