5.2 Evaluating and Graphing Polynomial Functions

DAY 1

Goal:

Evaluate and graph polynomial functions.

EVALUATING 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

Common Types of Polynomials

Degree Type Standard Form

0 Constant f(x) = a0 (f(x) = -5)

1 Linear f(x) = x (f(x) = 3x)

2 Quadratic f(x) = x2 + x + a0

3 Cubic f(x) = x3 + x2 + x + a0

4 Quartic f(x) = x4 + x3 + x2 + x + a0

Identifying Polynomial Functions

2 4

3

1. ( ) 3 7

2

. ( ) 3x

a f x x x

b f x x

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

Identifying Polynomial Functions

2 1

2

. ( ) 6 2

. ( ) 0.5 2

c f x x x x

d f x x x

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

Evaluating a Polynomial Function

One way to evaluate is to use direct substitution.

PLUG IT IN, PLUG IT IN

4 2

4 2

2 8 5 7 3

(3) 2(3) 8(3) 5(3) 7

162 72 15 7

98

Evaluate f x x x x when x

f

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

THEREMAINDER

Evaluate using Direct Substitution and Synthetic Substitution

31( ) , when x = 4

2f x x x

Direct Substitution Synthetic Substitution

Evaluate the Polynomial FunctionUsing Direct Substitution when x = -2

Evaluate the Polynomial FunctionUsing Synthetic Substitution

3 2( ) 5 4 8 1 when x = 2f x x x x

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x + 2

LinearFunction

Degree = 1

Max. Zeros: 1

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x2 + 3x + 2

QuadraticFunction

Degree = 2

Max. Zeros: 2

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x3 + 4x2 + 2

CubicFunction

Degree = 3

Max. Zeros: 3

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x4 + 4x3 – 2x – 1

QuarticFunction

Degree = 4

Max. Zeros: 4

POLYNOMIAL FUNCTIONS

END BEHAVIOR

Degree: Odd

Leading Coefficient: +

End Behavior:

As x -∞; f(x) -∞

As x +∞; f(x) +∞

f(x) = x3

POLYNOMIAL FUNCTIONS

END BEHAVIOR

Degree: Odd

Leading Coefficient: –

End Behavior:

As x -∞; f(x) +∞

As x +∞; f(x) -∞

f(x) = -x3

POLYNOMIAL FUNCTIONS

END BEHAVIOR

Degree: Even

Leading Coefficient: +

End Behavior:

As x -∞; f(x) +∞

As x +∞; f(x) +∞

f(x) = x4

POLYNOMIAL FUNCTIONS

END BEHAVIOR

Degree: Even

Leading Coefficient: –

End Behavior:

As x -∞; f(x) -∞

As x +∞; f(x) -∞

f(x) = -x4

Graphing Summary Degree -- highest power exponent or # of factors with variable termsNumber of turns -- if there are n zeros, then there are n-1 turns

Leading coefficient– coefficient of highest power variable or the number in front of all the factorsX-intercepts –the zeros, found by setting each factor equal to zeroY- intercept – where the graph crosses the y axis, found by evaluating the function for an x-value of 0

End behavior – four casesEven degree, + leading coefficient –both ends point upEven degree, - leading coefficient—both ends point downOdd degree, + leading coefficient—starts down, ends upOdd degree, - leading coefficient—starts up, ends down