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Polynomial Functions and their Graphs
Section 3.1
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General Shape of Polynomial Graphs
The graph of polynomials are smooth, The graph of polynomials are smooth, unbroken lines or curves, with no sharp unbroken lines or curves, with no sharp corners or cusps corners or cusps (see p. 251).
Every Polynomial function is defined and continuous for all real numbers.
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Review
General polynomial formula
a0, a1, … ,an are constant coefficients n is the degree of the polynomial Standard form is for descending powers of x aannxxn
n is said to be the “leading termleading term”
11 1 0( ) ...n n
n nP x a x a x a x a
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Family of Polynomials
Constant polynomial functions f(x) = a
Linear polynomial functions f(x) = mx + b
Quadratic polynomial functions f(x) = ax2 + bx + c
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Family of Polynomials
Cubic polynomial functions f(x) = a x3 + b x2 + c x + d 3rd degree polynomial
Quartic polynomial functions f(x) = a x4 + b x3 + c x2+ d x + e 4th degree polynomial
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Polynomial “End BehaviorEnd Behavior”
Consider what happens when x gets very large in positive and negative direction Called “end behavior” Also “long-run” behavior
Basically, the leading term Basically, the leading term aannxxnn
dominates the shape of graphdominates the shape of graph There are 4 possible scenarios:
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End Behavior
Discuss end behavior for the following graphs:
4 3( ) 2 5 4 7P x x x x
5 4 2( ) 2 6 10P x x x x 5 3( ) 3 5 2P x x x x
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Compare Graph Behavior
Consider the following graphs: f(x) = x4 - 4x3 + 16x - 16 g(x) = x4 - 4x3 - 4x2 +16x h(x) = x4 + x3 - 8x2 - 12x
Graph these on the window
-8 < x < 8 and 0 < y < 4000 Decide how these functions are alike or
different, based on the view of this graph
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Compare Graph Behavior
From this view, they appear very similar
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Compare “Short Run” Behavior
Now Change the window to be-5 < x < 5 and -35 < y < 15
How do the functions appear to be different from this view?
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Compare Short Run Behavior
Differences? Real zeros Local extrema Complex zeros
Note: The standard form of the polynomials does not give any clues as to this short run behavior of the polynomials:
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Using Zeros to Graph PolynomialsUsing Zeros to Graph Polynomials
Consider the following polynomial: p(x) = (x - 2)(2x + 3)(x + 5)
What will the zeros be for this polynomial? x = 2 x = -3/2 x = -5
How do you know? Zero-Factor Property: If a*b = 0 then, we know
that either a = 0 or b = 0 (or both)
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Guidelines to Graphing
ZerosZeros Test Points (like table of signs)Test Points (like table of signs) End BehaviorEnd Behavior Graph (a smooth curve through all Graph (a smooth curve through all
known points)known points)
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Intermediate Value Theorem
If P is a polynomial function and P(a) and P(b) have opposite signs, then there is at least one value c between a and b for which P(c) = 0.
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Theorem
Local Extrema of Polynomial Functions: A polynomial function of degree A polynomial function of degree nn has at most has at most nn - 1 local extrema. - 1 local extrema.
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Local Extrema (turning points)
Local Extrema – a point (x,y) on the graph where the graph changes from increasing to decreasing or vice-versa.
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