Essential Question: How do we decide for the degree of the polynomial with a variable? How do we...

Essential Question:

• How do we decide for the degree of the polynomial with a variable?

• How do we determine the end behavior of a polynomial function?

CCSS: F.IF.4

• For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Standards for Mathematical Practice

• 1. Make sense of problems and persevere in solving them.

• 2. Reason abstractly and quantitatively.

• 3. Construct viable arguments and critique the reasoning of others.

• 4. Model with mathematics.

• 5. Use appropriate tools strategically.

• 6. Attend to precision.

• 7. Look for and make use of structure.

• 8. Look for and express regularity in repeated reasoning.

A polynomial function is a function of the form:

on

nn

n axaxaxaxf 1

11

All of these coefficients are real numbers

n must be a positive integer

Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 …

The degree of one variable polynomial is the largest power on any x term in the polynomial.

x 0

2

1

xx

Not a polynomial because of the square root since the power is NOT an integer

xxxf 42

Determine which of the following are polynomial functions. If the function is a polynomial, state its degree.

A polynomial of degree 4.

2xg

12 xxh

23x

xxF

A polynomial of degree 0.

We can write in an x0 since this = 1.

Not a polynomial because of the x in the denominator since the power is negative 11 x

x

Graphs of polynomials are smooth and continuous.

No sharp corners or cusps No gaps or holes, can be drawn without lifting pencil from paper

This IS the graph of a polynomial

This IS NOT the graph of a polynomial

Let’s look at the graph of where n is an even integer.

nxxf

2xxf

4xxg 6xxh

and grows steeper on either

side

Notice each graph looks similar to x2 but is wider and flatter near the origin between –1 and 1

The higher the power, the flatter

and steeper

Let’s look at the graph of where n is an odd integer.

nxxf

3xxf

5xxg 7xxh

and grows steeper on either side

Notice each graph looks similar to x3 but is wider and flatter near the origin between –1 and 1

The higher the power, the flatter

and steeper

Let’s graph 24 xxf

Looks like x2 but wider near origin and steeper after 1 and -1

Reflects about the x-axis

Translates up 2

So as long as the function is a transformation of xn, we can graph it, but what if it’s not? We’ll learn some techniques to help us determine what the graph looks like in the next slides.

LEFT RIGHTand

HAND BEHAVIOUR OF A GRAPH

The degree of the polynomial along with the sign of the coefficient of the term with the highest power will tell us about the left and right hand behaviour of a graph.

http://images.google.com/imgres?imgurl=www.eslkidstuff.com/images/arms.gif&imgrefurl=http://www.eslkidstuff.com/Body.htm&h=326&w=324&prev=/images%3Fq%3Darms%26start%3D60%26svnum%3D10%26hl%3Den%26lr%3D%26ie%3DUTF-8%26oe%3DUTF-8%26sa%3DN

Even degree polynomials rise on both the left and right hand sides of the graph (like x2) if the coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behaviour determined by the highest powered term.

left hand behaviour: rises

right hand behaviour: rises

Even degree polynomials fall on both the left and right hand sides of the graph (like - x2) if the coefficient is negative.

left hand behaviour: falls right hand

behaviour: falls

turning points in the middle

Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x3) if the coefficient is positive.

left hand behaviour: falls

right hand behaviour: rises

turning Points in the middle

Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x3) if the coefficient is negative.

left hand behaviour: rises

right hand behaviour: falls

turning points in the middle

A polynomial of degree n can have at most n-1 turning points (so whatever the degree is, subtract 1 to get the most times the graph could turn).

doesn’t mean it has that many turning points but that’s the most it can have

3019153 234 xxxxxf

Let’s determine left and right hand behaviour for the graph of the function:

degree is 4 which is even and the coefficient is positive so the graph will look like x2 looks off to the left and off to the right.

The graph can have at most 3 turning points

How do we determine

what it looks like near the

middle?

x and y intercepts would be useful and we know how to find those. To find the y intercept we put 0 in for x.

3019153 234 xxxxxf

30300190150300 234 f

(0,30)

To find the x intercept we put 0 in for y.

51320 xxxx

Finally we need a smooth curve through the intercepts that has the correct left and right hand behavior. To pass through these points, it will have 3 turns (one less than the degree so that’s okay)

3019153 234 xxxxxf

We found the x intercept by putting 0 in for f(x) or y (they are the same thing remember). So we call the x intercepts the zeros of the polynomial since it is where it = 0. These are also called the roots of the polynomial.

51320 xxxx

321)( 23 xxxxgCan you find the zeros of the polynomial?

There are repeated factors. (x-1) is to the 3rd power so it is repeated 3 times. If we set this equal to zero and solve we get 1. We then say that 1 is a zero of multiplicity 3 (since it showed up as a factor 3 times).

What are the other zeros and their multiplicities?

-2 is a zero of multiplicity 2

3 is a zero of multiplicity 1

So knowing the zeros of a polynomial we can plot them on the graph. If we know the multiplicity of the zero, it tells us whether the graph crosses the x axis at this point (odd multiplicities CROSS) or whether it just touches the axis and turns and heads back the other way (even multiplicities TOUCH). Let’s try to graph:

221 xxxfYou don’t need to multiply this out but figure out what the highest power on an x would be if multiplied out. In this case it would be an x3. Notice the negative out in front.

What would the left and right hand behavior be?

What would the y intercept be? (0, 4)

Find the zeros and their multiplicity

1 of mult. 1(so crosses axis at 1)

-2 of mult. 2(so touches at 2)

Steps for Graphing a Polynomial•Determine left and right hand behaviour by looking at the highest power on x and the sign of that term.

•Determine maximum number of turning points in graph by subtracting 1 from the degree.

•Find and plot y intercept by putting 0 in for x

•Find the zeros (x intercepts) by setting polynomial = 0 and solving.

•Determine multiplicity of zeros.

•Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.

432 xxxxfLet’s graph:

•Determine left and right hand behavior by looking at the highest power on x and the sign of that term.

Multiplying out, highest power would be x4

•Determine maximum number of turns in graph by subtracting 1 from the degree.

Degree is 4 so maximum number of turns is 3

•Find and plot y intercept by putting 0 in for x

0403000 2 f•Find the zeros (x intercepts) by setting polynomial = 0 and solving. 430 2 xxx Zeros are: 0, 3, -4

•Determine multiplicity of zeros. 0 multiplicity 2 (touches)

3 multiplicity 1 (crosses)

-4 multiplicity 1 (crosses)

•Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behaviour as a guide.

Here is the actual graph. We did pretty good. If we’d wanted to be more accurate on how low to go before turning we could have plugged in an x value somewhere between the zeros and found the y value. We are not going to be picky about this though since there is a great method in calculus for finding these maxima and minima.

What is we thought backwards? Given the zeros and the degree can you come up with a polynomial? Find a polynomial of degree 3 that has zeros –1, 2 and 3.

What would the function look like in factored form to have the zeros given above?

321 xxxxf

Multiply this out to get the polynomial. FOIL two of them and then multiply by the third one.

64 23 xxxxf

Polynomial Functions

The largest exponent within the polynomial determines the degree of the polynomial.

Polynomial Function in

General Form

Degree Name of Function

1 Linear

2 Quadratic

3 Cubic

4 Quarticedxcxbxaxy 234

dcxbxaxy 23

cbxaxy 2

baxy

Explore Polynomials

Linear Function

Quadratic Function

Cubic Function

Quartic Function

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-60-55-50-45-40-35-30-25-20-15-10-5

510

Cubic PolynomialsLook at the two graphs and discuss the questions given below.

1. How can you check to see if both graphs are functions?

3. What is the end behavior for each graph?4. Which graph do you think has a positive leading coefficient? Why?

5. Which graph do you think has a negative leading coefficient? Why?

2. How many x-intercepts do graphs A & B have?

Graph B

Graph A

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Cubic Polynomials

EquationEquation

Factored form & Factored form & Standard formStandard form

X-InterceptsX-Intercepts Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x+1)(x+4)(x-2)

Standardy=x3+3x2-6x-8

-4, -1, 2 Positive

As x, y and x-,

y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x+1)(x+4)(x-2)

Standardy=-x3-3x2+6x+8

-4, -1, 2 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}

The following chart shows the properties of the graphs on the left.

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

Cubic Polynomials

EquationEquation






Factoredy=(x+3)2(x-1)

Standardy=x3+5x2+3x-9

-3, 1 Positive

As x, y and x-,

y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x+3)2(x-1)

Standardy=-x3-5x2-3x+9

-3, 1 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}


-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

Cubic Polynomials

EquationEquation






Factoredy=(x-2)3

Standardy=x3-6x2+12x-8

2 Positive

As x, y and x-, y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x-2)3

Standardy=-x3+6x2-12x+8

2 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}


-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

Quartic PolynomialsLook at the two graphs and discuss the questions given below.

1. How can you check to see if both graphs are functions?

3. What is the end behavior for each graph?

4. Which graph do you think has a positive leading coefficient? Why?

5. Which graph do you think has a negative leading coefficient? Why?

2. How many x-intercepts do graphs A & B have?

Graph B

Graph A

-5 -4 -3 -2 -1 1 2 3 4 5

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

Quartic Polynomials

EquationEquation

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading




Factoredy=(x-3)(x-1)(x+1)(x+2)

Standardy=x4-x3-7x2+x+6

-2,-1,1,3 Positive

As x, y and x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -12.95}

Factoredy=-(x-3)(x-1)(x+1)(x+2)

Standardy=-x4+x3+7x2-x-6

-2,-1,1,3 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 12.95}


-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

-10 -8 -6 -4 -2 2 4 6 8 10

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

Quartic Polynomials

EquationEquation






Factoredy=(x-4)2(x-1)(x+1)

Standardy=x4-8x3+15x2+8x-16

-1,1,4 Positive

As x, y and x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -16.95}

Factoredy=-(x-4)2(x-1)(x+1)

Standardy=-x4+8x3-15x2-8x+16

-1,1,4 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 16.95}


-5 -4 -3 -2 -1 1 2 3 4 5

-15

-12

-9

-6

-3

3

6

9

12

15

18

-5 -4 -3 -2 -1 1 2 3 4 5

-18

-15

-12

-9

-6

-3

3

6

9

12

15

Quartic Polynomials

EquationEquation







Factoredy=(x+2)3(x-1)

Standardy=x4+5x3+6x2-4x-8

-2,1 Positive

As x, y and x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -8.54}

Factoredy=-(x+2)3(x-1)

Standardy=-x4-5x3-6x2+4x+8

-2,1 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 8.54}


-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

Quartic Polynomials

EquationEquation







Factoredy=(x-3)4

Standardy=x4-12x3+54x2-108x+81

3 Positive

As x, y and x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ 0}

Factoredy=-(x-3)4

Standardy=-x4+12x3-54x2+108x-81

3 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 0}


-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

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