polynomial functions

POLYNOMIAL FUNCTIONSFebruary 27,2012

What is a polynomial function? What is a polynomial?

The sum or difference of two or more algebraic terms (monomials)

A polynomial equation used to represent a function is called a polynomial function

Example:f(x) = 4x2 – 3x + 2f(x) = 2x3 + 4x2 – 5x + 7

Quadratic Function

Cubic Function

Do Now: What does it mean to evaluate a function?

Evaluate p(2) p(x) = x2 + 3x + 4

p(2) = 22 + 3(2) + 4 = 14 Evaluate p(b+1)

p(x) = x2 + 3x + 4p(b+1) = (b+1)2 + 3(b+1) + 4

= b2 + 2b + 1 + 3b + 3 + 4 = b2 + 5b + 8

Plug in or substitute what is in the parenthesis for x in

the function

Vocabulary Degree - refers to the highest exponent

of a function Leading Coefficient - is the coefficient of

the term with the highest degreeExample:What is the degree and leading coefficient

of: 5x2 + 2x + 1

7x4 + 3x2 + x - 9

D: 2, LC: 5D: 4, LC: 7

Graphs of polynomial functions

• What relationship is there between the degree of the function and the maximum number of zeros?

The degree tells you how many roots you have• What do you notice about the degree and number of humps/ curves in the graph? The degree is one more than the number of humps

End Behavior What happens to the graph as x

approaches ±∞? This is represented by

x + ∞ and x - ∞x approaches positive infinityX approaches negative infinity

End behavior expressed f(x) –∞, as x –∞ f(x) + ∞, as x +∞

• For any polynomial function, the domain is all real numbers• For any polynomial function of odd degree, the range is all real numbers• The graph of any odd degree function has at least one x intercept

Put it all together

• f(x) - ∞, as x - ∞ and f(x) - ∞, as x + ∞• degree of 4 represents an even-degree polynomial function• 2 real zeros

Exit Slip

Agenda Do Now – Determine end behavior

through graph Review HW Notes – Determining end behavior

through equation Class work

Do Now

a) f(x) +∞ , as x +∞ f(x) +∞ , as x -∞b) Evenc) No real zeros

END BEHAVIOR

Degree: Even

Leading Coefficient: +

End Behavior: Up Up

END BEHAVIOR

Degree: Even

End Behavior: Down Down

Leading Coefficient:

END BEHAVIOR

Degree: Odd

Leading Coefficient: +

End Behavior: Down Up

END BEHAVIOR

Degree: Odd

End Behavior: Up Down

Leading Coefficient:

End Behavior

Function Degree

Leading Coeffici

ent

End Behavio

rGraph

f(x) = x2 Even + Up, Up

f(x) = – x2 Even — Down, Down

f(x) = x3 Odd + Down, Up

f(x) = – x3 Odd — Up, Down

END BEHAVIOR

PRACTICE Give the End Behavior:

END BEHAVIOR

PRACTICE Give the End Behavior:

Up Down

Up Up

Down Down

Down Up

Do Now What will the end behavior of the

following polynomial functions be? (hint: find the degree and leading coefficient)

y = – x2 + 6x – 5

y = x5 – 3x2 + 2x – 6

Agenda Do Now- Review Notes

Behavior on intervals Group Work Activity Explain Project Homework

Today’s Objective: Identify increasing, decreasing, and

constant intervals

Increasing, Decreasing, and Constant Intervals

An interval is the set of all real numbers between two given numbers

A function f is increasing on an interval if as x increases, then f(x) increases.

A function f is decreasing on an interval if as x increases, then f(x) decreases.

A function f is constant on an interval if as x increases, then f(x) remains the same.

Increasing, Decreasing, Constant Intervals

Find the interval(s) over which the interval is increasing, decreasing and constant?

Increasing, Decreasing, Constant Intervals

f(x) is decreasingin the interval (-1,1).

f(x) is increasingin the intervals

Today’s Agenda Do Now Review Home work

Intervals Review Notes

What do they say about… Class work

End behavior worksheet

Find the interval(s) over which the interval is increasing, decreasing and constant?

Do now

f(x) is decreasing over the intervalsf(x) is increasingover the interval (3,5).

f(x) is constantover the interval (-1,3).