6.2 Evaluating and Graphing Polynomial Functions.

6.2

Evaluating and Graphing Polynomial Functions

A polynomial function is a function of the form

f (x) = an x n + an – 1 x

n – 1 +· · ·+ a 1 x + a 0

Where an 0 and the exponents are all whole numbers.

A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

For this polynomial function, an is the leading coefficient,

a 0 is the constant term, and n is the degree.

an 0

an

an leading coefficient

a 0

a0 constant term n

n

degree

descending order of exponents from left to right.

n n – 1

Degree Type Standard Form

You are already familiar with some types of polynomialfunctions. Here is a summary of common types ofpolynomial functions.

4 Quartic f (x) = a4 x 4 + a 3 x

3 + a 2 x 2 + a 1 x + a

0 Constant f (x) = a

3 Cubic f (x) = a 3 x 3 + a 2 x

2 + a 1 x + a

2 Quadratic f (x) = a 2 x 2 + a 1 x + a

1 Linear f (x) = a1x + a

Identifying Polynomial Functions

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

f (x) = x 2

– 3x4 – 712

SOLUTION

The function is a polynomial function.

It has degree 4, so it is a quartic function.

The leading coefficient is – 3.

Its standard form is f (x) = – 3x 4

+ x 2 – 7. 1

2



The function is not a polynomial function because the

term 3

x does not have a variable base and an exponentthat is a whole number.

SOLUTION

f (x) = x 3 + 3

x



SOLUTION

f (x) = 6x 2 + 2 x

–1 + x

The function is not a polynomial function because the term2x

–1 has an exponent that is not a whole number.



SOLUTION

The function is a polynomial function.

It has degree 2, so it is a quadratic function.

The leading coefficient is .

Its standard form is f (x) = x2 – 0.5x – 2.

f (x) = – 0.5 x + x 2 – 2

f (x) = x 2 – 3 x

4 – 712


f (x) = x 3 + 3x

f (x) = 6x2 + 2 x– 1 + x

Polynomial function?

f (x) = – 0.5x + x2 – 2

Using Synthetic Substitution

One way to evaluate polynomial functions is to usedirect substitution. Another way to evaluate a polynomialis to use synthetic substitution.

Use synthetic division to evaluate

f (x) = 2 x 4 + 8 x

2 + 5 x 7 when x = 3.

Polynomial in standard form

Using Synthetic Substitution

2 x 4 + 0 x

3 + (–8 x 2) + 5 x + (–7)

2 6

6

10

18

35

30 105

98

The value of f (3) is the last number you write,In the bottom right-hand corner.

The value of f (3) is the last number you write,In the bottom right-hand corner.

2 0 –8 5 –7 CoefficientsCoefficients

3

x-value

3 •

SOLUTION

Polynomial instandard form

Now use direct substitution:

Use synthetic division to evaluate

f (x) = 2 x 4 + 8 x

2 + 5 x 7 when x = 3.

7)3(5)3(8)3(2)( 24 xf

98

HOMEWORK (DAY 1)

If “n” is even, the graph of the polynomial is “U-shaped” meaning it is parabolic (the higher the degree, the more curves the graph will have in it).

If “n” is odd, the graph of the polynomial is “snake-like” meaning looks like a snake (the higher the degree, the more curves the graph will have in it).

Let’s talk about the Leading Coefficient Test:

Leading Coefficient Test

Degree is evenDegree is odd

L.C. > 0 Start high,End high

L.C. < 0 Start low,End low

L.C. > 0 Start low,End high

L.C. < 0 Start high,End low

L.C. = Leading Coefficient

Determine the left and right behavior of the graph of each polynomial function.

f(x) = -x5 +3x4 – x

f(x) = x4 + 2x2 – 3x

f(x) = 2x3 – 3x2 + 5

Tell me what you know about the equation…




Fundamental Thm of AlgebraZeros of Polynomial Functions:1. The graph of f has at most n

real zeros

The “n” deals with the highest exponent!

How many zeros do these graphs have????

GRAPHING POLYNOMIAL FUNCTIONS

The end behavior of a polynomial function’s graphis the behavior of the graph as x approaches infinity(+ ) or negative infinity (– ). The expressionx + is read as “x approaches positive infinity.”

GRAPHING POLYNOMIAL FUNCTIONS

END BEHAVIOR

x

f(x)–3

–7

–2

3

–1

3

0

–1

1

–3

2

3

3

23

Graphing Polynomial Functions

Graph f (x) = x 3 + x

2 – 4 x – 1.

SOLUTION

To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.

x

f (x)

–3

–21

–2

0

–1

–1

0

0

1

3

2

–16

3

–105

Graphing Polynomial Functions

Graph f (x) = –x 4 – 2x

3 + 2x 2 + 4x.

SOLUTION

To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.

Assignment

6.2 Evaluating and Graphing Polynomial Functions.

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