Analyzing Graphs of Polynomials

Section 3.2

First a little review…

Given the polynomial function of the form:

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

011

1)( axaxaxaxf nn

nn

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 – This graph must not be a polynomial function.

A continuous curve is a curve that does not have a break or hole.

HoleBreak

This is not a continuous curve!

(Think of a line with positive slope!)

An < 0 , Odd Degree(Think of a line with negative slope!)

An > 0 , Even Degree(Think of a parabola graph… y = x2 .)

An < 0 , Even Degree(Think of a parabola graph… y = -x2 .)

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 Degree An < 0 , Even Degree

, ( )x f x , ( )x f x

, ( )x f x

, ( )x f x

, ( )x f x , ( )x f x , ( )x f x

, ( )x f x

Examples of End Behaviors1

. 2.

3.

4.

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 !4( ) ....P x x

What’s happening?

As x + , f(x)

As x -

, f(x)

Relative Maximums

Relative Minimums

The lowest degree of this polynomial is _____ .5

The leading coefficient is __________ .positive

The number of turning points is _____ .4

5( ) ....P x x

Example #1: Graph the function: f(x) = -(x + 4)(x + 2)(x - 3) and identify the following.

End Behavior: _________________________

Lowest Degree of polynomial: ______________

# Turning Points: _______________________

Graphing by hand

Step 1: Plot the x-interceptsStep 2: End Behavior? Number of Turning Points?Step 3: Check in Calculator!!!

x-intercepts

Negative-odd polynomial of degree 3 ( -x * x * x)

As x - , f(x) As x + , f(x)

2

3

You can check on your calculator!!

Example #2: Graph the function: f(x) = x4 – 4x3 – x2 + 12x – 2 and identify the following.

End Behavior: _________________________

Degree of polynomial: ______________

# Turning Points: _______________________

y-intercept: _______

Graphing with a calculator

Positive-even polynomial of degree 4

As x - , f(x) As x + , f(x)

3

4

1.Plug equation into y=

2.Find minimums and maximums using your calculator

Absolute minimum

Relative minimum

Relative max

Real Zeros

(0, -2)

Example #3: Graph the function: f(x) = x3 + 3x2 – 4x and identify the following.

End Behavior: _________________________

Degree of polynomial: ______________

# Turning Points: _______________________

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

2. Plot the zeros. Sketch the end behaviors.

f(x)=x(x2 + 3x – 4) = x(x - 4)(x + 1)

Zero Location TheoremGiven 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 number c between a and b such that P(c) = 0.

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 x = a & x = b!

Example #4: Use the Zero Location Theorem to verify that P(x) = 4x3 - x2 – 6x + 1 has a zero between a = 0 and b = 1.

0 4 -1 -6 1 0 0 0 4 -1 -6 1

1 4 -1 -6 1 4 3 -3 4 3 -3 -2

The graph of P(x) is continuous because P(x) is a polynomial function.

P(0)= 1 and P(1) = -2 Furthermore, -2 < 0 < 1

The Zero Location Theorem indicates there is a real zero

between 0 and 1!

Polynomial Functions: Real Zeros, Graphs, and Factors (x – c)

• If P is a polynomial function and c is a real root, then each of the following is equivalent.

• (x – c) __________________________________ .

• x = c __________________________________ .

• x = c __________________________________ .

• (c, 0) __________________________________ .

is a factor of P

is a real solution of P(x) = 0

is a real zero of P

is an x-intercept of the graph of y = P(x)

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

Assignment:

Write the questions and show all work for each.

pp. 301-302

#1-13 ODD, 17 & 19 (TI-83),

21-29 ODD, 33, 35, 41, & 43