analyzing graphs of polynomials
DESCRIPTION
Analyzing Graphs of Polynomials. Section 3.2. First a little review…. Given the polynomial function of the form: f(x) = a n x n + a n−1 x n−1 + . . . + a 1 x + a 0 If k is a zero, Zero: __________ Solution: _________ Factor: _________ - PowerPoint PPT PresentationTRANSCRIPT
Analyzing Graphs of Polynomials
Section 3.2
First a little review…
Given the polynomial function of the form: f(x) = anx
n + an−1xn−1 + . . . + 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
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.
HoleBreak
(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)
As x + , f(x)
As x - , f(x)
As x + , f(x)As x + , f(x)As x + , f(x)
As x - , f(x)As x - , f(x)As x - , f(x)
End Behavior
An > 0 , Odd DegreeAn > 0 , Odd Degree
y
x
y
x
y
x
y
x
An < 0 , Odd Degree An > 0 , Even DegreeAn < 0 , Even Degree
What happens in the middle?
The graph “turns”
The graph “turns”
** 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.
Relative maximum
Relative m
inimums
** The lowest degree of a polynomial is (# turning points + 1).
So, the lowest degree of this
polynomial is 4 !
What’s happening?
As x - , f(x)
As x + , f(x) click
click
Relative MaximumsAlso called Local Maxes
Relative MinimumsAlso called Local Mins
The lowest degree of this polynomial is 5
The leading coefficient is positive
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: ______________
Graphing by hand
Step 1: Plot the x-interceptsStep 2: End Behavior? Number of Turning Points?Step 3: Plot points in between the x-intercepts.
X-intercepts
Negative-odd polynomial of degree 3
As x - , f(x) As x + , f(x)
2
3
Try some points in the middle.
(-3, -6), (-1, 12), (1, 30), (2, 24)
2
You can check on your calculator!
Example #2: Graph the function: f(x) = x4 – 4x3 – x2 + 12x – 2 and identify the following.
End Behavior: _________________________
# Turning Points: _______________________
Degree of polynomial: ______________
Graphing with a calculator
Positive-even polynomial of degree 4
As x - , f(x) As x + , f(x)
3
4
Plug equation into y=
Absolute minimum
Relative minimum
Relative max
Real Zeros
Example #3: Graph the function: f(x) = x3 + 3x2 – 4x and identify the following.
End Behavior: _________________________
# Turning Points: _______________________
Degree of polynomial: ______________
Graphing without a calculator
Positive-odd polynomial of degree 3
As x - , f(x) As x + , f(x)
2
3
1. Factor and solve equation to find x-intercepts
Where are the maximums and minimums?(Check on your calculator!)
2. Try some points in the around the Real Zeros
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(a) is negative. (The y-value is negative.)
P(b) is positive. (The y-value is positive.)
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