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3-3 Polynomial Graphs
Unit 3 Quadratic and Polynomial Functions
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Concepts and Objectives
Objective #11
Identify and interpret vertical and horizontal
translations
Identify the end behavior of a function Identify the number of turning points of a function
Use the Intermediate Value Theorem and the
Boundedness Theorem to locate zeros of a function
Use the calculator to approximate real zeros
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Graphing Polynomial Functions
If we look at graphs of functions of the form ,
we can see a definite pattern: n f x ax
2 f x x 3 g x x
4h x x 5 j x x
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Graphing Polynomial Functions
For a polynomial function of degree n
Ifn is even, the function is an even function.
An even function has a range of the form , k] or
[k, for some real number k. Ifn is odd, the function is an odd function.
The range of an odd function is the set of all real
numbers, , .
For odd functions, the graph will have at least onereal zero (x-intercept).
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Graphing Polynomial Functions
Compare the graphs of the two functions:
2 f x x
2 2 g x x
2h x x
2
1 j x x
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Graphing Polynomial Functions
Vertical translation
The graph of is shifted kunits up if
k> 0 and |k| units down ifk< 0.
Horizontal translation
The graph of is shifted h units to the
right ifh > 0 and |h| units to the left ifh < 0.
n f x ax k
n
f x a x h
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Graphing Polynomial Functions
Example: Write the equation of the function of degree 3
graphed below.
This is an odd function.The vertex has been shifted up 3
units and to the right2 units.
So, its going to be something like:
3
2 3 f x a x
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Graphing Polynomial Functions
Example (cont.):
To determine whata is, we can picka point and plug in values:
3 4f
3
3 3 2 3 4f a
3 4a
1a
3
2 3 f x x
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Multiplicity and Graphs
What is the multiplicity of ?
The zero 4 has multiplicity 5
The multiplicity of a zero and whether the function iseven or odd determines what the graph does at a zero.
A zero of multiplicity one crosses thex-axis.
A zero of even multiplicity turns or bounces at the
x-axis . A zero of odd multiplicity greater than one crosses
thex-axis and wiggles.
5
4 g x x
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Turning Points and End Behavior
The point where a graph changes direction (bounces
or wiggles) is called a turning pointof the function.
A function of degree n will have at mostn 1 turning
points, with at least one turning point between each
pair of adjacent zeros.
The end behaviorof a polynomial graph is determined by
the term with the largest exponent (the dominating
term).
For example, has the same end
behavior as . 32 8 9 f x x x
32 f x x
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End Behavior
Example: Use symbols for end behavior to describe the
end behavior of the graph of each function.
1.
2.
3.
4 22 8
f x x x x even functionopens downward
3 23 5 g x x x x odd functionincreases
5 3
1h x x x odd functiondecreases
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Intermediate Value Theorem
This means that if we plug in two numbers and the
answers have different signs (one positive and one
negative), the function has to have crossed thex-axisbetween the two values.
Iffx defines a polynomial function with only real
coefficients, and if for real numbers a and b, the
valuesfa andfb are opposite in sign, then there
exists at least one real zero between a and b.
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Intermediate Value Theorem
Example: Show that has a real
zero between 2 and 3.
You can either plug the values in, or you can usesynthetic division to evaluate each value.
Since the sign changes, there must be a real zero
between 2 and 3.
3 22 1 f x x x x
2 1 2 1 1
1
2
0
0
1
2
1
3 1 2 1 1
1
3
1
3
2
6
7
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Intermediate Value Theorem
Iffa andfb are not opposite in sign, it does not
necessarily mean that there is no zero between a and b.
Consider the function, , at1 and 3: 2 2 1 f x x x
f1 = 2 > 0 andf3= 2 >0
This would imply that there is no
zero between 1 and 3, but we cansee thatfhas two zeros between
those points.
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Boundedness Theorem
Letfx be a polynomial function of degree n 1 with
real coefficients and with a positive leading coefficient.
Iffx is divided synthetically byx c, and
(a) if c > 0 and all numbers in the bottom row are
nonnegative, thenfx has no zeros greater than c;
(b) if c < 0 and the numbers in the bottom row
alternate in sign, thenfx has no zero less than c.
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Boundedness Theorem
Example: Show that the real zeros of
satisfy the following conditions,
a) No real zero is greater than 1b) No real zero is less than 2
4 25 3 7 f x x x x
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Boundedness Theorem
Example: Show that the real zeros of
satisfy the following conditions,
a) No real zero is greater than 1
Since the bottom row numbers are all 0,fx has
no zero greater than 1.
4 25 3 7 f x x x x
1 1 0 5 3 7
9
9
6
6
1
1
1
1 2
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Boundedness Theorem
Example: Show that the real zeros of
satisfy the following conditions,
b) No real zero is less than 2
Since the signs of the bottom numbers alternate,fx
has no zero less than 2.
4 25 3 7 f x x x x
2 1 0 5 3 7
30
15
18
9
4
2
2
1 23
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Approximating Real Zeros
Example: Approximate the real zeros of
Step 1: Enter the function into
3 28 4 10 f x x x x
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Approximating Real Zeros
Example: Approximate the real zeros of
Step 2: Press and then
3 28 4 10 f x x x x
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Approximating Real Zeros
Example: Approximate the real zeros of
Step 3: Position the cursor at the farleft above thex-axis and press
Step 4: Move the cursor below the
x-axis and press
3 28 4 10 f x x x x
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Approximating Real Zeros
Example: Approximate the real zeros of
Step 5: Our first zero is at8.33594
3 28 4 10 f x x x x
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Approximating Real Zeros
Example: Approximate the real zeros of
Repeat steps 1-5 to find the next twozeros
#2: 0.9401088
3 28 4 10 f x x x x
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Approximating Real Zeros
Example: Approximate the real zeros of
Repeat steps 1-5 to find the next twozeros
#2: 0.9401088
#3: 1.2760488
3 28 4 10 f x x x x
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Homework
College Algebra
Page 352: 21-27 (3), 48-69 (3), 81
Turn In: 24, 48, 54, 60, 63, 66