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 Lesson 3.2 Polynomial Functions and Models

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A. What are the characteristics of graphs of polynomial

functions? Video tutorial A-D 

B. Even and Odd Functions

C. The Leading Coefficient Test

  Whether a graph eventually rises or falls can be determined by the functions degree

(odd or even) and by applying the leading coefficient test.

  When the degree is odd

  The beginning and end behavior is similar to y = x3.

  If the leading coefficient is positive, the graph falls to the left and rises to theright.

  If the leading coefficient is negative, the graph rises to the left and falls to

the right.  When the degree is even

  The beginning and end behavior is similar to y = x2.

  If the leading coefficient is positive, the graph rises to the left and rises to

the right.

  If the leading coefficient is negative, the graph falls to the left and falls to

the right.

1.  Polynomial functions are continuous.

 

This means that the graphs have no breaks, holes, or gaps.

  See pg. 195

2.  Polynomial functions are smooth.  This means there are no sharp turns, like in the graph of y = |x|.

  See pg. 195

  You can say that a function is continuous if its graph can be drawn with a pencilwithout lifting the pencil from the paper.

 

If the degree, n, of a function is even, the graph will be similar to the graph of y = x2 

  If the degree, n, of a function is odd, the graph will be similar to the graph of y = x3.

http://www.youtube.com/watch?v=mt0S7hvz930&hd=1http://www.youtube.com/watch?v=mt0S7hvz930&hd=1http://www.youtube.com/watch?v=mt0S7hvz930&hd=1http://www.youtube.com/watch?v=mt0S7hvz930&hd=1

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Example: Use the LCT to determine the end behavior

of the function. Sketch.

D. Zeros

If a is a real number, then the following statements are equivalent:

1.  x = a is a zero of the function f.

  Remember, that zeros are where the graph crosses the x-axis.

2.  x = a is a solution of the equation f(x) = 0.3.  (x –  a) is a factor of the polynomial f(x).

4.  (a, 0) is an x-intercept of the graph of f.

Multiplicities:

  If a function has a zero that repeats an even number of times (like x = 2 twice), then the

graph will bounce off the zero instead of cross though it.  If a function has a zero that repeats an odd number of times (like x = 2 three times), then

the graph will cross through the zero.

 -2 0 2

Remember, wecan factor thispolynomial.

 

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E. Steps for graphing polynomial functions by hand.

Sketch a graph of the polynomial function. Make sure

your graph shows all intercepts and exhibits the proper

end behavior.

1.  Use the Leading Coefficient Test to determine end behavior.  Is the degree even or odd?

  Is the leading coefficient positive or negative?

 

Sketch the end behavior… 2.  Find the zeros of the function algebraically.

  Factor and set each group equal to zero and solve.

  These numbers are the x-intercepts of the graph.3.   Now, plot your zeros on the x-axis and sketch the end-behavior that you found from

the LCT.

4.  To find out what happens in between your zeros, you need to plug in a point that is

 between each pair of x-intercepts.  Make a table of values

  Plug in points that are between each pair of zeros

  Plot your points

5. 

Connect your points in one continuous smooth curve.

3. ( ) 9a f x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Video tutorial 

2. Find the zeros of the function algebraically. Factor and set

equal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each

 pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve.

x y

http://www.youtube.com/watch?v=zCDDjsQn108&hd=1http://www.youtube.com/watch?v=zCDDjsQn108&hd=1http://www.youtube.com/watch?v=zCDDjsQn108&hd=1

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3 2. 2b f x x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Video tutorial 

2. Find the zeros of the function algebraically. Factor and set

equal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each

 pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve.

x y

3 2. ( ) 3 4 12c f t x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Video tutorial 

2. Find the zeros of the function algebraically. Factor and set

equal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each

 pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve.

x y

http://www.youtube.com/watch?v=vE3TbdgA8OQ&hd=1http://www.youtube.com/watch?v=vE3TbdgA8OQ&hd=1http://www.youtube.com/watch?v=3EmZcR6J7Dc&hd=1http://www.youtube.com/watch?v=3EmZcR6J7Dc&hd=1http://www.youtube.com/watch?v=3EmZcR6J7Dc&hd=1http://www.youtube.com/watch?v=vE3TbdgA8OQ&hd=1

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4 3. 2 8 16d f x x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Vid#eo tutorial 

2. Find the zeros of the function algebraically. Factor and setequal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve.

x y

6 3. ( ) 2 1e f x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Video tutorial 

2. Find the zeros of the function algebraically. Factor and set

equal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each

 pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve.

x y

http://www.youtube.com/watch?v=5DGqMx8dTgc&hd=1http://www.youtube.com/watch?v=5DGqMx8dTgc&hd=1http://www.youtube.com/watch?v=l3JI5MgTbCU&hd=1http://www.youtube.com/watch?v=l3JI5MgTbCU&hd=1http://www.youtube.com/watch?v=l3JI5MgTbCU&hd=1http://www.youtube.com/watch?v=5DGqMx8dTgc&hd=1

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. 2 1 1 3 f P x x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Video tutorial 2. Find the zeros of the function algebraically. Factor and set

equal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each

 pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve. x y

21

. ( ) 55

 g P x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Video tutorial 

2. Find the zeros of the function algebraically. Factor and set

equal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each

 pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve.

x y

http://www.youtube.com/watch?v=bX2w7y2IzDM&hd=1http://www.youtube.com/watch?v=bX2w7y2IzDM&hd=1http://www.youtube.com/watch?v=ysd9U9du8ng&hd=1http://www.youtube.com/watch?v=ysd9U9du8ng&hd=1http://www.youtube.com/watch?v=ysd9U9du8ng&hd=1http://www.youtube.com/watch?v=bX2w7y2IzDM&hd=1

8/20/2019 Lesson 3.2 Video Tutorials Polynomial Functions

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 Lesson 3.2 Polynomial Functions and Models

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2 3

. 1 2h P x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Video tutorial 

2. Find the zeros of the function algebraically. Factor and setequal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each

 pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve.x y

23. 3 4i P x x x x  

1. From the Leading Coefficient Test, what is the end behavior?

Video tutorial 

2. Find the zeros of the function algebraically. Factor and set

equal to zero.

The zeros are : __________________________________

3. Plot your zeros on the x-axis and sketch the end behavior of

the function that comes from the Leading Coefficient Test.

4. Make a table of values. Plug in points that are between each

 pair of zeros.

5. Plot your points.

6. Connect your points in one continuous smooth curve.

x y

http://www.youtube.com/watch?v=WleDd8qnX1Q&hd=1http://www.youtube.com/watch?v=WleDd8qnX1Q&hd=1http://www.youtube.com/watch?v=TahUvXpWIAw&hd=1http://www.youtube.com/watch?v=TahUvXpWIAw&hd=1http://www.youtube.com/watch?v=TahUvXpWIAw&hd=1http://www.youtube.com/watch?v=WleDd8qnX1Q&hd=1