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Section 3.2Polynomial Functions and
Their Graphs
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Smooth, Continuous Graphs
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Polynomial functions of degree 2 or higher have graphs that are smooth and continuous. By smooth, we mean that the graphs contain only rounded curves with no sharp corners. By continuous, we mean that the graphs have no breaks and can be drawn without lifting your pencil from the rectangular coordinate system.
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Notice the breaks and lack of smooth curves.
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End Behavior of Polynomial Functions
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Odd-degree polynomial
functions have graphs with
opposite behavior at each end.
Even-degree polynomial
functions have graphs with the
same behavior at each end.
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Example
Use the Leading Coefficient Test to determine the end behavior of the graph of f(x)= - 3x3- 4x + 7
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Example
Use the Leading Coefficient Test to determine the end behavior of the graph of f(x)= - .08x4- 9x3+7x2+4x + 7
This is the graph that you get with the standard viewing window. How do you know that you need to change the window to see the end behavior of the function? What viewing window will allow you to see the end behavior?
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Zeros of Polynomial Functions
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If f is a polynomial function, then the values of x for which f(x) is equal to 0 are called the zeros of f. These values of x are the roots, or solutions, of the polynomial equation f(x)=0. Each real root of the polynomial equation appears as an x-intercept of the graph of the polynomial function.
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Find all zeros of f(x)= x3+4x2- 3x - 12
3 2
2
2
2
By definition, the zeros are the values of x
for which f(x) is equal to 0. Thus we set
f(x) equal to 0 and solve for x as follows:
(x 4 ) (3 12) 0
x (x 4) 3(x 4) 0
x+4 x - 3 0
x+4=0 x - 3=0
x=-4
x x
2 x 3
x = 3
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Example
Find all zeros of x3+2x2- 4x-8=0
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Multiplicity of x-Intercepts
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2 2For f(x)=-x ( 2) , notice that each
factor occurs twice. In factoring this equation
for the polynomial function f, if the same
factor x- occurs times, but not +1 times,
we call a zero with multip
x
r k k
r
licity . For the
polynomial above both 0 and 2 are zeros with
multiplicity 2.
k
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3 2
3 2
2
2
Find the zeros of 2 4 8 0
2 4 8 0
x 2 4( 2) 0
2 4 0
x x x
x x x
x x
x x
2 2 2 0
2 has a multiplicity of 2, and 2 has a multiplicity of 1.
Notice how the graph touches at -2 (even multiplicity),
but crosses at 2 (odd multiplicity).
x x x
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Graphing Calculator- Finding the Zerosx3+2x2- 4x-8=0
One of the zeros
The other zero
Other zero
One zero of the function
The x-intercepts are the zeros of the function. To find the zeros, press 2nd Trace then #2. The zero -2 has multiplicity of 2.
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Example
Find the zeros of f(x)=(x- 3)2(x-1)3 and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.
Continued on the next slide.
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Example
Now graph this function on your calculator. f(x)=(x- 3)2(x-1)3
x
y
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The Intermediate Value Theorem
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Show that the function y=x3- x+5 has a zero between - 2 and -1.
3
3
f(-2)=(-2) ( 2) 5 1
f(-1)=(-1) ( 1) 5 5
Since the signs of f(-1) and f(-2) are opposites then
by the Intermediate Value Theorem there is at least one
zero between f(-2) and f(-1). You can also see th
ese values
on the table below. Press 2nd Graph to get the table below.
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Example
Show that the polynomial function f(x)=x3- 2x+9 has a real zero between - 3 and - 2.
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Turning Points of Polynomial functions
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The graph of f(x)=x5- 6x3+8x+1 is shown below. The graph has four smooth turning points. The polynomial is of degree 5. Notice that the graph has four turning points. In general, if the function is a polynomial function of degree n, then the graph has at most n-1 turning points.
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A Strategy for Graphing Polynomial Functions
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Example
Graph f(x)=x4- 4x2 using what you have learned in this section.
x
y
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Example
Graph f(x)=x3- 9x2 using what you have learned in this section.
x
y
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(a)
(b)
(c)
(d)
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function f(x)=x3- 9x2 +27
falls left, rises right
rises left, falls right
rises left, rises right
falls left, falls right
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(a)
(b)
(c)
(d)
-3 touches, 1 touches
-3 crosses, 1 crosses
-3 touches, 1 crosses
-3 crosses, 1 touches
State whether the graph crosses the x-axis, or touches the x-axis and turns around at the zeros of 1, and - 3.
f(x)=(x-1)2(x+3)3