(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

2-2: Evaluate and Graph Polynomial Functions

What is a polynomial?-A monomial or sum of monomials with whole number exponents. Degree of a polynomial:- The highest exponent of the polynomial

How do we write polynomials?-Standard form: terms in descending order of exponents from left to right.

P(x) = 4x3 + 3x2 - 6x + 7

cubic term

quadratic term

linear termconstant

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

Example of a polynomial:f (x) = 4x4 + 2x3 - x + 7

What is the constant?

What is the degree?

0 constant 71 linear ½x-32 quadratic 8x2 -7 3 cubic 2x3-4x2+x-8 4 quartic 2x4 + 3x3 - 5x-11

5 quintic x5-7x4+x3-4x2+x-8

Degree Name Example

Classification of a polynomial by degree:

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

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

a. f (x) = 2x2 - x-2

b. f (x) = -0.8x3 + x4 - 5

Decide whether the function is a polynomial function. If it is, write the function in standard form, classify the polynomial by degree and by the number of terms.

a. f (x) = x + 5x3 b. f (x) = 6 - 4x + πx4

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

Try these: Classify each polynomial by degree and number of terms.

a) x2 + 4 - 8x - 2x3

b) 3x3 + 2 - x3 - 6x5

Evaluating Functions: -To evaluate, just substitute the number into the function.

Evaluate f (x) = 4x4 + 2x3 - x + 7 when x = -2

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

Exploring End Behavior

What is end behavior?-the behavior of the graph as x approaches positive or negative infinity.

Polynomial Function Graphs

Graph: y = 2x2 - 3x + 4

leading coefficient _________degree __________

From the Left: _________From the Right: _________

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

When the function's degree is odd, the ends will go in_____________ directions.

If the leading coefficient is positive, the graph ________ to the right.

If the leading coefficient is negative, the graph ________ to the right.

Summary:

When the function's degree is even the ends go in the __________ direction.

rises

falls

opposite

same

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

Describe the end behavior of the graph. Explain your answer.

f (x) = x5 + 2x2 - x + 4

f (x) = 4x4 + 2x3 - x + 7

as x approaches f(x) approaches

as x approaches f(x) approaches

as x approaches f(x) approaches

as x approaches f(x) approaches

f (x) = -x3 + 3x2 + 6x - 2

f (x) = -x6 - 4x3 + 8x2 + 7

Describe the end behavior of the graph. Explain your answer.

as x approaches f(x) approaches

as x approaches f(x) approaches

as x approaches f(x) approaches

as x approaches f(x) approaches

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

local maximum - the highest point of the function in that area of the graph

local minimum - the lowest point of the function in that area of the graph

Turning Points

*A cubic function has at most 2 turning points. *A quartic function has at most 3 turning points, and so on.

minimum

maximum

x-intercepts:

Turning Points:

Increasing:

Decreasing:

f(x) = x3 - 3x2 + 6

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

f(x) = x4 - 6x3 + 3x2 + 10x - 3

x-intercepts:

Turning Points:

Increasing:

Decreasing:

f(x) = x4 + 3x3 - x2 - 4x - 5

x-intercepts:

Turning Points:

Increasing:

Decreasing:

(2) ­ Evaluate and Graph Polynomial Functions.notebook March 05, 2015

You are making a rectangular box out of a 16-inch by 20-inch piece of cardboard. The box will be formed by making the cuts shown in the diagram and folding up the sides. You want to make the box to have the greatest volume possible.

How long should you make the cuts?

What is the maximum volume?

What will the dimensions of the finished box be?

Assignment:

p. 99 # 4, 6,12p. 100 # 28 - 36 even (no calculator) p. 148 # 22 (calculator)