Graphs of Polynomial Functions

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.

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Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps.

Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions.

x

y

x

y

continuous not continuous continuoussmooth not smooth

polynomial not polynomial not polynomial

x

y f (x) = x3 – 5x2 + 4x + 4

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A polynomial function is a function of the form1

1 1 0( ) n nn nf x a x a x a x a

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 Degree5 3( ) 2 3 5 1f x x x x

3 2( ) 6 7f x x x x

( ) 14f x

-2 5

1 3

14 0

Classification of a Polynomial

Degree Name Example

5

Y = -2x5+3x4–x3+3x2–2x+6

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

constant Y = 3

linear Y = 5x + 4

quadratic Y = 2x2 + 3x - 2

cubic Y = 5x3 + 3x2 – x + 9

quartic Y = 3x4 – 2x3 + 8x2 – 6x + 5 quintic

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.

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Polynomial functions of the form f (x) = x n, n 1 are called power functions.

If n is even, their graphs resemble the graph of

f (x) = x2.

If n is odd, their graphs resemble the graph of

f (x) = x3.

x

y

x

y

f (x) = x2

f (x) = x5

f (x) = x4

f (x) = x3

Moreover, the greater the value of n, the flatter the graph near the origin

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.

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Leading Coefficient TestAs x grows positively or negatively without bound, the value f (x) of the polynomial function

f (x) = anxn + an – 1x

n – 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

n odd n even

an positive

an negative

Using our calculator examine the behavior of the polynomials

10

823)( .5

123.4

14)(.3

23.2

23.1

6

25

34

3

2

xxxf

xxy

xxxf

xxy

xxy

Zeros of Polynomial FunctionsIt 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.

Given the polynomials below, answer the following

A. What is the degree?B. What is its leading coefficient?C. How many “turns”(relative maximums or minimums) could it have (maximum)?D. How many real zeros could it have (maximum)?E. How would you describe the left and right behavior of the graph of the equation?F. What are its intercepts (y for all, x for 1 & 2 only)?

Equations:

12

144)(.4

123.3

82)(.2

23.1

6

25

24

23

xxxf

xxy

xxxf

xxxy

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Example: Determine the multiplicity of the zeros of f (x) = (x – 2)3(x +1)4.

Zero Multiplicity Behavior

2

–1

3

4

odd

even

crosses x-axis at (2, 0) touches x-axis at (–1, 0)

Repeated ZerosIf 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).

x

y

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.

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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.

These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0).

The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible.

y

x

f (x) = x4 – x3 – 2x2

Turning pointTurning point

Turning point

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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)