1. Analyzing Graphs of Polynomials
Section 3.2

2. Given the polynomial function of the form:
First a little review…
Given the polynomial function of the form:
f(x) = anxn + an−1xn− a1x + a0
If k is a zero,
Zero: __________ Solution: _________
Factor: _________
If k is a real number, then k is also a(n) __________________.
x = k
x = k
(x – k)
x - intercept

3. 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

4. End Behavior (think a positive slope line!) An < 0 , Odd Degree
(think a negative slope line!)
An > 0 , Even Degree
(think of an x2 parabola graph)
An < 0 , Even Degree
(think of an -x2 parab. graph)
An > 0 , Odd Degree
An > 0 , Odd Degree
An < 0 , Odd Degree
An > 0 , Even Degree
An < 0 , Even Degree y x y x y x y x
As x  +
, f(x)
As x  -

5. What happens in the middle?
** This graph is said to have
3 turning points.
Relative maximum
The graph “turns”
** The turning points happen when
the graph changes direction.
This happens at the vertices.
** Vertices are
minimums and maximums.
Relative minimums
The graph “turns”
** The lowest degree of a polynomial is
(# turning points + 1).
So, the lowest degree of this
polynomial is 4 !

6. What’s happening? 5 Relative Maximums Also called Local Maxes
Relative Minimums
Also called Local Mins
As x  -
, f(x)
As x  +
click
The lowest degree of this polynomial is 5
click
The leading coefficient is
positive

7. Graphing by hand 2 3 Step 1: Plot the x-intercepts
Step 2: End Behavior? Number of Turning Points?
Step 3: Plot points in between the x-intercepts.
Negative-odd polynomial
of degree 3
Example #1:
Graph the function: f(x) = -(x + 4)(x + 2)(x - 3)
and identify the following.
End Behavior: _________________________
# Turning Points: _______________________
Lowest Degree of polynomial: ______________
As x  +
, f(x)
As x  -
, f(x) 2 3 2
Try some points in the middle.
(-3, -6), (-1, 12), (1, 30), (2, 24)
You can check on your calculator!
X-intercepts

8. Graphing with a calculator
Example #2:
Graph the function: f(x) = x4 – 4x3 – x2 + 12x – 2
and identify the following.
Positive-even polynomial
of degree 4
End Behavior: _________________________
# Turning Points: _______________________
Degree of polynomial: ______________
As x  +
, f(x)
As x  -
, f(x) 3 4
Relative max
Plug equation into y=
Real Zeros
Relative minimum
Absolute minimum

9. Graphing without a calculator
Example #3:
Graph the function: f(x) = x3 + 3x2 – 4x
and identify the following.
Positive-odd polynomial
of degree 3
End Behavior: _________________________
# Turning Points: _______________________
Degree of polynomial: ______________
As x  +
, f(x)
As x  -
, f(x) 2 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!)

10. 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!

11. 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