1. 5.2: Graphing Polynomial Functions

2. Monomial Binomial Trinomial Polynomial
What are the differences between these words?
What do you think the prefixes mean?
Monomial
Binomial
Trinomial
Polynomial
One term
Ex’s: 4, 2𝑥, 3 𝑥 3
Two terms
Ex’s:
5𝑥+3, 4 𝑥 2 +1
Three terms
Ex:
5 𝑥 2 +𝑥−3
Many terms
EVERY ONE OF THESE IS A POLYNOMIAL!

3. Constant Linear Quadratic Cubic
Indicates the degree of my function
Constant Linear Quadratic Cubic
Ex: Ex: 𝑥 Ex: 𝑥 Ex: 𝑥 3 −2𝑥
A degree is the highest exponent in the polynomial.
Ex: What’s the degree of 4 𝑥 2 +2𝑥?
Degree = Degree = Degree = Degree = 3

4. Ex #1:
Write each polynomial in standard form. What is the classification of each polynomial by degree? By number of terms?
3𝑥+9 𝑥 b) 4𝑥−6 𝑥 2 + 𝑥 𝑥 2 −4𝑥
c) 𝑥 3 −𝑥+5 𝑥 d) 3−4𝑥+2 𝑥 2 +10
Let them do the last two on their own

5. Definition
End behavior: The directions of the graph to the far left and to the far right
4 types:
Up & Up
Down & Down
Down & Up
Up & Down

6. You determine end behavior from the leading term
You determine end behavior from the leading term. If the leading term is 𝑎 𝑥 𝑛 ,
End Behavior

7. Ex #2 What is the end behavior of these graphs? 4 𝑥 3 −3𝑥
−2 𝑥 4 +8 𝑥 3 −8 𝑥 2 +2
8𝑥−7+4 𝑥 2

8. On Your Own What is the end behavior of these graphs?
− 𝑥 3 +2 𝑥 2 −𝑥−2
5 𝑥 4 −7 𝑥 2 +2 𝑥 5 +1
Have them just say this out loud

9. Ex #3
Factor 𝑥 3 −2 𝑥 2 −15𝑥 completely. Hint: Is there a GCF? If so, factor it out first. Now solve 𝑥 𝑥−5 𝑥+3 =0 What is the end behavior of 𝑥 3 −2 𝑥 2 −15𝑥?

10. Zeros X-intercepts Roots
Definition
The solutions of a polynomial equation 𝑷 𝒙 =𝟎 are the zeros of the function
Zeros X-intercepts Roots

11. Back to Ex #3:
When we factored 𝑥 3 −2 𝑥 2 −15𝑥 as 𝑥 𝑥−5 𝑥+3 , we found it’s factored form. When we solved 𝑥 𝑥−5 𝑥+3 =0, we got 𝑥=0, 5, −3. Those were the zeros of the polynomial function.

12. Ex #4
What are the zeros of 𝑦=(𝑥+2)(𝑥−1)(𝑥−3)? To find the zeros, we set 𝑦=0. So we now need to solve 𝑥+2 𝑥−1 𝑥−3 =0 We set each to 0 and get 𝒙=−𝟐, 𝟏, 𝟑

13. On Your Own
What are the zeros of 𝑦=(𝑥−2)(𝑥+4)(𝑥+1)? To find the zeros, we set 𝑦=0. So we now need to solve 𝑥−2 𝑥+4 𝑥+1 =0 We set each to 0 and get 𝒙=𝟐, −𝟒, −𝟏

14. Ex #5
What are the zeros of 𝑦= 𝑥 2 ( 𝑥 2 −3)? If 𝑥 2 𝑥 2 −3 =0, 𝑥 2 =0 and 𝑥 2 −3=0 We then get 𝒙=𝟎 and 𝑥 2 =3. This means that 𝒙=± 𝟑

15. Ex #6: What are the zeros of 𝑦=( 𝑥 2 +3𝑥−4)( 𝑥 2 +6𝑥+8)?
If ( 𝑥 2 +3𝑥−4) 𝑥 2 +6𝑥+8 =0, we would need to solve 𝑥 2 −3𝑥+4=0 and 𝑥 2 +6𝑥+8=0. We can try factoring both to get: 𝑥+4 𝑥−1 =0 𝑥+4 𝑥+2 =0 For the first one, we get 𝑥=−4, 1. For the second one, we get 𝑥=−4, −2. Note: -4 has a multiplicity of 2 since it appeared as an answer twice Therefore, our zeros are: -4, 1, and -2

16. Definition
When we write a polynomial function in factored form, we can find each zero’s multiplicity. The multiplicity is how many times the linear factor appears.
Ex: 𝑦= 𝑥−2 𝑥+3 2
Zeros
Multiplicity 2 1 -3

17. Multiplicity The multiplicity helps us graph because: Back to Ex #4:
If the multiplicity is odd, we cross
If the multiplicity is even, we bounce
Back to Ex #4:
What was the end behavior of
𝑥 3 −2 𝑥 2 −15𝑥? f
Y-int:
Graph.
Zeros
Multiplicity
Bounce/Cross

18. Ex #7 Factor 𝑦= 𝑥 4 −2 𝑥 3 −8 𝑥 2 b) c) End behavior: Y-int: e) Graph.
Since there is a GCF of 𝑥 2 , when we factor out 𝑥 2 , we get
𝑥 2 𝑥 2 −2𝑥−8 = 𝑥 2 𝑥−4 𝑥+2
The zeros are found by setting y to be 0. We get:
𝑥 2 𝑥−4 𝑥+2 =0
𝑥=0, 4, −2
Since our leading term is 𝑥 4 , the end behavior is up and up
(0,0)
Zeros
Multiplicity
Bounce/Cross
Zeros
Multiplicity
Bounce/Cross 2
Cross 4 1
Bounce -2

19. On Your Own Factor 𝑦= 𝑥 3 +6 𝑥 2 +5𝑥 b) c) End behavior: Y-int: Graph.
Zeros
Multiplicity
Bounce/Cross

20. Definition
If b is a zero of a polynomial function, then 𝑥−𝑏 is a linear factor.
Ex: If 2 is a zero, then 𝑥−2 is a linear factor
Question: If -4 is a zero, what is its linear factor?

21. 𝑥−𝑏 is a factor 𝑏 is a zero
Factor Theorem
𝑥−𝑏 is a factor 𝑏 is a zero

22. Ex #8
What is a cubic polynomial function in standard form with zeros -2, 2, and 3?

23. On Your Own:
What is a cubic polynomial function in standard form with zeros -3, -1 and 5?

24. Ex #9
Find an equation of a polynomial in standard form that has the following roots:
𝑥=5, 4𝑖, −4𝑖
𝑦=(𝑥−5)(𝑥−4𝑖)(𝑥+4𝑖)
𝑦=(𝑥−5)( 𝑥 2 +4𝑖𝑥−4𝑖𝑥−16 𝑖 2 )
𝑦=(𝑥−5)( 𝑥 2 +16)
𝑦= 𝑥 3 +16𝑥−5 𝑥 2 −80
𝑦= 𝑥 3 −5 𝑥 2 +16𝑥−80

25. Ex #10
Write the equation for the following polynomial given its graph.
There could be a constant in front of the equation so that is why I put an “a”. We will figure out what that could be later. Also, since all the of zeros cross, they must have an odd multiplicity. Let’s just use a multiplicity of one since the problem doesn’t specify.
𝑦=𝑎 𝑥+3 𝑥−1 𝑥−2
𝑦=𝑎 𝑥 2 −𝑥+3𝑥−3 𝑥−2
𝑦=𝑎 𝑥 2 +2𝑥−3 𝑥−2
𝑦=𝑎( 𝑥 3 −2 𝑥 2 +2 𝑥 2 −4𝑥−3𝑥+6)
𝑦=𝑎( 𝑥 3 −7𝑥+6)
Since we have a y-intercept of -6, if we set x to be zero, we should get have y to be -6. Hence:
−6=𝑎 −7 0 +6
−6=𝑎(6)
𝑎=−1
So we have the equation: 𝑦=−1( 𝑥 3 −7𝑥+6)
𝑦=− 𝑥 3 +7𝑥−6

26. Ex #11: Sketch a polynomial that has these characteristics:
It has degree 4;
It has roots at -2, 1, and 4
As 𝑥→−∞, 𝑦→∞
As 𝑥→∞, 𝑦→∞

27. Ex #12
Write a polynomial function in factored form that has these characteristics:
Have degree 4
Have roots at –4, –2, and 3
As 𝑥→−∞, 𝑦→−∞
As 𝑥→∞, 𝑦→−∞
Include the point (0, 96)
There could be a constant in front of the equation so that is why I put an “a”. We will figure out what that could be later. However, I do know that a should be negative so we can get the end behavior to match. We can assume that the function crosses at -4 and 3 but bounces at -2.
𝑦=𝑎 𝑥+4 (𝑥+2) 2 𝑥−3
Since we have a y-intercept of 96, if we set x to be zero, we should get have y to be 96. Hence:
96=𝑎 0+4 (0+2) 2 0−3
96=𝑎(4)(4)(−3)
96=𝑎(−48)
𝑎=−2
So we have the equation: 𝑦=−2 𝑥+4 (𝑥+2) 2 𝑥−3