Polynomial Functions

2.1 (M3)

Make sure you have book and working calculator EVERY day!!!

EXAMPLE 1 Identify polynomial functions

4

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

a. h (x) = x4 – x2 + 31a. Yes it’s a Polynomial. It is in standard form.

Degree 4 – Quartic Trinomial Its leading coefficient is 1.

237)(. xxxgb b. Yes it’s a Polynomial. Standard form is Degree 2 – Quadratic Trinomial Leading Coefficient is

37)( 2 xxxg

EXAMPLE 1 Identify polynomial functions

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

c. f (x) = 5x2 + 3x –1 – x c. Not a polynomial function

d. k (x) = x + 2x – 0.6x5

d. Not a polynomial function

e. f (x) = 13 – 2x e. polynomial function; f (x) = –2x + 13; degree 1linear binomial,leading coefficient: –2

Graph Trend based on Degree

• Even degree - end behavior going the same direction

• Odd degree – end behavior (tails) going in opposite directions

Leading Coefficient

Leading Coefficient

Symmetry: Even/Odd/Neither• First look at degree• Even if it is symmetric respect to y-axis

– When you substitute -1 in for x, all signs stay the SAME

• Odd if it is symmetric with respect to the origin– When you substitute -1 in for x, all of the signs CHANGE

• Neither if it is NOT symmetric around the y-axis or origin

Tell whether it is even/odd/neither1. f(x)= x2 + 2

2. f(x)= x2 - 4x

3. f(x)= x3

4. f(x)= x3 + x

5. f(x)= x3 + 5x +1

Additional Vocabulary to Review• End Behavior:

Left side x– ∞, f(x) ____

Right side x+∞, f(x) ____

Additional Vocabulary to Review• Domain: set of all possible x values• Range: set of all possible y values• Symmetry: even (across y), odd (around origin),

or neither• Interval of increase (where graph goes up to the

right)• Interval of decrease (where the graph goes down

to the right)• End Behavior:

Left side x– ∞, f(x) ____

Right side x+∞, f(x) ____

Polynomial Functions and Their Graphs

There are several different elements to examine on the graphs of polynomial functions:

Local minima and maxima:

On the graph above: A local maximum: f(x) = A local minimum: f(x) =

Give the Local Maxima and Minima

Must use y to describe High and Low

Finding a local max and/or local min is EASY with the calculator!

Graph each of the following and find all local maxima or minima:

2) ( ) 3 2A f x x x 4 2) ( ) 4 1B g x x x 3 2) ( ) 2 4 9C h x x x

Now describe their end behavior.

yxA ,) ,x y

) ,B x y ,x y

) ,C x y ,x y

Describe the Interval of Increasing and Decreasing

Increasing when ___________

Decreasing when _____________

Increasing when ___________

Must use x to describe Left to Right

x

y

(Left to Right) The graph is:

, 3

3, 5

5,

Assignment

• Page 69#1-3 Classify

#8-10 End Behavior

#11-13 Symmetry (Even/Odd/Neither)

#14-16 Max/Min, Domain/Range, Intervals of

Increasing and Decreasing