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