polynomial functions 2.1 (m3) make sure you have book and working calculator every day!!!
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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