Chapter 3 Section 3.2 Polynomial Functions and Models

10/29/2010 Section 3.2 v5.0 2 Polynomials Example where f(x) is the U.S. consumption of natural gas in trillion cubic feet from 1965 to 1980 and x is the number of years after 1960 Source: U.S. Department of Energy f(x) is called a polynomial function The expression for f(x) is called a polynomial Questions: Is f(x) linear ? What is f(10) ? Polynomial Functions f(x) = x 4 – x x 2 – x What does this mean in terms of the model ? f(10) = 0.706

10/29/2010 Section 3.2 v5.0 3 Polynomial Functions Polynomial Terminology Polynomial a n x n + a n-1 x n-1 + … + a 1 x + a 0 a polynomial of degree n Polynomial Function f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 a polynomial function of degree n Polynomial Equation a n x n + a n-1 x n-1 + … + a 1 x + a 0 = 0 an n th degree polynomial equation in standard form

10/29/2010 Section 3.2 v5.0 4 Polynomial Functions Polynomials and Their Graphs Turning Point A point where the graph changes from increasing to decreasing or vice versa Occurs at local minimum or local maximum x y ● ● ● ● How many turning points ? How many local extrema ? How many x-intercepts ? How many y-intercepts ?

10/29/2010 Section 3.2 v5.0 5 Polynomial Functions Polynomials By Degree Zero Degree Constant function: f(x) = a, a ≠ 0 If a = 0, degree is undefined (i.e. no degree) Graph: Horizontal line No turning points No x-intercepts First Degree Linear function: f(x) = a x + b, a ≠ 0 Graph: Non-vertical, non-horizontal line No turning points One x-intercept End behavior: opposite directions x y x y ●

10/29/2010 Section 3.2 v5.0 6 Polynomial Functions Polynomials By Degree Second Degree Quadratic function/equation: f(x) = a x 2 + b x + c, a ≠ 0 Graph: Parabola One turning point At most two x-intercepts  End behavior: same direction Third Degree Cubic function/equation: f(x) = a x 3 + b x 2 + c x + d, a ≠ 0 Graph: Two or no turning points One to three x-intercepts End behavior: opposite directions x y x y ● ● ● ● ● ● ● ● ● ● ●

10/29/2010 Section 3.2 v5.0 7 x y Polynomial Functions Polynomials By Degree Fourth Degree Quartic function/equation: f(x) = a x 4 + b x 3 + c x 2 + d x + e, a ≠ 0 Graph: At most three turning points At most four x-intercepts End behavior: same direction Fifth Degree Quintic function/equation: f(x) = a x 5 + b x 4 + c x 3 + d x 2 + e x + k, a ≠ 0 Graph: At most four turning points At most five x-intercepts End behavior: opposite direc tions x y ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●

10/29/2010 Section 3.2 v5.0 8 Polynomials By Degree n th Degree Degree n function/equation: f(x) = a n x n + a n–1 x n–1 + … + a 2 x 2 + a 1 x + a 0, a n ≠ 0 Graph: At most n – 1 turning points At most n x-intercepts End behavior: depends on a and n For a > 0 : For a < 0 : opposite behavior from above Polynomial Functions f(x) ∞ if n odd or even if n odd if n even as x ∞ ∞ – f(x) ∞ – as x ∞ – f(x ) ∞

10/29/2010 Section 3.2 v f(x) = –3x f(x) = –x 3 + x Sketch the graphs of the following: 1. f(x) = 3x f(x) = x 3 – x Polynomial Function Examples x y x y x y x y

10/29/2010 Section 3.2 v Sketch the graphs (continued): 5. f(x) = x 4 – 5x f(x) = x 5 – 5x 3 + 4x 6. f(x) = –x 4 + 5x 2 – 4 8. f(x) = –x 5 + 5x 3 – 4x Polynomial Function Examples x y x y x y x y

10/29/2010 Section 3.2 v Think about it !