End Behavior of Polynomial Functions

End Behavior of a Polynomial Function

Standard Form of a Polynomial

• List the terms by degrees in DESCENDING order

• The following 2 minute video shows an example

Multiple Choice

Which of the following polynomials is written in STANDARD form?

• $$3x^2-12x+4x^3$$

• $$4x^3+3x^2-12x$$

Multiple Choice

Which of the following polynomials is written in STANDARD form?

• $$-4x^5+2$$

• $$2-4x^5$$

Leading Coefficient

• Once a polynomial is in Standard Form, we can easily identify the Leading Coefficient

• As the name suggests, the Leading Coefficient is the coefficient of the first term (which is the term with the highest exponent)

•  $8x^4-3x^3+12x_{ }^2+5x-16$  is a 4th degree polynomial (aka Quartic) whose LEADING COEFFICIENT is 8.

• The sign of the  leading coefficient combined with the degree will give us a lot of information about the END BEHAVIOR of the polynomial.

Poll

What is the leading coefficient   $$6x^3-2x^2+12x-7$$  

• 6

• 3

• -2

• 12

• 1

Open Ended

Name the leading coefficient
$$-7x^2+3x-5$$  

Even Degree Polynomials

• Polynomials with an EVEN degree will always end in the same direction.

• UP if the Leading Coefficient is POSITIVE

• DOWN if the Leading Coefficient is NEGATIVE

Odd Degree Polynomials

• Polynomials with an ODD degree will always in in opposite directions (one up, one down).

• If the Leading Coefficient is POSITIVE, the polynomial will go up to the right and down to the left.

• If the Leading Coefficient is NEGATIVE, the polynomial will go down to the right and up to the left.

Multiple Choice

Which polynomial has an ODD degree?

• $$3x^2-5x+1$$

• $$2x^3+4x^2+6x-8$$

Multiple Select

Select all the polynomials that have an odd degree.

• $$6x^3+8x-1$$  

• $$4x^6-5x^3+2x^2+8x-1$$  

• $$x^2+7x-3$$  

• $$x+5$$  

• $$x^5-2x^2$$  

End Behavior

The guy on this 3 minute long video is super goofy, but his information is good!

Poll

What is the degree of this polynomial?

$$-3x^4\ +\ 2x^3-5x^2+10x-7$$

• 1
  

• 2

• 3

• 4

• 5

Poll

Is the degree of this polynomial ODD or EVEN?

$$-3x^4\ +\ 2x^3-5x^2+10x-7$$

• ODD

• EVEN

Poll

Is the leading coefficient of this polynomial POSITIVE or NEGATIVE?

$$-3x^4\ +\ 2x^3-5x^2+10x-7$$

• POSITIVE

• NEGATIVE

Poll

Describe the END BEHAVIOR of this polynomial

$$-3x^4\ +\ 2x^3-5x^2+10x-7$$

• Both sides go up

• Both sides go down

• Left up and right down

• Left down and right up

Examples

The following slides show examples of polynomials of various degrees and leading coefficients.

Multiple Choice

A polynomial with an EVEN degree will have...

• the ends going the same direction

• the ends going in different direction

Multiple Choice

A polynomial with an ODD degree will have...

• the ends going the same direction

• the ends going in different direction

Multiple Choice

A Quadratic equation with a negative Leading Coefficient will have

• both end going up

• both ends going down

• the left down and the right up

• the left up and the right down

Multiple Choice

Does this graph represent a polynomial with an ODD or EVEN degree?

• Odd

• Even

Multiple Choice

Is the Leading Coefficient of this CUBIC polynomial POSITIVE or NEGATIVE?

• Positive

• Negative

Multiple Choice

Does this graph represent a polynomial with an ODD or EVEN degree?

• Odd

• Even

Multiple Choice

Is the Leading Cofficient of the QUARTIC polynomial POSITIVE or NEGATIVE?

• Positive

• Negative

Multiple Choice

Which of the following graphs represents a polynomial that has a POSITIVE leading coefficient and EVEN degree?

Multiple Choice

Which of the following graphs represents a polynomial that has a NEGATIVE leading coefficient and EVEN degree?

Multiple Choice

Which of the following graphs represents a polynomial that has a POSITIVE leading coefficient and ODD degree?

Multiple Choice

Which of the following graphs represents a polynomial that has a NEGATIVE leading coefficient and ODD degree?

End Behavior Notation

• It looks crazy but you really only have to fill in the part that with the yellow underline.

• If the graph is going up, use  $+\infty$  

• If the graph is going down, use  $-\infty$ 

• Remember that  $+\infty$  means RIGHT and  $-\infty$  means LEFT when we are talking about x.

Multiple Choice

Please complete the end behavior function notation $$as\ x\rightarrow+\infty,\ f\left(x\right)\ \rightarrow$$

• $$+\infty$$  

• $$-\infty$$  

Multiple Choice

Please complete the end behavior function notation $$as\ x\rightarrow-\infty,\ f\left(x\right)\ \rightarrow$$

• $$+\infty$$  

• $$-\infty$$  

Multiple Choice

Please complete the end behavior function notation $$as\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow$$

• $$+\infty$$  

• $$-\infty$$  

Multiple Choice

Please complete the end behavior function notation $$as\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow$$

• $$+\infty$$  

• $$-\infty$$  

Multiple Choice

Please complete the end behavior function notation $$as\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow$$

• $$+\infty$$  

• $$-\infty$$  

Multiple Choice

Please complete the end behavior function notation $$as\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow$$

• $$+\infty$$  

• $$-\infty$$  

REAL Zeroes

• Indicates how many times the function crosses the x axis.

• There are 6 real zeros in this example.

REAL Zeroes, continued

• This polynomial with a positive Leading Coefficient and odd degree (d=5) has 3 real zeroes.

• There are 2 imaginary zeroes

• The total number of zeroes (real + imaginary) will equal the degree

Multiple Choice

How many REAL zeroes are shown in this polynomial function?

• 0

• 1

• 2

• 3

• 4

Multiple Choice

How many REAL zeroes are shown in this polynomial function?

• 0

• 1

• 2

• 3

• 4

Open Ended

How many REAL zeroes are shown in this polynomial function?