1. Polynomial Functions of Higher Degree
Section 2.2
Precalculus PreAP/Dual, Revised ©2017
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§2.2: Polynomial Functions

2. §2.2: Polynomial Functions
End Behavior
Every graph is continuous. There are no gaps, jumps, holes, or sharp corners.
End Behavior
The leading coefficient focuses on the RIGHT side.
If it is POSITIVE, the graph will RISE TO THE RIGHT
If it is NEGATIVE, the graph will FALL TO THE RIGHT
The highest degree focuses on the LEFT side.
EVEN: The left behavior is the SAME as the right behavior; EQUAL  EVEN
ODD: The left behavior is the OPPOSITE of the right behavior; OPPOSITE  ODD
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§2.2: Polynomial Functions

3. §2.2: Polynomial Functions
Example 1
Describe the right-hand and left-hand behavior of the graph below.
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§2.2: Polynomial Functions

4. §2.2: Polynomial Functions
Example 2
Identify the end behavior of 𝒇 𝒙 =−𝟐 𝒙 𝟓 +𝟑 𝒙 𝟐 −𝟒𝒙−𝟏 and describe increasing, decreasing, or constant.
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§2.2: Polynomial Functions

5. §2.2: Polynomial Functions
Example 3
Identify the end behavior of 𝒇 𝒙 = 𝒙 𝟒 −𝟓 𝒙 𝟐 +𝟒 and describe increasing, decreasing, or constant.
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§2.2: Polynomial Functions

6. §2.2: Polynomial Functions
Your Turn
Identify the end behavior of 𝒇 𝒙 = −𝒙 𝟑 +𝟒𝒙 and describe increasing, decreasing, or constant.
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§2.2: Polynomial Functions

7. Establishing Real Zeros & Turning Points
Make the equation and solve for the zeros
Establish all multiple roots through the power of the exponent
To determine the Highest Degree, it is the amount of “peaks” or “valleys” and subtract 1.
ODD graphs are symmetrical through the origin.
EVEN graphs are always symmetrical about the vertical axis (that is, we have a mirror image through the 𝒚-axis)
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§2.2: Polynomial Functions

8. §2.2: Polynomial Functions
Example 4
Find all zeros and maximum turns of 𝒇 𝒙 =−𝟐 𝒙 𝟒 +𝟐 𝒙 𝟐 .
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§2.2: Polynomial Functions

9. §2.2: Polynomial Functions
Example 5
Find all zeros and maximum turns of 𝒇 𝒙 = 𝒙+𝟏 𝟐 𝒙−𝟐 𝒙−𝟑 𝟑 .
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§2.2: Polynomial Functions

10. §2.2: Polynomial Functions
Your Turn
Find all zeros and maximum turns of 𝒇 𝒙 = 𝒙 𝟑 −𝟏𝟐 𝒙 𝟐 +𝟑𝟔𝒙.
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§2.2: Polynomial Functions

11. Sketching the Polynomial Graph
Apply the end behavior rules
Determine the real zeros
Plot additional points by making a table and testing points
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§2.2: Polynomial Functions

12. §2.2: Polynomial Functions
Example 6
Sketch the graph 𝒇 𝒙 =𝟑 𝒙 𝟒 −𝟒 𝒙 𝟑
−∞,𝟎
𝟎, 𝟒 𝟑
𝟒 𝟑 ,∞
𝒙=−𝟏
𝒚=𝟕
𝒙=𝟏
𝒚=−𝟏
𝒙=𝟐
𝒚=𝟏𝟔
−∞,0
0, 4 3
4 3 ,∞
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§2.2: Polynomial Functions

13. §2.2: Polynomial Functions
Example 7
Sketch the graph 𝒇 𝒙 =− 𝒙 𝟒 +𝟑 𝒙 𝟑
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§2.2: Polynomial Functions

14. §2.2: Polynomial Functions
Your Turn
Sketch the graph 𝒇 𝒙 =𝟐 𝒙 𝟑 −𝟔 𝒙 𝟐
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§2.2: Polynomial Functions

15. §2.2: Polynomial Functions
Assignment
Page all, odd, odd (A to C and then sketch the graph), 43, 47
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§2.2: Polynomial Functions