5.2 Notes: Graphing of Polynomial Functions

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5.2 Notes Graphing of Polynomial Functions
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Features of Graphs of Polynomial Functions
1. Graphs are continuous
2. At most, n 1 turns, where n degree of
polynomial
Ex. x4 has at most ____ turns
3
y 2x3 x
y x4 9x2
3

• If the leading coefficient is positive, the graph
  rises to the right. If it negative, the graph
  falls to the right.

y 2x3 x
y x4 9x2
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Example 1 State the maximum number of turns and
describe the right behavior in the graph of each
function
a.
Positive coefficient
n 1
3 1
right end behavior ____
?
turns ____
2
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Example 1 State the maximum number of turns and
describe the right behavior in the graph of each
function
b.
n 1
2 1
Negative coefficient
right end behavior ____
?
turns ____
1
6
Example 1 State the maximum number of turns and
describe the right behavior in the graph of each
function
c.
x4
Positive coefficient
n 1
4 1
right end behavior ____
?
turns ____
3
7
Example 1 State the maximum number of turns and
describe the right behavior in the graph of each
function
d.
x3
Positive coefficient
n 1
3 1
right end behavior ____
?
turns ____
2
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Even and Odd functions
1. If the degree of the polynomial is even, the
graph has the same behavior to the left and right
(rise-rise or fall-fall)
y x4 9x2
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Even and Odd functions
2. If the degree of the polynomial is odd, the
graph has the opposite behavior to the left and
right (rise-fall or fall-rise)
y 2x3 x
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Example 3 Describe the left and right
behaviors of the graphs.
a.
Negative coefficient
Odd
Left end behavior ____
Right end behavior ____
?
?
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Example 3 Describe the left and right
behaviors of the graphs.
b.
Positive coefficient
Even
Left end behavior ____
Right end behavior ____
?
?
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Example 3 Describe the left and right
behaviors of the graphs.
c.
Negative coefficient
Even
Left end behavior ____
Right end behavior ____
?
?
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Example 3 Describe the left and right
behaviors of the graphs.
d.
Negative coefficient
Odd
Left end behavior ____
Right end behavior ____
?
?
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Intercepts
The y-intercept of the graph of a function occurs
when ________.
x 0.
(0, ___)
zeros
The x-intercepts are the _______ of the function.
At an x-intercept, the value of the function is
______.
0
y 0 (____, 0)
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Example 4 Sketch the graph of
x3
2

1. Max of turns _________
2. Right End Behavior _____
3. Left End Behavior ______
4. x intercepts __________
5. y intercept(s) __________

(0-3)(02)(0-1)
6
?
?
3,
2,
1
6
16
2

1. Max of turns _________
2. Right End Behavior _____
3. Left End Behavior ______
4. x intercepts __________
5. y intercept(s) __________

?
?
3,
2,
1
6
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Example 5 Sketch the graph of
x3
2

1. Max of turns _________
2. Right End Behavior _____
3. Left End Behavior ______
4. x intercepts __________
5. y intercept(s) __________

(0)(0-3)(03)
?
0
?
0,
3,
1
0
18
2

1. Max of turns _________
2. Right End Behavior _____
3. Left End Behavior ______
4. x intercepts __________
5. y intercept(s) __________

?
?
0,
3,
1
0
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Example 6 Sketch the graph of
x4
3

1. Max of turns _________
2. Right End Behavior _____
3. Left End Behavior ______
4. x intercepts __________
5. y intercept(s) __________

-(0-2)(01)(0-4)(03)
?
24
?
2,
1,
4,
3
24
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3

1. Max of turns _________
2. Right End Behavior _____
3. Left End Behavior ______
4. x intercepts __________
5. y intercept(s) __________

?
?
2,
1,
4,
3
24
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Example 5 Sketch the graph of
2

• Max of turns _________
• Right End Behavior _____
• Left End Behavior ______
• x intercepts __________

?
?
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( )( )
x2
(x 3)
4(
x 3)
(x2 4)(x 3)
x
2
x
2
2x
2x
(x 2)(x 2)(x 3)
2,
2,
3
x intercepts _________
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Example 5 Sketch the graph of
2

1. Max of turns _________
2. Right End Behavior _____
3. Left End Behavior ______
4. x intercepts __________
5. y intercept(s) __________

?
?
2,
2,
3
12
24
2

1. Max of turns _________
2. Right End Behavior _____
3. Left End Behavior ______
4. x intercepts __________
5. y intercept(s) __________

?
?
2,
2,
3
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