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(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)