7.4 Guided Notes – Graphing Polynomial Functions
Name: ______________________
Date: _____________ Period: ___
Objective: I can determine end behavior of a function, if a function is even or odd and use this information to
help me sketch a graph. I can identify multiplicity in polynomial graphs and equations and apply to graphing.
The graph of a polynomial function is ______________ and _________________, (meaning it
can be drawn without lifting the pencil off the page).
_____________________ - Direction the arrows point at the start and at the end of the graph.
Polynomials will always have ONE of the following four end behaviors:
EVEN-degree FUNCTION
ODD-degree FUNCTION
Leading Coefficient: POSITIVE
Leading Coefficient: POSITIVE
End Behavior:
End Behavior:
Domain:
Range:
Domain:
Range:
EVEN-degree FUNCTION
ODD-degree FUNCTION
Leading Coefficient: NEGATIVE
Leading Coefficient: NEGATIVE
End Behavior:
End Behavior:
Domain:
Range:
Domain:
Range:
EXAMPLE: Describe the end behavior. State the domain and range.
𝑓(𝑥) = −𝑥 5 + 4𝑥 3 – 5𝑥 – 2
𝑔(𝑥) = 2𝑥 3 + 𝑥 2 – 𝑥 – 2
ℎ(𝑥) = −3𝑥 4 + 𝑥 3 + 2𝑥 2
REAL ZEROS – zeros are where the graph __________ /__________ the x-axis (as in x = ____ and y=0)
Even-degree functions have an even number of real zeros. Odd-degree functions have odd number of real zeros
EXAMPLES:
Degree:
Degree:
End Behavior:
End Behavior:
Zeros:
Zeros:
Domain:
Range:
Domain:
Range:
SKETCHING QUICK GRAPHS - End behaviors, zeros, and degrees help us sketch quick graphs!!
EXAMPLES: Sketch the graph of the polynomial functions with the following characteristics.
C) an odd function
real zeros at -3, 2, and 4
positive leading coefficient
B) an even function
real zeros at -3, -1, 1, 3
negative leading coefficient
A) an odd function
real zeros at -5, -2, 0, 1, 3
starts in quadrant II.
Now we are ready to get a little more specific with the
graphs of polynomial functions, starting with the
different types of
_________= _________= _________ = _________
Figure A
Remember: a “zero” is where the graph crosses or
____________ the x-axis.
MULTIPLICITY: same solution exists multiple times.
Figure A has zeros at x = ______ with multiplicity _____
and at x = ______ with multiplicity _____
x = ______ written as a factor would be (
).
x = ______,
written as a factor (
MULTIPLICITY GUIDE
Even multiplicity  ______________ .
EXAMPLES
Solution(s):
Degree:
Multiplicity of:
Direction:
Possible Equation of this graph:
Odd multiplicity  ______________ .
As a Factor:
).
Solution(s):
Degree:
Multiplicity of:
Direction:
Possible Equation of this graph:
As a Factor:
Solution(s):
Degree:
Multiplicity of: As a Factor:
Direction:
Possible Equation of this graph:
Now that you can pull factors and the equation out of a graph, reverse the process to sketch graphs with much more accuracy.
EXAMPLES: Graph the following equations in factored form.
𝑓(𝑥) = 𝑥(𝑥 + 5)(𝑥 – 3)
𝑔(𝑥) = −(𝑥 + 4)2 (𝑥 + 1)(𝑥 – 2)3
Degree:
Degree:
Direction:
Direction: