Polynomial Functions: Graphs, Applications, and Models

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3.4
Polynomial Functions Graphs, Applications, and
Models
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Graphing Functions of the Form f(x) axn
Sketch the graph of f(x) x2
Sketch the graph of f(x) x3
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Graphing Functions of the Form f(x) axn
Sketch the graph of f(x) -x3
Sketch the graph of f(x) - x2
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End Behavior of Graphs of Polynomial Functions
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End Behavior of Graphs of Polynomial Functions
continued
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Odd Degree

• Typical graphs of polynomial functions of odd
  degree suggest that for every polynomial function
  f of odd degree there is at least one real value
  of x that make f(x) 0. The zeros are the
  x-intercepts of the graph.

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Even Degree

• A polynomial of even degree has a range of the
  form (??, k or k, ?) for some real number k.

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Turning Points (Relative Maxima or Relative
Minima)

• A polynomial function of degree n has at most n
  ? 1 turning points, with at least one turning
  point between each pair of successive zeros.
• The end behavior of a polynomial graph is
  determined by the dominating term, that is, the
  term of greatest degree.

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Graphing Techniques

• A comprehensive graph of a polynomial function
  will show the following characteristics.
• all x-intercepts (zeros)
• the y-intercept
• all turning points (maxima and minima)
• enough of the domain to show the end behavior
• When you are finished, you should be able to
  determine the intervals for which the function is
  increasing or decreasing and determine the domain
  and the range.

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Example

• Graph f(x) 3x3 4x2 ? 5x ? 2.

x-intercepts
(-2,0)
(-1/3,0)
(1,0).
y-intercepts
(0, -2)
(-1.31, 4.67)
Maxima Minima
(0.42, -3.17)
(-8, 8)
Domain
y1 3x3 4x2 5x 2 y2 0
(-8, 8)
Range
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x-intercepts, Zeros, Solutions, and Factors

• If (c,0) is an x-intercept of the graph of y
  f(x), then c is a zero of f, f(c) 0, x c is a
  solution of f(x) 0, and x ? c is a factor of
  f(x).
• Intermediate Value Theorem for Polynomials
• If f(x) defines a polynomial function with only
  real coefficients, and if for real numbers a and
  b, the values f(a) and f(b) are opposite in sign,
  then there exists at least one real zero between
  a and b.

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Applying the Intermediate Value Theorem (IVT)

• Use the Intermediate Value Theorem to show that
  f(x) 2x3 ? 8x2 x 15 has a zero between ?1
  and ?2.

x -1.158
f (x) 0
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Homework

• Pg 343
• 1 61 eoo