1. Graphs of Polynomial Functions

2. The Polynomial Functions
The key features of a polynomial graph
Leading Coefficient Test to determine the end behavior of graphs of polynomial functions.
Finding zeros of polynomial functions
Determine a polynomial equation given the zeros of the function.

3. Graphs of Polynomial Functions
Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps. x y
f (x) = x3 – 5x2 + 4x + 4 x y x y
continuous
not continuous
continuous
smooth
not smooth
polynomial
not polynomial
not polynomial
Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions.
Graphs of Polynomial Functions

4. A polynomial function is a function of the form
where n is a nonnegative integer and a1, a2, a3, … an are real numbers.
The polynomial function has a leading
coefficient an and degree n.
Examples: Find the leading coefficient and degree of each polynomial function.
Polynomial Function Leading Coefficient Degree
Polynomial Function

5. Classification of a Polynomial
Degree
Name
Example
n = 0
constant
Y = 3
n = 1
linear
Y = 5x + 4
n = 2
quadratic
Y = 2x2 + 3x - 2
n = 3
cubic
Y = 5x3 + 3x2 – x + 9
n = 4
quartic
Y = 3x4 – 2x3 + 8x2 – 6x + 5
n = 5
quintic
Y = -2x5+3x4–x3+3x2–2x+6

6. Graphs of Polynomial Functions
The polynomial functions that have the simplest graphs are monomials of the form f (x) = xn, where n is an integer greater than zero.

7. If n is odd, their graphs resemble the graph of f (x) = x3.
Polynomial functions of the form f (x) = x n, n  1 are called power functions.
f (x) = x5 x y
f (x) = x4 x y
f (x) = x3
f (x) = x2
If n is odd, their graphs resemble the graph of f (x) = x3.
If n is even, their graphs resemble the graph of f (x) = x2.
Moreover, the greater the value of n, the flatter the graph near the origin
Power Functions

8. The Leading Coefficient Test
Polynomial functions have a domain of all real numbers. Graphs eventually rise or fall without bound as x moves to the right.
Whether the graph of a polynomial function eventually rises or falls can be determined by the function’s degree (even or odd) and by its leading coefficient, as indicated in the Leading Coefficient Test.

9. Leading Coefficient Test
As x grows positively or negatively without bound, the value f (x) of the polynomial function
f (x) = anxn + an – 1xn – 1 + … + a1x + a0 (an  0)
grows positively or negatively without bound depending upon the sign of the leading coefficient an and whether the degree n is odd or even. x y x y
an positive
an negative
n even
n odd

10. Using our calculator examine the behavior of the polynomials

11. Zeros of Polynomial Functions
It can be shown that for a polynomial function f of degree n, the following statements are true.
1. The function f has, at most, n real zeros.
2. The graph of f has, at most, n – 1 turning points. (Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa.)
Finding the zeros of polynomial functions is one of the most important problems in algebra.

12. Given the polynomials below, answer the following
What is the degree?
What is its leading coefficient?
How many “turns”(relative maximums or minimums) could it have (maximum)?
How many real zeros could it have (maximum)?
How would you describe the left and right behavior of the graph of the equation?
What are its intercepts (y for all, x for 1 & 2 only)?
Equations:

13. Repeated Zeros
If k is the largest integer for which (x – a) k is a factor of f (x) and k > 1, then a is a repeated zero of multiplicity k.
1. If k is odd the graph of f (x) crosses the x-axis at (a, 0).
2. If k is even the graph of f (x) touches, but does not cross
through, the x-axis at (a, 0).
Example: Determine the multiplicity of the zeros of f (x) = (x – 2)3(x +1)4. x y
Zero Multiplicity Behavior
crosses x-axis at (2, 0) 2 3
odd –1 4
even
touches x-axis at (–1, 0)

14. Example - Finding the Zeros of a Polynomial Function
Find all real zeros of f (x) = –2x4 + 2x2. Then determine the number of turning points of the graph of the function.

15. Another example: Find all the real zeros and turning points of the graph of f (x) = x 4 – x3 – 2x2.
Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2).
The real zeros are x = –1, x = 0, and x = 2. y x
f (x) = x4 – x3 – 2x2
These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0).
Turning point
The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible.
Turning point
Turning point

16. Example continued: Sketch the graph of f (x) = 4x2 – x4.
3. Since f (–x) = 4(–x)2 – (–x)4 = 4x2 – x4 = f (x), the graph is
symmetrical about the y-axis.
4. Plot additional points and their reflections in the y-axis:
(1.5, 3.9) and (–1.5, 3.9 ), ( 0.5, 0.94 ) and (–0.5, 0.94)
5. Draw the graph. x y
(1.5, 3.9)
(–1.5, 3.9 )
(– 0.5, 0.94 )
(0.5, 0.94)