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Unit 6: Polynomial Functions

Unit 6: Polynomial Functions

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•

Mathematics

•

9th - 12th Grade

•

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Created by

Natalie McIntosh

Used 14+ times

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15 Slides • 15 Questions

1

Unit 6: Polynomial Functions

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2

Poll

Current Mood? (Cat Edition)

3

About Unit 6: Polynomial Functions

  • One-day Lesson, One-day Review, Canvas Quiz

  • This Quizizz is the lesson and is graded for effort and completeness.

  • After we return from Spring Break, we will do a review and a practice quiz in Canvas.

  • Then we will move on to Unit 7: Rational Functions

4

What you need to be able to do with Polynomial Functions:

  • Determine the degree

  • Determine the zeros/roots/solutions

  • Determine the end behavior

  • Determine the increasing and decreasing intervals

5

Multiple Select

Which of these are polynomial functions?

1

f(x)=5x+6f\left(x\right)=5x+6

2

f(x)=x3−5x2+x−9f\left(x\right)=x^3-5x^2+x-9

3

f(x)=x+8f\left(x\right)=\sqrt{x+8}

4

f(x)=∣x−4∣f\left(x\right)=\left|x-4\right|

5

f(x)=x2+8x−9f\left(x\right)=x^2+8x-9

6

Polynomial Functions

  • No radicals

  • No absolute values

  • No variables in denominators

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7

Multiple Select

Which of these are polynomial functions?

1

 f(x)=5x+67f\left(x\right)=\frac{5x+6}{7} 

2

 f(x)=x3−5x2+x−9x+9f\left(x\right)=\frac{x^3-5x^2+x-9}{x+9} 

3

 f(x)=x2−3x3f\left(x\right)=\sqrt[3]{x^2-3x} 

4

 f(x)=13f\left(x\right)=13 

5

 f(x)=2x−3f\left(x\right)=2x-3 

8

Multiple Choice

What is the degree of the following polynomial?  f(x)=6x3−5x2+x+10f\left(x\right)=6x^3-5x^2+x+10  

1

4

2

3

3

10

4

6

9

Degree of a Polynomial

For polynomials with one variable, the degree is the highest exponent on the variable.

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10

Multiple Choice

What is the degree of the following polynomial?  f(x)=7x2+8x5−x3+14x−1+2x4f\left(x\right)=7x^2+8x^5-x^3+14x-1+2x^4  

1

6

2

5

3

7

4

2

11

Multiple Choice

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What are the approximate roots of the following function?

1

−6, −1, 2.5-6,\ -1,\ 2.5

2

−4, 2.3-4,\ 2.3

3

−6, −4, −1, 2.3, 2.5-6,\ -4,\ -1,\ 2.3,\ 2.5

4

2.32.3

12

Zeros/Roots/Solutions

The x-values where the graph crosses or meets the x-axis.


For example, the roots of the graph shown are 1 and 4.

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13

Multiple Choice

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What are the approximate roots of the following function?

1

 −3, 0, 1, 2-3,\ 0,\ 1,\ 2 

2

 −3, −1, 0, 0.5-3,\ -1,\ 0,\ 0.5 

3

 −3, 1, 2-3,\ 1,\ 2 

4

 0, 1, 20,\ 1,\ 2 

14

Multiple Select

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What is the end behavior of this graph? (Check all that apply.)

1

As 

 x→+∞, y→+∞x→+∞,\ y→+∞ 

2

As  x→−∞, y→+∞x→-∞,\ y→+∞  

3

As  x→+∞, y→−∞x→+∞,\ y→-∞  

4

As  x→−∞, y→−∞x→-∞,\ y→-∞  

15

End Behavior Symbols

  • As  x⟶−∞x\longrightarrow-\infty  "On the LEFT side..." 

  • As  x⟶+∞x\longrightarrow+\infty  "On the RIGHT side..." 

  • As  y⟶−∞y\longrightarrow-\infty  "The end points DOWN."

  • As  y⟶+∞y\longrightarrow+\infty  "The end points UP."

16

End Behavior in Words

  • "Where are the ends of the graph pointing?"

  • I. DOWN-DOWN

  • II. UP-UP 

  • III. DOWN-UP 

  • IV. UP-DOWN

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17

End Behavior In Symbols

  • I.  As x⟶−∞, y⟶−∞As\ x\longrightarrow-\infty,\ y\longrightarrow-\infty        As x⟶+∞, y⟶−∞As\ x\longrightarrow+\infty,\ y\longrightarrow-\infty  

  • II.   As x⟶−∞, y⟶+∞As\ x\longrightarrow-\infty,\ y\longrightarrow+\infty          As x⟶+∞, y⟶+∞As\ x\longrightarrow+\infty,\ y\longrightarrow+\infty  

  • III.   As x⟶−∞, y⟶−∞As\ x\longrightarrow-\infty,\ y\longrightarrow-\infty         As x⟶+∞, y⟶+∞As\ x\longrightarrow+\infty,\ y\longrightarrow+\infty  

  • IV.   As x⟶−∞, y⟶+∞As\ x\longrightarrow-\infty,\ y\longrightarrow+\infty         As x⟶+∞, y⟶−∞As\ x\longrightarrow+\infty,\ y\longrightarrow-\infty  

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18

Multiple Select

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Which of the following describes an end behavior of the function graphed? Select all that apply.

1

As x→−∞, y→−∞x→-∞,\ y→-∞

2

As x→−∞, y→+∞x→-∞,\ y→+∞

3

As x→+∞, y→−∞x→+∞,\ y→-∞

4

As x→+∞,y→+∞x→+∞,y→+∞

19

Multiple Select

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Which of the following describes an end behavior of the function graphed? Select all that apply.

1

As x→−∞, y→−∞x→-∞,\ y→-∞

2

As x→−∞, y→+∞x→-∞,\ y→+∞

3

As x→+∞, y→−∞x→+∞,\ y→-∞

4

As x→+∞,y→+∞x→+∞,y→+∞

20

Multiple Select

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Which of the following describes an end behavior of the function graphed? Select all that apply.

1

As x→−∞, y→−∞x→-∞,\ y→-∞

2

As x→−∞, y→+∞x→-∞,\ y→+∞

3

As x→+∞, y→−∞x→+∞,\ y→-∞

4

As x→+∞,y→+∞x→+∞,y→+∞

21

Multiple Choice

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When is this function increasing?

1

Interval: (-3, 3)

2

Interval: (-5, 5)

3

Interval: (-55, 55)

4

Interval: (0, 0)

22

Increasing and Decreasing Intervals

Remember the Roller Coaster?

This is review from Unit 1

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23

Increasing Intervals

  • The roller coaster always travels from left to right.

  • "For what values of x is the roller coaster going uphill?"

  • From x=-4 to x=1, the roller coaster is going upwards.

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24

Increasing Intervals

  • Answer is written as the interval  (−4,1)(-4,1)   or  −4<x<1-4<x<1  

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25

Multiple Select

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When is this function increasing? (Select all that apply.)

1

Interval:

 (−∞, 1)\left(-\infty,\ 1\right)  

2

Interval: 

 (1, 3)\left(1,\ 3\right)  

3

Interval: 
 (3, 4)\left(3,\ 4\right)  

4

Interval:  
 (4,+∞)\left(4,+\infty\right)  

26

Increasing Intervals

Intervals  (1,3) and (4,∞)\left(1,3\right)\ and\ \left(4,\infty\right)  
or  1<x<3 and 4<x<∞1<x<3\ and\ 4<x<\infty  

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27

Multiple Select

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When is this function decreasing? (Check all that apply.)

1

Interval:

(−∞,−1)\left(-\infty,-1\right)

2

Interval: (−∞,1)\left(-\infty,1\right)

3

Interval: (−1,1)\left(-1,1\right)

4

Interval: (1,+∞)\left(1,+\infty\right)

5

Interval: (−2,2)\left(-2,2\right)

28

When is the function decreasing?

Intervals:

 (−∞,−1) and (1,+∞)\left(-\infty,-1\right)\ and\ \left(1,+\infty\right)  

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29

Multiple Select

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When is this function decreasing? (Check all that apply.)

1

Interval:

 (−∞,−4)\left(-\infty,-4\right) 

2

Interval:  (−4,−2)\left(-4,-2\right) 

3

Interval:  (−2, 2)\left(-2,\ 2\right) 

4

Interval:  (2, 5)\left(2,\ 5\right) 

5

Interval:  (5, +∞)\left(5,\ +\infty\right) 

30

Next Class - Friday, March 12th

  • Friday is an Early Release Day. Classes are 50 minutes long and we get out at 11:15am.

  • Log into Meet, take attendance, listen to any announcements.

  • I will then stay on Meet to help anyone who has questions about any assignments from the third quarter.

pattern-tertiary

Unit 6: Polynomial Functions

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