Let's Graph a Polynomial Function

Let's Graph a Polynomial Function

Step by Step Review

 $f\left(x\right)\ =\ x^3-4x^2-4x+16$  

• This lesson will guide you step by step to graph the above function

Multiple Choice

What is the degree of the function

 $f\left(x\right)=x^3-4x^2-4x+16$  

• 1

• 2

• 3

• 4

• 16

Multiple Choice

Is the Degree of

 $f\left(x\right)=x^3-4x^2-4x+16$  Even or Odd?

• Even

• Odd

Multiple Choice

What is the sign of the Leading Coefficient?

 $f\left(x\right)=x^3-4x^2-4x+16$  

• Positive  (+)

• Negative  (-)

Multiple Choice

Which graph will be the End Behaviors of

 $f\left(x\right)=x^3-4x^2-4x+16$  

Multiple Choice

How many x-intercepts should there be for the function

 $f\left(x\right)\ =\ x^3-4x^2-4x+16$  

• 1

• 2

• 3

• 4

• 16

Multiple Choice

What is the y-intercept of

 $f\left(x\right)\ =\ x^3-4x^2-4x+16$  

• 1

• 2

• 3

• 4

• 16

Let's review what we know about the function
 

 $f\left(x\right)\ =\ x^3-4x^2-4x+16$  

• We have a third degree (odd) function with a positive leading coefficient

• We should cross the x-axis 3 times

• Our End Behaviors are shown here

We need to find our x-intercepts by factoring the function

 $f\left(x\right)\ =\ x^3-4x^2-4x+16$  

Multiple Choice

Factor

 $x^3-4x^2-4x+16=0$  to find the x-intercepts (zeros)
Hint - Use Grouping

•  $x\left(x+4\right)\left(x-4\right)=0$  

•  $\left(x-4\right)\left(x-4\right)=0$  

•  $\left(x+2\right)\left(x-2\right)\left(x-4\right)=0$  

•  $\left(x+2\right)\left(x-2\right)\left(x+4\right)=0$  

Factor by Grouping
 $x^3-4x^2-4x+16=0$  

•  $x^2\left(x-4\right)-4\left(x-4\right)=0$  

•  $\left(x-4\right)\left(x^2-4\right)=0$  (Difference of Perfect Squares)

•  $\left(x-4\right)\left(x+2\right)\left(x-2\right)=0$  

Multiple Choice

Now that it is in factored form, what are the zeros?

 $\left(x-4\right)\left(x+2\right)\left(x-2\right)=0$  

• -2, 2, 4

• -2, 2, -4

We have identified our x-intercepts by factoring and placed them on the graph

We also have identified our end behaviors, so we should be ready to sketch the graph

Multiple Choice

Which is the correct graph for

 $f\left(x\right)=x^3-4x^2-4x+16$  

Congratulations!

you just graphed

 $f\left(x\right)=x^3-4x^2-4x+16$