Unit 2 Class 5: Multiplicity

Unit 2 Class 5: Multiplicity

Learning Target: Let’s sketch some polynomial functions.

Multiple Choice

State the degree of the following polynomial function:

 $f\left(x\right)=\left(x-2\right)^2\left(x^4-1\right)\left(x+6\right)$  

• 6

• 8

• 7

• 3

What do you notice and what do you wonder?

Vocabulary

Multiplicity

Single Multiplicity

 $\left(x-a\right)$  

Creates a linear graph around the x-axis.

Double Multiplicity

 $\left(x-a\right)^2$  Creates a Quadratic graph around the x-axis.

Triple Multiplicity

 $\left(x-a\right)^3$  Creates a cubic graph around the x-axis.

Multiple Choice

What is the multiplicity of

 $x-1$  in the following polynomial:  $p\left(x\right)=\left(x-1\right)^3\left(x+3\right)^2$  

• 3

• 2

• 1

• No Multiplicity

Multiple Choice

What is the multiplicity of x-intercept 2?

• Single

• Double

• Triple

• No Multiplicity

Key Features of Polynomial Functions

Here are the features need to be able to graph a polynomial function:

Degree, End Behaviors, X-Intercepts, Y-Intercepts , and Domain

Graph the following function:

 $p\left(x\right)=\left(x+3\right)^2\left(x-4\right)^3\left(x-7\right)$  

Graph the following function:

Degree: 6

End Behaviors:  $x\rightarrow\infty\ y\rightarrow\infty\$    $x\rightarrow-\infty\ y\rightarrow\infty$  

X-Intercepts:  $\left(-3,0\right),\left(4,0\right),\left(7,0\right)$  
Y-Intercept:  $\left(0,4032\right)$  
Domain:  $D=\left\{-\infty<x<\infty\right\}$  

Graph the following function:

 $p\left(x\right)=\left(x+5\right)^2\left(x-2\right)^2\left(x-4\right)$  

Graph the following function:

Degree: 6

End Behaviors:  $x\rightarrow\infty\ y\rightarrow\infty\$    $x\rightarrow-\infty\ y\rightarrow-\infty$  

X-Intercepts:  $\left(-5,0\right),\left(2,0\right),\left(4,0\right)$  
Y-Intercept:  $\left(0,-400\right)$  
Domain:  $D=\left\{-\infty<x<\infty\right\}$  

Multiple Choice

What is the domain for any polynomial function?

• $D=\left\{0\le x\le\infty\right\}$

• $D=\left\{-\infty\le x\le\infty\right\}$

• $D=\left\{-\infty<x<\infty\right\}$

• They are never the same

Homework: Unit 2 Class 5 Homework

Do It.