1. 2.1 Day 2 Homework Answers D: −2,∞ 𝑅: [0, ∞) x-int: -2, y-int: 7.35
End Behavior: 𝑙𝑖𝑚 𝑥→∞ 𝑓 𝑥 = ∞
continuous, increasing −2, ∞
D: −3,∞ 𝑅: (−∞, −3]
x-int: none, y-int: -3.71
End Behavior: lim 𝑥→∞ 𝑓 𝑥 =− ∞
continuous, decreasing −3, ∞

2. D: (−∞,0.25] 𝑅: [−16, ∞)
x-int: , y-int: -15
End Behavior: lim 𝑥→−∞ 𝑓 𝑥 = ∞
continuous, increasing −∞, 0.25
D: (−∞,∞) 𝑅:(−∞,∞)
x-int: , y-int: -6.5
End Behavior: lim 𝑥→−∞ 𝑓 𝑥 = ∞, lim 𝑥→∞ 𝑓 𝑥 = −∞
continuous, decreasing (−∞,∞)

3. 2.2 – Polynomial Functions
Objective: graph polynomial functions & model real-world data with polynomial functions.

4. Polynomial Functions:
Let n be a nonnegative integer and let 𝑎 0 , 𝑎 1 , 𝑎 2 … 𝑎 𝑛−1 , 𝑎 𝑛 be real numbers with 𝑎 𝑛 ≠0, then: 𝑓 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +…+ 𝑎 2 𝑥 2 + 𝑎 1 𝑥+ 𝑎 0 is called a polynomial function of degree n. The leading coefficient of a polynomial function is the coefficient of the variable with the greatest exponent.

5. Example of a Polynomial Function
𝑓 𝑥 =−3 𝑥 5 +4 𝑥 3 −7 𝑥 2 +𝑥−18 Degree: 5 Leading Coefficient: −3

6. Polynomial Graphs:

7. Example 1:
Describe the transformation of the graph, then give domain, range, intercepts, end behavior, continuity, & increasing and/or decreasing intervals. A. f (x) = (x – 3)5 B. f (x) = x 6 – 1

8. Leading Term Test Used to Describe End Behavior of Polynomials
𝒏 𝒐𝒅𝒅, 𝒂 𝒏 positive
 odd degree, leading coefficient positive
𝒏 𝒐𝒅𝒅, 𝒂 𝒏 negative
 odd degree, leading coefficient negative

9. Leading Term Test Used to Describe End Behavior of Polynomials
𝒏 𝒆𝒗𝒆𝒏, 𝒂 𝒏 positive
 even degree, leading coefficient positive
𝒏 𝒆𝒗𝒆𝒏, 𝒂 𝒏 negative
 even degree, leading coefficient negative

10. Example 2:
Without a calculator, describe the end behavior of the graph using limits. Explain your reasoning using the leading term test.
A. f (x) = 3x 4 – x 3 + x 2 + x – 1
B. f (x) = –3x 2 – 2x 5 – x 3

11. Turning Points
Turning Points: where the graph of a function changes from increasing to decreasing and vice versa.
A polynomial of degree n has at most n – 1 turning points

12. Zeros Where the graph crosses the x-axis
A polynomial of degree n has at most n real zeros

13. Example 3:
State the number of possible real zeros and turning points of the given function.. Then determine all of the real zeros by factoring.
A. f (x) = x 3 + 5x 2 + 4x B. f (x) = x 4 – 13x

14. Quadratic Form
A polynomial expression in x is in quadratic form if it is written 𝑎 𝑢 2 +𝑏𝑢+𝑐
Example: 𝑥 4 +5 𝑥 2 −14 can be written as ( 𝑥 2 ) 2 −5 𝑥 2 −14
If we let 𝑢= 𝑥 2 then the expression becomes 𝑢 2 −5𝑢−14

15. Example 4:
State the number of possible real zeros and turning points. Then determine all of the real zeros by factoring.
A. g (x) = x 4 – 4x B. h (x) = x 5 - 5x 3 - 6x

16. Factoring by Grouping (2.2 Notes Cont.)
𝑓 𝑥 = 𝑥 3 +2 𝑥 2 −𝑥−2

17. Example 5 (2.2 Notes Cont.)
State the number of possible real zeros and turning points of h (x) = x 4 + 5x 3 + 6x 2. Then determine all of the real zeros by factoring.

18. Repeated Zeros
If a zero c has odd multiplicity, then the graph crosses the x-axis at x = c
If a zero c has even multiplicity, then the graph is tangent to the x-axis at x = c
Example:
𝑓 𝑥 =𝑥(𝑥−1)(𝑥+2 ) 2

19. Example 6:
For f (x) = x(3x + 1)(x – 2) 2: A. Apply the leading-term test. B. Determine the zeros and state the multiplicity of any repeated zeros.

20. Example 6:
C. Sketch the graph (without a calculator).

21. Example 6: A) 0, –2 (multiplicity 2), (multiplicity 3)
Determine the zeros and state the multiplicity of any repeated zeros for f (x) = 3x(x + 2)2(2x – 1)3.
A) 0, –2 (multiplicity 2), (multiplicity 3)
B) 2 (multiplicity 2), – (multiplicity 3)
C) 4 (multiplicity 2), (multiplicity 3)
D) –2 (multiplicity 2), (multiplicity 3)

22. Exit Slip Find the real zeros of by FACTORING:
(you may check graphically)