6.7/6.8 Analyzing Graphs of Polynomials

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6.7/6.8 Analyzing Graphs of Polynomials

• How do you find the local maximum and minimum on
  a polynomial graph?
• What is the maximum number of turning points
  based on the degree of polynomial?
• How do you find the equation of polynomial of the
  least degree given the x-intercepts and another
  point on the graph?

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The Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n where n gt 0,
then the equation has at least one root in the
set of complex numbers.
This means that the degree of polynomial will
tell you the number of solutions to look for.
Some of the solutions may be repeating solutions.
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Zeros, Factors, Solutions, and Intercepts
If f(x) is a polynomial function, then these
statements are equivalent.

• Zero k is a zero of the polynomial.
• Factor x k is a factor of the polynomial.
• Solution k is a solution of the polynomial
  equation f(x).
• Intercept If k is a real number then k is an
  x-intercept of the graph of the polynomial.

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Turning Points of Polynomial Functions
The graph of every polynomial function of degree
n has at most n 1 turning points. Moreover, if
a polynomial function has n distinct real zeros,
then its graph has exactly n -1 turning points.
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Local Maximum and Minimum

• The y-coordinate of a turning point is a local
  maximum of the function if the point is higher
  than all nearby points.
• The y-coordinate of a turning point is a local
  minimum of the funct if the point is lower than
  all nearby points.

Local maximum
Local minimum
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Writing a Cubic Function

• Two points determine a line.
• Three points determine a parabola
• Four points determine a cubic function

Remember
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Given Three Intercepts and a Fourth Point on the
Graph
Example Given x-intercepts (-2, 0), (-1, 0),
(1, 0) and a fourth point on the graph (0, 2),
find the equation of a polynomial with the least
degree.
1. Change the x-intercept to factor form.
Write the polynomial function in factor form with
the leading coefficient as a.
2. Substitute the x and y coordinates of the
fourth point into the equation.
3. Solve for a.
4. Substitute a into the equation.