1. 4.3 - End Behavior and Graphing
Worksheet Key 1)
4x3 – 17x2 – 39x – 18 2)
x4 – 9x3 + 39x2 – 225x + 350 3)
x4 + 4x3 + 4x2 + 4x + 3 4)
x5 – 4x4 - 2x3 + 8x2 - 24x + 96 5)
{ +3/2i, -1, 3 } 6)
{ + 𝒊 𝟑 , 1, -4 } 7)
{ 1 + i, -3, 4 } 8)
{ + 2i, -6 } 9)
omit 1)
x3 + 2x2 + 13x + 10 2)
x3 + 4x2 + 3x 3)
x3 - 10x2 + 29x - 20 4)
x3 - 7x2 + 36 5)
x3 - 12x2 + 44x - 48 6)
x3 - 25x
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4.3 - End Behavior and Graphing

2. Graphs of Polynomial Functions
Section 4.3
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4.3 - End Behavior and Graphing

3. 4.3 - End Behavior and Graphing
Graphing End Behavior
Every graph is continuous. There are no gaps, jumps, holes, or sharp corners.
End Behavior
When a polynomial function has an ODD degree, one end of its graph shoots upward and the other end downwards.
When a polynomial function has an EVEN degree, both ends will shoot upwards or downwards
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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4.3 - End Behavior and Graphing

4. 4.3 - End Behavior and Graphing
Example 1
How would you describe the graph from negative x to positive x?
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4.3 - End Behavior and Graphing

5. 4.3 - End Behavior and Graphing
Your Turn
How would you describe the graph from negative x to positive x?
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4.3 - End Behavior and Graphing

6. 4.3 - End Behavior and Graphing
Example 2
Identify the end behavior of P(x) = 2x5 + 3x2 – 4x – 1
As x  –∞, P(x) –∞ As x  +∞, P(x) +∞
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4.3 - End Behavior and Graphing

7. 4.3 - End Behavior and Graphing
Your Turn
Identify the end behavior of P(x) = –0.6x5 + 4x2 + x – 4
As x  –∞, P(x) +∞ As x  +∞, P(x) –∞
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4.3 - End Behavior and Graphing

8. 4.3 - End Behavior and Graphing
Multiplicity
Intercepts
The graph of any polynomial function of a degree n has only one y-intercept, which is equal to the constant term
Has at most n x-intercepts
Multiplicity is the amount of many times a particular number is a zero for a given polynomial
Given (x – k)n
If n is ODD, the zero CROSSES the x-axis
If n is EVEN, the zero TOUCHES the x-axis
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4.3 - End Behavior and Graphing

9. 4.3 - End Behavior and Graphing
Example 3
Given the graph below, state the zero and identify whether each function touches or crosses the x-axis crosses the x-axis and identify whether it is odd or even, y-intercept, and sketch the graph
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4.3 - End Behavior and Graphing

10. 4.3 - End Behavior and Graphing
Example 4
Find all the zeros of f(x) = (x + 1)2(x + 2)(x – 3)3, state the multiplicity of each zero, and identify whether each function touches or crosses the x-axis and identify whether it is odd or even, y-intercept, and sketch the graph
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4.3 - End Behavior and Graphing

11. 4.3 - End Behavior and Graphing
Your Turn
Find all the zeros of f(x) = (x – 1)4(x + 1)(x + 3)5, state the multiplicity of each zero, and identify whether each function touches or crosses the x-axis and identify whether it is odd or even and sketch the graph
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4.3 - End Behavior and Graphing

12. 4.3 - End Behavior and Graphing
Functions
To determine the Highest Degree, it is the amount of “peaks” or “valleys” and ADD 1. Take the number.
ODD graphs are symmetrical through the origin.
EVEN graphs is always symmetrical about the vertical axis (that is, we have a mirror image through the y-axis)
The LEADING COEFFICIENT is the right side of the graph. If it rises, it is positive. If it falls, it is negative.
To determine the SMALLEST Degree, it is the amount of “peaks” or “valleys” and ADD 1.
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4.3 - End Behavior and Graphing

13. 4.3 - End Behavior and Graphing
Example 5
Given the graph below, A) Is the highest degree even or odd? B) Is the leading coefficient positive or negative? C) What are its real zeros? D) What is the smallest degree of the function?
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4.3 - End Behavior and Graphing

14. 4.3 - End Behavior and Graphing
Example 6
Given the graph below, A) Is the highest degree even or odd? B) Is the leading coefficient positive or negative? C) What are its real zeros? D) What is the smallest degree of the function?
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4.3 - End Behavior and Graphing

15. 4.3 - End Behavior and Graphing
Example 7
Given the graph below, A) Is the highest degree even or odd? B) Is the leading coefficient positive or negative? C) What are its real zeros? D) What is the smallest degree of the function?
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4.3 - End Behavior and Graphing

16. 4.3 - End Behavior and Graphing
Your Turn
Given the graph below, A) Is the highest degree even or odd? B) Is the leading coefficient positive or negative? C) What are its real zeros? D) What is the smallest degree of the function?
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4.3 - End Behavior and Graphing

17. 4.3 - End Behavior and Graphing
Assignment
Page odd and include end behavior, 15 and 17 label multiplicity, zeros crosses or touches, and if multiplicity is odd/even, all, 43, 44
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4.3 - End Behavior and Graphing