Unit 6: Polynomial Functions

Unit 6: Polynomial Functions

Poll

Current Mood? (Cat Edition)

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

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

Multiple Select

Which of these are polynomial functions?

• $f\left(x\right)=5x+6$

• $f\left(x\right)=x^3-5x^2+x-9$

• $f\left(x\right)=\sqrt{x+8}$

• $f\left(x\right)=\left|x-4\right|$

• $f\left(x\right)=x^2+8x-9$

Polynomial Functions

• No radicals

• No absolute values

• No variables in denominators

Multiple Select

Which of these are polynomial functions?

•  $f\left(x\right)=\frac{5x+6}{7}$ 

•  $f\left(x\right)=\frac{x^3-5x^2+x-9}{x+9}$ 

•  $f\left(x\right)=\sqrt[3]{x^2-3x}$ 

•  $f\left(x\right)=13$ 

•  $f\left(x\right)=2x-3$ 

Multiple Choice

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

• 4

• 3

• 10

• 6

Degree of a Polynomial

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

Multiple Choice

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

• 6

• 5

• 7

• 2

Multiple Choice

What are the approximate roots of the following function?

• $-6,\ -1,\ 2.5$

• $-4,\ 2.3$

• $-6,\ -4,\ -1,\ 2.3,\ 2.5$

• $2.3$

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.

Multiple Choice

What are the approximate roots of the following function?

•  $-3,\ 0,\ 1,\ 2$ 

•  $-3,\ -1,\ 0,\ 0.5$ 

•  $-3,\ 1,\ 2$ 

•  $0,\ 1,\ 2$ 

Multiple Select

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

• As 

   $x→+∞,\ y→+∞$ 

• As  $x→-∞,\ y→+∞$  

• As  $x→+∞,\ y→-∞$  

• As  $x→-∞,\ y→-∞$  

End Behavior Symbols

• As  $x\longrightarrow-\infty$  "On the LEFT side..." 

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

• As  $y\longrightarrow-\infty$  "The end points DOWN."

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

End Behavior in Words

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

• I. DOWN-DOWN

• II. UP-UP 

• III. DOWN-UP 
  

• IV. UP-DOWN

End Behavior In Symbols

• I.  $As\ x\longrightarrow-\infty,\ y\longrightarrow-\infty$        $As\ x\longrightarrow+\infty,\ y\longrightarrow-\infty$  

• II.   $As\ x\longrightarrow-\infty,\ y\longrightarrow+\infty$          $As\ x\longrightarrow+\infty,\ y\longrightarrow+\infty$  

• III.   $As\ x\longrightarrow-\infty,\ y\longrightarrow-\infty$         $As\ x\longrightarrow+\infty,\ y\longrightarrow+\infty$  
  

• IV.   $As\ x\longrightarrow-\infty,\ y\longrightarrow+\infty$         $As\ x\longrightarrow+\infty,\ y\longrightarrow-\infty$  

Multiple Select

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

• As $x→-∞,\ y→-∞$

• As $x→-∞,\ y→+∞$

• As $x→+∞,\ y→-∞$

• As $x→+∞,y→+∞$

Multiple Select

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

• As $x→-∞,\ y→-∞$

• As $x→-∞,\ y→+∞$

• As $x→+∞,\ y→-∞$

• As $x→+∞,y→+∞$

Multiple Select

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

• As $x→-∞,\ y→-∞$

• As $x→-∞,\ y→+∞$

• As $x→+∞,\ y→-∞$

• As $x→+∞,y→+∞$

Multiple Choice

When is this function increasing?

• Interval: (-3, 3)

• Interval: (-5, 5)

• Interval: (-55, 55)

• Interval: (0, 0)

Increasing and Decreasing Intervals

Remember the Roller Coaster?

This is review from Unit 1

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.

Increasing Intervals

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

Multiple Select

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

• Interval:

   $\left(-\infty,\ 1\right)$  

• Interval: 

   $\left(1,\ 3\right)$  

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

• Interval:  
   $\left(4,+\infty\right)$  

Increasing Intervals

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

Multiple Select

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

• Interval:

  $\left(-\infty,-1\right)$

• Interval: $\left(-\infty,1\right)$

• Interval: $\left(-1,1\right)$

• Interval: $\left(1,+\infty\right)$

• Interval: $\left(-2,2\right)$

When is the function decreasing?

Intervals:

 $\left(-\infty,-1\right)\ and\ \left(1,+\infty\right)$  

Multiple Select

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

• Interval:

   $\left(-\infty,-4\right)$ 

• Interval:  $\left(-4,-2\right)$ 

• Interval:  $\left(-2,\ 2\right)$ 

• Interval:  $\left(2,\ 5\right)$ 

• Interval:  $\left(5,\ +\infty\right)$ 

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.