3.4 - 1 example 1 graphing functions of the form (x) = ax n solution choose several values for x,...
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Example 1 GRAPHING FUNCTIONS OF THE FORM (x) = axn
Solution Choose several values for x, and find the corresponding values of (x), or y.
a.Graph the function.
3( )x xf
x (x)
– 2 – 8– 1 – 10 0
1 1
2 8
3( )x xf
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If the zero has even multiplicity, the graph is tangent to the x-axis at the corresponding x-intercept (that is, it touches but does not cross the x-axis there).
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If the zero has odd multiplicity greater than one, the graph crosses the x-axis and is tangent to the x-axis at the corresponding x-intercept. This causes a change in concavity, or shape, at the x-intercept and the graph wiggles there.
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Turning Points and End Behavior
The previous graphs show that polynomial functions often have turning points where the function changes from increasing to decreasing or from decreasing to increasing.
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Turning Points
A polynomial function of degree n has at most n – 1 turning points, with at least one turning point between each pair of successive zeros.
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End Behavior
The end behavior of a polynomial graph is determined by the dominating term, that is, the term of greatest degree. A polynomial of the form
11 0( ) n n
n nx a x a x a f
has the same end behavior as . ( ) nnx a xf
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End Behavior of Polynomials
Suppose that axn is the dominating term of a polynomial function of odd degree.1.If a > 0, then as and as Therefore, the end behavior of the graph is of the type that looks like the figure shown here.
We symbolize it as .
, ( ) ,x x f, ( ) .x x f
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End Behavior of Polynomials
Suppose that axn is the dominating term of a polynomial function of odd degree.2. If a < 0, then as and as
Therefore, the end behavior of the graph looks like the graph shown here.
We symbolize it as .
, ( ) ,x x f, ( ) .x x f
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End Behavior of Polynomials
Suppose that axn is the dominating term of a polynomial function of even degree.1.If a > 0, then as Therefore, the end behavior of the graph looks like the graph shown here.
We symbolize it as .
, ( ) .x x f
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End Behavior of Polynomials
Suppose that is the dominating term of a polynomial function of even degree.2. If a < 0, then as Therefore, the end behavior of the graph looks like the graph shown here.
We symbolize it as .
, ( ) .x x f
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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL
Match each function with its graph.4 2( ) 5 4x x x x f
Solution Because is of even degree with positive leading coefficient, its graph is C.
A. B. C. D.
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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL
Match each function with its graph.6 2( ) 3 4x x x x g
Solution Because g is of even degree with negative leading coefficient, its graph is A.
A. B. C. D.
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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL
Match each function with its graph.3 2( ) 3 2 4x x x x h
Solution Because function h has odd degree and the dominating term is positive, its graph is in B.
A. B. C. D.
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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL
Match each function with its graph.7( ) 4x x x k
Solution Because function k has odd degree and a negative dominating term, its graph is in D.
A. B. C. D.
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Graphing Techniques
We have discussed several characteristics of the graphs of polynomial functions that are useful for graphing the function by hand. A comprehensive graph of a polynomial function will show the following characteristics:1. all x-intercepts (zeros)2. the y-intercept3. the sign of (x) within the intervals formed by the x-intercepts, and all turning points4. enough of the domain to show the end behavior.
In Example 4, we sketch the graph of a polynomial function by hand. While there are several ways to approach this, here are some guidelines.