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Chapter 3

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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

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Chapter 3

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  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

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