Sect' 7'2 Graphing Polynomial Functions

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Sect. 7.2 Graphing Polynomial Functions
Goal 1 Graph Polynomial Functions and
Locate their Zeros Goal 2 Find the Maxima
and Minima of Polynomial Functions
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Graphs of Polynomial Functions

• We have learned how to graph functions with the
  following degrees
• 0 Example f(x) 2
  horizontal line
• 1 Example f(x) 2x 3 line
• 2 Example f(x) 2x2 2x 3 parabola
• How do you graph polynomial functions with
  degrees higher than 2?

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Graph f(x) 3x3 9x 1

• Well make a table of values, then graph...

x f(x)
Notice that the graph rises to the right.
-3 -2 -1 0 1 2 3
-53 -5 7 1 -5 7 55
Notice that the graph has two turns.
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Graphs of Polynomial Functions

• are continuous (there are no breaks)
• have smooth turns
• with degree n, have at most n 1 turns (In the
  previous example, the polynomial had a degree of
  3, so there were 2 turns)
• rise to the right if the leading coefficient is
  positive (In the previous example, the leading
  coefficient was 3, so the graph rose to the
  right).
• fall to the right if the leading coefficient is
  negative.

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Graph f(x) x3 4x
Prior to graphing, predict the number of turns
and the left and right behaviors of the
graph The graph will have at most 2 turns (since
it is degree 3) The graph will fall to the right
(since the leading coefficient is 1)
x f(x)
-4 -3 -2 -1 0 1 2 3 4
48 15 0 -3 0 3 0 -15 -48
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Location Principle
Suppose y f(x) represents a polynomial function
and a and b are two numbers such that f(a) lt 0
and f(b) gt 0. Then the function has at least
one real zero between a and b.
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Use the Location Principle to locate between
which values of x each Real Zero is located for
the following function.
f(x) x3 3x2 3
x f(x)
-3 -2 -1 0 1 2 3
-51 -17 -1 3 1 -1 3
Therefore zeros are between x 1 and x 0,
between x 1 and x 2, and between x 2 and x
3.
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Relative Maximum
Relative Maximum Point on the graph of a function
that has no other nearby function with a greater
y-value
Relative Minimum Point at which no other nearby
points have lesser y-coordinates
These points are also called Turning points.
Polynomials have at most n 1 turning points
Relative Minima
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Graph f(x) x3 4x2 5
Estimate the x-coordinates at which the Relative
Maxima and Relative Minima occur.
Relative Maximum at x 0
Relative Mimimum at x? 2.66
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Graphing a Polynomial Function
Graph f(x) x4 - 2x3 - x2 2x and determine
the following
a) the real zeros b) the y-intercepts c) the
intervals where f(x) gt 0 d) relative maximum or
minimums
x lt -1
x gt 2
0 lt x lt 1
a) Zeros
-1, 0, 1, 2
d) Relative maximum or minimums
b) y-intercept
0
Relative maximum
(0.5, 0.6)
c) Intervals where f(x) gt 0
Relative minimums
x lt -1
(-0.5, -1) and (1.5, -1)
0 lt x and x lt 1
(Can also be called Absolute Minimum points,
because no other point with lesser y-value.)
x gt 2
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The weight, w, in pounds, of a patient during a
7-week illness is modeled by the cubic equation
w(n) 0.1n3 0.6n2 110, where n is the
number of weeks since the patient became ill.
a) Graph the function
b) Describe the turning points and its end
behavior
Relative Minimum point at week 4. For end
behavior, w(n) increases as n increases.
c) What trends in the patients weight does the
graph suggest?
The patient lost weight for each of the 4 weeks
after becoming ill. After 4 weeks, the patient
gained weight and continues to gain.