Section 1'6 Powers, Polynomials, and Rational Functions

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Section 1.6Powers, Polynomials, and Rational
Functions
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• A quantity y is directly proportional to a power
  of x if
• A quantity y is inversely proportional to a power
  of x if
• Functions of this form are called power functions

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• Which of the following are power functions and
  identify the k and the n (recall )

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• Power functions can be odd, even or neither
• How can we decide?
• What about the following?
• What about the end behavior of a power function
  versus an exponential
• Which grows faster?

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• What happens if we add or subtract power
  functions?
• A polynomial is a sum (or difference) of power
  functions whose exponents are nonnegative
  integers
• What determines the degree of a polynomial?
• For example
• What is the leading term in this polynomial?

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• We have the general form of a polynomial which
  can be written as
• Where n is a positive integer called the degree
  of p
• Each power function is called a term
• The constants an , an-1, a0,are called
  coefficients
• The term a0 is called the constant term
• The term anxn is called the leading term

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End Behavior The shape of the graph of a
polynomial function depends on the degree. Degree
EVEN Degree ODD
angt0
anlt0
angt0
anlt0
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• What are the zeros (or roots) of a polynomial?
• Where the graph hits the x-axis
• The input(s) that make the polynomial equal to 0
• How can we find zeros of a polynomial?
• For example, what are the zeros of
• Notice this polynomial is in its factored form
• It is written as a product of its linear factors
• A polynomial of degree n can have at most n real
  zeros

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• When a polynomial, p, has a repeated linear
  factor, then it has a multiple root
• If the factor (x - k) is repeated an even number
  of times, the graph does not cross the x-axis at
  x k. It bounces off. The higher the (even)
  exponent, the flatter the graph appears around x
  k.
• If the factor (x - k) is repeated an odd number
  of times, the graph does cross the x-axis at x
  k. It appears to flatten out. The higher the
  (odd) exponent, the flatter it appears around x
  k.

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• If r can be written as the ratio of polynomial
  functions p(x) and q(x),
• then r is called a rational function
• The long-run behavior is determined by the
  leading terms of both p and q
• These functions often have horizontal asymptotes
  which define their long run behavior

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• We have three cases
• The degree of p lt the degree of q
• The horizontal asymptote is the line y 0
• The degree of p gt the degree of q
• There is no horizontal asymptote
• The degree of p the degree of q
• The horizontal asymptote is the ratio of the
  coefficients of the leading terms of p and q

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• Lets consider the following functions
• How do we find their x-intercepts?
• What are they?
• What happens if the denominators equal 0?
• What are their horizontal asymptotes?