Polynomial and Rational Functions

1
Polynomial and Rational Functions

• Chapter 3

2
Polynomial Functions and Models

• 3.2

3
Polynomial Function

• A polynomial function is a function of the form
• F(x) anxn an-1xn-1a1xa0
• Where the as are real numbers and n is a
  non-negative integer.

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Degree

• The largest power of x that appears in a
  polynomial of one variable.

5
Sample Polynomials

• f(x) 0
• zero function
• no degree
• f(x) a0
• constant function
• 0 degree

6
Sample Polynomials

• f(x) a1x a0
• linear function
• degree 1
• f(x) a2x2a1xa0
• quadratic function
• degree 2

7
Polynomial or Not? Give Degree
Sample problems in class.
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Graph a Power Function

• Graph the three common points
• Graph three other points on each side of the
  origin that are symmetric

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A Power Function of Degree n

• F(x) axn
• Where a is a real number and not zero.
• Where ngt0 is an integer.

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Sample Power Functions

• F(x) 3x
• F(x) -5x2
• F(x) 8x3
• F(x) -5x4

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Properties of Even Power Functions

• The graph is symmetric with respect to
• The y-axis
• The domain is
• All reals
• The range is
• All non-negative reals
• The graph always contains the points
• (0,0), (1,1), (-1,1)

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Properties of Even Power Functions

• As the exponent n increases in magnitude, the
  graph becomes
• More vertical beyond 1 and -1
• Between 1 and -1 it lies closer to the x-axis

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Properties of Odd Power Functions

• The graph is symmetric with respect to
• The origin so f is odd
• The domain is
• All reals
• The range is
• All reals
• The graph always contains the points
• (0,0), (1,1), (-1,-1)

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Properties of Odd Power Functions

• As the exponent n increases in magnitude, the
  graph becomes
• More vertical beyond 1 and -1
• Between 1 and -1 it lies closer to the x-axis

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Create a Polynomial given the zeros and the
degree.
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Zeroes or Roots of f

• If f is a polynomial function and r is a real
  number for which f(r)0, then r is called a real
  zero of f, or root of f. If r is a real zero of
  f, then
• r is an x-intercept of the graph of f
• (x-r) is a factor of f.

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Multiple Zeros

• If the same factor occurs more than once, then r
  is called a repeated, or multiple, zero of f.

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Multiple Zeros

• If (x-r)m is a factor of a polynomial f and
  (x-r)(m1) is not a factor of f, then r is
  called a zero of multiplicity m of f.

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Given the polynomial

• List each real zero and its multiplicity.
• Determine whether the graph crosses or touches
  the x-axis at each x-intercept.
• Find the power function that f resembles for
  large values of x and x.
• Determine local maxima or minima, if they exist
  using the calculator.
• State where the graph is increasing and where it
  is decreasing.
• State the domain and range of f.

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If r is a zero of Even Multiplicity

• Sign of f(x) does not change from one side of r
  to the other side.
• Graph touches the x-axis at r.

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If r is a zero of Odd Multiplicity

• Sign of f(x) changes from one side of r to the
  other side.
• Graph crosses the x-axis at r.

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Turning Points

• Points on the graph where the graph changes from
  increasing to decreasing.

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Turning Points Theorem

• If f is a polynomial function of degree n, then f
  has at most n-1 turning points.

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End Behavior

• For large values of x, a polynomial resembles the
  graph of the power function that is the first
  term of the polynomial when placed in descending
  order of the powers of x.

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Between the Zeros

• Between the zeros, the graph of a polynomial
  function is either above or below the x-axis.

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Examples

• 46,52
• 58,80
• 96 d,e,f,g
• Homework pg. 183-185 37,43
  45,51,57,79,81,