Polynomials and Rational Functions 2'1

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Polynomials and Rational Functions (2.1)

• The shape of the graph of a polynomial function
  is related to the degree of the polynomial

2
Shapes of Polynomials

• Look at the shape of the odd degree polynomials

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Graph of Odd polynomial
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Graph of Odd Polynomial
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Graphs of even degree polynomials

• Now, look at the shape of the even degree
  polynomial

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Graph of even degree polynomial

• Here is another example of an even degree
  polynomial

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Generalization

• The graphs of odd-degree polynomials start
  negative, end positive and cross the x-axis at
  least once.
• The even-degree polynomial graphs start positive,
  end positive, and may not cross the x axis at all

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Characteristics of polynomials

• Graphs of polynomials are continuous. One can
  sketch the graph without lifting up the pencil.
• 2. Graphs of polynomials have no sharp corners.
• 3. Graphs of polynomials usually have turning
  points, which is a point that separates an
  increasing portion of the graph from a decreasing
  portion.

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Turning points and x intercepts

• Theorem 1 Turning points and x Intercepts of
  Polynomials
• The graph of a polynomial function of positive
  degree n can have at most n-1 turning points and
  can cross the x axis at most n times.

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Largest value of the roots of a polynomial

• Theorem 2 Maximum value of an x-intercept of a
  polynomial. If r is a zero of the polynomial
  P(x) this means that P(r) 0. For example,
• is a second degree polynomial . and
• , so r 4 is a zero of the polynomial as well as
  being an x-intercept of the graph of p(x).

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Cauchys Theorem

• A theorem by a French mathematician named Cauchy
  allows one to determine the maximum value of a
  zero of a polynomial (maximum value of the
  x-intercept).
• Lets take an example the polynomial

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Cauchys Theorem

• According to this theorem
• The numbers within the absolute value symbols are
  the coefficients of the polynomial p(x).

lt 1 maximum value of
1 4 5
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Result of application of Cauchys theorem

• From this result we have , which means -5 lt r lt
  5 . This tells us that we should look for any
  potential x intercepts within the range of -5 and
  5 on the x axis. In other words, no intercepts
  (roots) will be found that are greater than 5 nor
  less than -5.

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Conclusion
, we find that the other zero is located at
(0,0). Thus, the two zeros , 0 , -4, are within
the range of -5 to 5 on the x-axis. Now, lets
try another example

• From the graph of

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An Example

• Example Approximate the real zeros of
• First step Coefficient of cubic term must equal
  one, so divide each term by three to get a new
  polynomial Q(x)

Roots of new polynomial are the same as the roots
of P(x).
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Example, continued

• Step 2 Use the theorem

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Example, continued

• Step3 We know that all possible x intercepts
  (roots) are found along the x-axis between -5 and
  5. So we set our viewing rectangle on our
  calculator to this window and graph the
  polynomial function.
• Step 4. Use the zero command on our calculator to
  determine that the root is approximately -3.19
  (there is only one root).

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Rational Functions

• Definition Rational function a quotient of two
  polynomials, P(x) and Q(x), is a rational
  function for all x such that Q(x) is not equal
  to zero.

Example Let P(x) x 5 and Q(x) x 2 then
R(x) is a rational function
that is defined for all real values of x with the
exception of 2 (Why?)
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Domain of rational functions

• Domain
• and x is a real number. This is read as the set
  of all numbers, x , such that x is not equal to
  2.
• X intercepts of a rational function To
  determine the x-intercepts of the graph of any
  function, we find the values of x for which y 0
  . In our case y 0 implies that 0
• This implies that x 5 0 or x -5 .

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Y-intercept of a rational function

• Y intercept The y intercept of a function is the
  value of y for which x 0 . Setting x 0 in the
  equation we have y , or -5/2. So, the
  y-intercept is located at ( 0, -2.5). Notice that
  the y-intercept is a point described by an
  ordered pair, not just a single number. Also,
  remember that a function can have only one y
  intercept but more than one x-intercept
• ( Why?)

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Graph of a Rational function

• Plot points near the value at which the function
  is undefined. In our case, that would be near x
  2. Plot values such as 1.5, 1.7. 1.9 and 2.1,
  2.3, 2.5. Use your calculator to evaluate
  function values and make a table.
• Determine what happens to the graph of f(x) if x
  increases or decreases without bound. That is,
  for x approaching positive infinity or x
  approaching negative infinity.
• Sketch a graph of a function through these
  points.
• 4. Confirm the results using a calculator and a
  proper viewing rectangle.

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Graph of rational function
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Conclusions

• From the graph we see that there is a vertical
  asymptote at x 2 because the graph approaches
  extremely large numbers as x approaches 2 from
  either side.
• We also see that y 0 is a horizontal asymptote
  of the function since y tends to go to zero as x
  tends to either a very large positive number or
  very small negative number.