1. Evaluate and Graph Polynomial Functions
Section 5.2
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5.2: Evaluate and Graph Polynomial Functions

2. Common Polynomial Functions
POLYNOMIAL is an expression made with constants, variables and exponents, which are combined using an operation.
The exponents can only be WHOLE numbers. Another word for exponents is a degree.
Leading Coefficient is the number in front of the highest polynomial
Types of coefficient:
First Degree polynomial is a LINEAR function
Second Degree polynomial is a QUADRATIC function
Third Degree polynomial is a CUBIC function
Fourth Degree polynomial is a QUARTIC function
Fifth Degree polynomials and beyond are known by their degree functions.
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5.2: Evaluate and Graph Polynomial Functions

3. 5.2: Evaluate and Graph Polynomial Functions
Example 1
Is 4x3 + 6x4 + 5x2 – 7x + 10 a polynomial? If so, write it in standard form and identify its degree, type, leading coefficient, and constant term.
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5.2: Evaluate and Graph Polynomial Functions

4. 5.2: Evaluate and Graph Polynomial Functions
Example 2
Is a polynomial? If so, write it in standard form and identify its degree, type, leading coefficient, and constant term.
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5.2: Evaluate and Graph Polynomial Functions

5. 5.2: Evaluate and Graph Polynomial Functions
Example 3
Is 3 + x3 – 1/4x2 a polynomial? If so, write it in standard form and identify its degree, type, leading coefficient, and constant term.
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5.2: Evaluate and Graph Polynomial Functions

6. 5.2: Evaluate and Graph Polynomial Functions
Your Turn
Is a polynomial? If so, write it in standard form and identify its degree, type, leading coefficient, and constant term.
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5.2: Evaluate and Graph Polynomial Functions

7. 5.2: Evaluate and Graph Polynomial Functions
End Behavior
FIRST DEGREE polynomial is a straight-line
SECOND DEGREE polynomial is a parabola
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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5.2: Evaluate and Graph Polynomial Functions 7

8. 5.2: Evaluate and Graph Polynomial Functions
Example 4
How would you describe the graph from negative x to positive x?
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5.2: Evaluate and Graph Polynomial Functions

9. 5.2: Evaluate and Graph Polynomial Functions
Example 5
How would you describe the graph from negative x to positive x?
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5.2: Evaluate and Graph Polynomial Functions

10. 5.2: Evaluate and Graph Polynomial Functions
Your Turn
How would you describe the graph from negative x to positive x?
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5.2: Evaluate and Graph Polynomial Functions

11. 5.2: Evaluate and Graph Polynomial Functions
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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5.2: Evaluate and Graph Polynomial Functions 11

12. 5.2: Evaluate and Graph Polynomial Functions
Example 6
Identify the end behavior of f(x) = 2x5 + 3x2 – 4x – 1
What is the leading coefficient?
Is it positive or negative?
As x  +∞, f(x) +∞
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5.2: Evaluate and Graph Polynomial Functions

13. 5.2: Evaluate and Graph Polynomial Functions
Example 6
Identify the end behavior of f(x) = 2x5 + 3x2 – 4x – 1
What is the highest degree?
Is it odd or even?
As x  +∞, f(x) +∞
As x  –∞, f(x) –∞
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5.2: Evaluate and Graph Polynomial Functions

14. 5.2: Evaluate and Graph Polynomial Functions
Example 6
Identify the end behavior of f(x) = 2x5 + 3x2 – 4x – 1
As x  –∞, f(x) –∞ As x  +∞, f(x) +∞
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5.2: Evaluate and Graph Polynomial Functions

15. Example 7 Identify the end behavior of f(x) = –0.6x5 + 4x2 + x – 4
As x  –∞, f(x) +∞ As x  +∞, f(x) –∞
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5.2: Evaluate and Graph Polynomial Functions

16. Example 8 Identify the end behavior of f(x) = x4 – 3x3 + 5x
As x  –∞, f(x) +∞ As x  +∞, f(x) +∞
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5.2: Evaluate and Graph Polynomial Functions

17. Your Turn Identify the end behavior of f(x) = –2x4 + 8x3 – 8x2
As x  –∞, f(x) –∞ As x  +∞, f(x) –∞
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5.2: Evaluate and Graph Polynomial Functions

18. Steps in Identifying Graphs
Plug in every x with a negative x and solve
EVEN FUNCTION
Even Functions are functions that are symmetric to the y–axis.
When substituting x, the original equation is EQUAL to the new equation
C. ODD FUNCTION
Odd Functions are functions that cross the origin.
When substituting x, the original equation is OPPOSITE to the new equation
For an odd function, rotating the graph through 180 degrees would give the same graph.
D. NEITHER
Functions that is not symmetric on the y–axis and at the origin.
OPPOSITE  ODD EQUAL  EVEN
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5.2: Evaluate and Graph Polynomial Functions

19. 5.2: Evaluate and Graph Polynomial Functions
Example 9
Given f(x) = x2 – 5, is the function even, odd, or neither?
EVEN FUNCTION
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5.2: Evaluate and Graph Polynomial Functions

20. 5.2: Evaluate and Graph Polynomial Functions
Example 9
Given f(x) = x2 – 5, is the function even, odd, or neither?
EVEN FUNCTION
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5.2: Evaluate and Graph Polynomial Functions

21. 5.2: Evaluate and Graph Polynomial Functions
Example 10
Given f(x) = –5x3 + x, is the function even, odd, or neither?
ODD FUNCTION
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5.2: Evaluate and Graph Polynomial Functions

22. 5.2: Evaluate and Graph Polynomial Functions
Example 10
Given f(x) = –5x3 + x, is the function even, odd, or neither?
ODD FUNCTION
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5.2: Evaluate and Graph Polynomial Functions

23. 5.2: Evaluate and Graph Polynomial Functions
Example 11
Given f(x) = x3 – 1, is the function even, odd, or neither?
NEITHER
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5.2: Evaluate and Graph Polynomial Functions

24. 5.2: Evaluate and Graph Polynomial Functions
Example 11
Given f(x) = x3 – 1, is the function even, odd, or neither?
NEITHER
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5.2: Evaluate and Graph Polynomial Functions

25. 5.2: Evaluate and Graph Polynomial Functions
Your Turn
Given f(x) = 2x3 – 3x2 – 4x + 4, is the function even, odd, or neither?
NEITHER
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5.2: Evaluate and Graph Polynomial Functions

26. 5.2: Evaluate and Graph Polynomial Functions
Example 12
Is this graph an odd function, even function, or neither?
ODD FUNCTION
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5.2: Evaluate and Graph Polynomial Functions

27. 5.2: Evaluate and Graph Polynomial Functions
Example 13
Is this graph an odd function, even function, or neither?
EVEN FUNCTION
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5.2: Evaluate and Graph Polynomial Functions

28. 5.2: Evaluate and Graph Polynomial Functions
Example 14
Is this graph an odd function, even function, or neither?
NEITHER
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5.2: Evaluate and Graph Polynomial Functions

29. 5.2: Evaluate and Graph Polynomial Functions
Your Turn
Is this graph an odd function, even function, or neither?
ODD FUNCTION
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5.2: Evaluate and Graph Polynomial Functions

30. 5.2: Evaluate and Graph Polynomial Functions
Assignment
Worksheet
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5.2: Evaluate and Graph Polynomial Functions