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Analyzing Graphs
of Polynomials
Section 3.2
First a little review…
Given the polynomial function of the form:
f(x) = anxn + an−1xn−1 + . . . + a1x + a0
If k is a zero,
Zero: __________
Solution: _________
x = k
x = k
Factor: _________
(x – k)
x - intercept
If k is a real number, then k is also a(n) __________________.
What kind of curve?
All polynomials have graphs that are smooth continuous curves.
A smooth curve is a curve that does not have sharp corners.
Sharp corner – must not be a polynomial function
A continuous curve is a curve that does not have a break or hole.
Hole
Break
End Behavior
An > 0 , Odd Degree
(think a positive
slope line!)
y
Ann < 0 , Odd Degree
(think a negative
slope line!)
y
x
As x  +  , f(x)

y
y
x
x
 , f(x) 
As x  +  , f(x)   As x  +

As x  -  , f(x)
As x  -  , f(x) 
Ann < 0 , Even Degree
(think of an -x2
parab. graph)
An > 0 , Even Degree
(think of an x2
parabola graph)

As x  -  , f(x) 
x

As x  +  , f(x)  

As x  -  , f(x)  
What happens in the middle?
** This graph is said to have
3 turning points.
** The turning points happen when
the graph changes direction.
This happens at the vertices.
** Vertices are
minimums and maximums.
** The lowest degree of a polynomial is
(# turning points + 1).
So, the lowest degree of this
polynomial is
4!
What’s happening?
Relative Maximums
Also called Local Maxes
Relative Minimums
Also called Local Mins
As x  +  , f(x)
click

As x  -  , f(x)
click

The lowest degree of this polynomial is
The leading coefficient is
positive
5
Graphing by hand
Step 1: Plot the x-intercepts
Step 2: End Behavior? Number of Turning Points?
Step 3: Plot points in between the x-intercepts.
Example #1:
Graph the function:
and identify the following.
Negative-odd polynomial
of degree 3
f(x) = -(x + 4)(x + 2)(x - 3)
, f(x) 

As x  +  , f(x)   As x  - 
End Behavior: _________________________
2
# Turning Points: _______________________
3
Lowest Degree of polynomial: ______________
Try some points in the middle.
(-3, -6), (-1, 12), (1, 30), (2, 24)
You can check on your calculator!
X-intercepts
2
Graphing with a calculator
Example #2:
Graph the function:
and identify the following.
f(x) = x4 – 4x3 – x2 + 12x – 2
  As x  -  , f(x)  
As x  + 
End Behavior: _________________________
, f(x)
3
# Turning Points: _______________________
Degree of polynomial: ______________
4
Plug equation into y=
Relative max
Real Zeros
Relative minimum
Absolute minimum
Positive-even polynomial
of degree 4
Graphing without a calculator
Example #3:
Graph the function:
and identify the following.
f(x) = x3 + 3x2 – 4x

As x  + 


End Behavior: _________________________
, f(x)
As x  -
, f(x)
2
# Turning Points: _______________________
Degree of polynomial: ______________
3
1. Factor and solve equation to find x-intercepts
2. Try some points in the around the Real Zeros
Where are the maximums and minimums?
(Check on your calculator!)
Positive-odd polynomial
of degree 3
Zero Location Theorem
Given a function, P(x) and a & b are real numbers.
If P(a) and P(b) have opposite signs,
then there is at least one real zero (x-intercept) in between x = a & b.
a
b
P(b) is positive.
(The y-value is positive.)
P(a) is negative.
(The y-value is negative.)
Therefore, there must be
at least one real zero in between a & b!
Even & Odd Powers of (x – c)
The exponent of the factor tells if that zero crosses over the x-axis or is a vertex.
If the exponent of the factor is ODD, then the graph CROSSES the x-axis.
If the exponent of the factor is EVEN, then the zero is a VERTEX.
Try it. Graph y = (x + 3)(x – 4)2
Try it. Graph y = (x + 6)4 (x + 3)3