1. College Algebra Chapter 3 Polynomial and Rational Functions
Section 3.2
Introduction to Polynomial Functions
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2. Concepts Determine the End Behavior of a Polynomial Function
Identify Zeros and Multiplicities of Zeros
Sketch a Polynomial Function
Apply the Intermediate Value Theorem

3. Concept 1 Determine the End Behavior of a Polynomial Function

4. Determine the End Behavior of a Polynomial Function (1 of 5)
To analyze polynomial functions with multiple terms, one important characteristic is the “end behavior” of the function. This is, what general direction does the function follow as x approaches  or –?

5. Determine the End Behavior of a Polynomial Function (2 of 5)
Notation for Infinite Behavior of y = f(x) x  ∞ “x approaches infinity” (x becomes infinitely large in the positive direction) x  - ∞ “x approaches negative infinity” ( x becomes infinitely “large” in the negative direction) f(x)  ∞ “f(x) approaches infinity” (the y value becomes infinitely large in the positive direction) f(x)  - ∞ “f(x) approaches negative infinity” (the y value becomes infinitely “large” in the negative direction)

6. Determine the End Behavior of a Polynomial Function (3 of 5)
we can observe the general direction the function follows by looking at the leading term

7. Determine the End Behavior of a Polynomial Function (4 of 5)
As x  -∞, f(x)  ∞.
As x ∞, f(x)  ∞.
As x  -∞, f(x)  -∞.
As x  ∞, f(x)  -∞.

8. Determine the End Behavior of a Polynomial Function (5 of 5)
As x  -∞, f(x)  -∞.
As x ∞, f(x)  ∞.
As x  -∞, f(x)  ∞.
As x  ∞, f(x)  -∞.

9. Example 1
Use the leading term to determine the end behavior of the graph of the function.

10. Example 2
Use the leading term to determine the end behavior of the graph of the function.

11. Skill Practice 1
Use the leading term to determine the end behavior of the graph of the function.

12. Concept 2 Identify Zeros and Multiplicities of Zeros

13. Identify Zeros and Multiplicities of Zeros (1 of 2)
The values of x in the domain of f for which f (x) = 0 are called the zeros of the function.
These are the real solutions (or roots) of the equation f(x) = 0 and correspond to the x-intercepts of the graph of y = f(x).
If a polynomial function has a factor (x – c) that appears exactly k times, then c is a zero of multiplicity k.
If c is a zero of ODD multiplicity, then the graph CROSSES the x-axis at c.
If c is a zero of EVEN multiplicity, then the graph TOUCHES the x-axis at c.

14. Identify Zeros and Multiplicities of Zeros (2 of 2)
The turning points of a polynomial function are the points where the function changes from increasing to decreasing or vice versa.
The turning points correspond to points where the function has relative maxima and relative minima.
Turning points: The graph of a polynomial function of degree n has at most n – 1 turning points.

15. Example 3
Determine the zeros of the function defined by

16. Skill Practice 2
Find the zeros of the function defined by

17. Skill Practice 3
Find the zeros of the function defined by

18. Example 4
Determine the zeros and their multiplicities for

19. Skill Practice 4
Determine the zeros and their multiplicities for the given functions.

20. Example (1 of 3)
What effect do the multiplicities of the zeros have on the graph? (Note how the lead term changes behavior.) g(x) = x(x – 1)(x + 3)

21. Example (2 of 3)
What effect do the multiplicities of the zeros have on the graph? (Note how the lead term changes behavior.)

22. Example (3 of 3)
What effect do the multiplicities of the zeros have on the graph? (Note how the lead term changes behavior.)

23. Example 5
Given f(x), identify the zeros of the function and whether there is a cross point or a touch point at each zero.

24. Example 6
Give the maximum number of turning points for

25. Concept 3 Sketch a Polynomial Function

26. Sketch a Polynomial Function (1 of 3)
To graph a polynomial function defined by y = f(x),
Use the leading term to determine the end behavior of the graph. 
Determine the y-intercept by evaluating f(0).
Determine the real zeros of f and their multiplicities (these are the x-intercepts of the graph of f).
Plot the x- and y-intercepts and sketch the end behavior.

27. Sketch a Polynomial Function (2 of 3)
Draw a sketch starting from the left-end behavior. Connect the x- and y-intercepts in the order that they appear from left to right.
Remember: cross the x-axis if the zero has an odd multiplicity, touch but not cross if the zero has an even multiplicity.
If the test for symmetry is easy to apply, plot additional points.
f is an even function (symmetric to the y-axis) if f(–x) = f(x).
f is an odd function (symmetric to the origin) if f(–x) = –f(x).

28. Sketch a Polynomial Function (3 of 3)
Plot more points if a greater level of accuracy is desired. In particular, to estimate the location of turning points, find several points between two consecutive x-intercepts.

29. Example 7 Sketch the function.
General Shape? Turns? Intercepts? Cross or touch?

30. Example 8
Sketch the function.

31. Concept 4 Apply the Intermediate Value Theorem

32. Apply the Intermediate Value Theorem
Intermediate Value Theorem: Let f be a polynomial function. For a < b, if f (a) and f (b) have opposite signs, then f has at least one zero on the interval [a, b].

33. Example 9
Show that f(x) has a zero on the interval [–1, 0].

34. Skill Practice 5
Show that f(x) has a zero on the interval [-4, -3].

35. Skill Practice 6
Graph

36. Skill Practice 7
Graph