1. Warm Up
What are the zeros of the function?
𝑓 𝑥 =𝑥(𝑥+2)(𝑥−5)

2. 3.4 Evaluating and Graphing Polynomials
𝑓 𝑥 = 𝑥 4 − 2𝑥 2 +𝑥
EQ: How do you sketch the graph of a polynomial using relative mins, and maxs and end behavior?

4. Number of Zeros POSSIBLE Number of Relative Max/Min POSSIBLE
Polynomial of degree
Number of Zeros POSSIBLE
Number of Relative Max/Min POSSIBLE
TWO
ONE
THREE N
N – 1

5. Exit Ticket
How many relative max/mins and zeros could a polynomial of degree 4 have?

6. Warm Up Domain: Range: Y-intercept: Zeros: # Relative maximums:
# Relative minimums:
# Absolute maximums:
# Absolute minimums:
Endpoint behavior:

7. Relative Maximum and Minimums are the “turning points” in a graph.
Absolute Maximum and Minimums are the definitive highest/lowest point of the graph. Absolute maximums and minimums occur only in even functions.

8. B A D C E Relative Maximum: Relative Minimum: Absolute Maximum:
Absolute Minimum: B A D C E

9. Standard Form (Quadratic) Standard Form (Polynomial of n degree)
𝒇 𝒙 = 𝒂𝒙 𝟐 +𝒃𝒙+𝒄
Standard Form (Polynomial of n degree)
𝑃 𝑥 = 𝑎𝑥 𝑛 + 𝑏𝑥 𝑛−1 + 𝑐𝑥 𝑛−2 …𝑑

10. The Domain and Range
The Y-intercept: (0,c)
The X-intercepts (the Zeros, aka Roots): (x,0)
End Behavior
Minimum or Maximum
a. Relative and Absolute min/max
Intervals of increase or decrease

11. “Even” Functions (n is even)
𝑎𝑥 𝑛
Domain: (−∞,∞)
Range: [𝒎𝒊𝒏,∞)
End Behavior:
𝐚𝐬 𝐱 →−∞, 𝒚 → ∞
𝐚𝐬 𝐱 →∞, 𝒚 → ∞
−𝑎𝑥 𝑛
Domain:(−∞,∞)
Range: (−∞,𝒎𝒂𝒙]
𝐚𝐬 𝐱 →−∞, 𝒚 →−∞
𝐚𝐬 𝐱 →∞, 𝒚 →−∞
“Odd” Functions (m is odd)
𝑏𝑥 𝑚
Range: (−∞,∞)
−𝑏𝑥 𝑚

12. Example 1
Graph f(x) by using the zeros.
𝑓 𝑥 =𝑥(𝑥+2)(𝑥−5)

13. Example 2
Zeros:
Degree:

14. Example 2
Zeros: -1, 1, 2
Degree: 4
𝑃 𝑥 = 𝑥 𝑥−1 𝑥−2
Multiplicity is how many times 𝑥−𝑎 appears as a factor.

15. Example 3
What is the multiplicity of each zero?
𝑥(𝑥−2) 2
𝑥 2 (𝑥+2)(𝑥−4)
𝑥−2 2 (𝑥−1)

16. Example 2
What is the multiplicity of each factor?
𝑥(𝑥−2) 2
𝑥 2 (𝑥+2)(𝑥−4)
𝑥−2 2 (𝑥−1)
Zeros: x= 1 (multiplicity 1), 2 (multiplicity 2)
Zeros: x= 0 (multiplicity 2), -2 (multiplicity 1), 4 (multiplicity 1)
Zeros: x= 2 (multiplicity 2), 1 (multiplicity 1)

17. Exit Ticket
Write a polynomial function f in standard form that has 5 and -3 as zeros of multiplicity 1 and 2, respectively
P(x)=

18. Warm Up
Use the zeros, max/mins, end behavior, domain, range and y intercept? Then find the Polynomial expression.
Zeros:
End Behavior:
Domain:
Range:
Y-intercept:
Max and min:
𝑃 𝑥 =

19. Practice Problems
Graph the given polynomials
1.) 𝑃 𝑥 =𝑥 2 (𝑥+1)(𝑥−2)
2.) 𝑃 𝑥 =(𝑥−1)(𝑥+1)(𝑥−3)
3.) 𝑃 𝑥 = 𝑥−3 2 (2𝑥−1)(𝑥+1)

20. Practice Problems
Create the polynomial expression, in standard form, of the given graph?

21. Exit Ticket