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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: _________

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slide2

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: _________

Factor: _________

If k is a real number, then k is also a(n) __________________.

x = k

x = k

(x – k)

x - intercept

slide3

Sharp corner – must not be a polynomial function

Hole

Break

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.

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

slide4

y

y

y

y

x

x

x

x

, f(x)

, f(x)

, f(x)

, f(x)

, f(x)

, f(x)

, f(x)

, f(x)

As x  -

As x  +

As x  +

As x  +

As x  -

As x  -

As x  +

As x  -

End Behavior

An < 0 , Even Degree

An > 0 , Odd Degree

An > 0 , Odd Degree

An < 0 , Odd Degree

An > 0 , Even Degree

slide5

Relative maximum

The graph “turns”

Relative minimums

The graph “turns”

What happens in the middle?

** This graph is said to have

3turning 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 !

slide6

The lowest degree of this polynomial is

The leading coefficient is

, f(x)

, f(x)

As x  -

As x  +

What’s happening?

Relative Maximums

Also called Local Maxes

Relative Minimums

Also called Local Mins

click

5

click

positive

slide7

Negative-odd polynomial

of degree 3

, f(x)

As x  -

, f(x)

As x  +

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: f(x) = -(x + 4)(x + 2)(x - 3)

and identify the following.

End Behavior: _________________________

# Turning Points: _______________________

Lowest Degree of polynomial: ______________

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

slide8

Positive-even polynomial

of degree 4

Relative max

Relative minimum

Absolute minimum

, f(x)

As x  -

, f(x)

As x  +

Graphing with a calculator

Example #2:

Graph the function: f(x) = x4 – 4x3 – x2 + 12x – 2

and identify the following.

End Behavior: _________________________

# Turning Points: _______________________

Degree of polynomial: ______________

3

4

Plug equation into y=

Real Zeros

slide9

Positive-odd polynomial

of degree 3

, f(x)

As x  -

, f(x)

As x  +

Graphing without a calculator

Example #3:

Graph the function: f(x) = x3 + 3x2 – 4x

and identify the following.

End Behavior: _________________________

# Turning Points: _______________________

Degree of polynomial: ______________

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!)

slide10

P(b) is positive.

(The y-value is positive.)

a

b

P(a) is negative.

(The y-value is negative.)

Therefore, there must be

at least one real zero in between a & b!

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.

slide11

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