1 / 11

Chapter 3

Chapter 3. Section 3.2 Polynomial Functions and Models. Polynomial Functions. 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

tallulah-lynch
Download Presentation

Chapter 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 3 Section 3.2 Polynomial Functions and Models

  2. Polynomial Functions • 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 yearsafter1960 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) ? f(x) = 0.0001234x4 – 0.005689x3+0.08792x2 – 0.5145x +1.514 f(10) = 0.706 What does this mean in terms of the model ? Section 3.2 v5.0

  3. Polynomial Functions • Polynomial Terminology • Polynomial anxn+an-1xn-1+… +a1x+a0 a polynomial of degreen • Polynomial Function f(x) = anxn+an-1xn-1+… +a1x+a0 a polynomial function of degreen • Polynomial Equation anxn+an-1xn-1+… +a1x+a0 = 0 an nthdegreepolynomial equation in standard form Section 3.2 v5.0

  4. y x 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 ● How many turning points ? ● How many local extrema ? How many x-intercepts ? ● ● How many y-intercepts ? Section 3.2 v5.0

  5. y y x x 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) = ax +b , a ≠ 0 • Graph: Non-vertical, non-horizontal line No turning points One x-intercept • End behavior: opposite directions ● Section 3.2 v5.0

  6. y y x x Polynomial Functions • Polynomials By Degree • Second Degree • Quadratic function/equation: f(x) = ax2+bx +c , a ≠ 0 • Graph: Parabola One turning point At most two x-intercepts • End behavior: same direction • Third Degree • Cubic function/equation: f(x) = ax3+bx2+cx +d , a ≠ 0 • Graph: Two or no turning points One to three x-intercepts • End behavior: opposite directions ● ● ● ● ● ● ● ● ● ● ● Section 3.2 v5.0

  7. x y y x Polynomial Functions • Polynomials By Degree • Fourth Degree • Quartic function/equation: f(x) = ax4+bx3+cx2+dx+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) = ax5+bx4+cx3+dx2 +ex +k, a ≠ 0 • Graph: At most four turning points At most five x-intercepts • End behavior: opposite directions ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Section 3.2 v5.0

  8. ∞ as x ∞ f(x) ∞ f(x) ∞ – – as x ∞ ∞ as x f(x) – Polynomial Functions • Polynomials By Degree • nth Degree • Degree n function/equation: f(x) = anxn+an–1xn–1+ … +a2x2 +a1x +a0, an ≠ 0 • Graph: At mostn – 1 turning points At mostn x-intercepts • End behavior: depends on aandn • For a > 0 : • For a < 0 : opposite behavior from above ifn odd or even ifn odd ifn even Section 3.2 v5.0

  9. y y y y x x x x Polynomial Function Examples • Sketch the graphs of the following: • 1.f(x) = 3x2+ 1 • 3. f(x) = x3 – x • 2. f(x) = –3x2+ 1 • 4. f(x) = –x3+ x Section 3.2 v5.0

  10. y y y y x x x x Polynomial Function Examples • Sketch the graphs (continued): • 5.f(x) = x4 – 5x2 + 4 • 7. f(x) = x5 – 5x3+ 4x • 6. f(x) = –x4+ 5x2 – 4 • 8. f(x) = –x5+ 5x3 – 4x Section 3.2 v5.0

  11. Think about it ! Section 3.2 v5.0

More Related