2. Who Am i?
I am a number with a couple of friends, quarter a dozen, and you’ll find me again. Who am I?

3. PBIS and Classroom Expectations
Parkside High School
PBIS and Classroom Expectations

4. #RAMFAM OBJECTIVES
All students will use skills to interact effectively with diverse populations
All students will help create a climate that is welcoming, encouraging, and supportive
All students will learn in a positive learning environment

5. What is PBIS? systems of support that include:
positive strategies that define and teach appropriate student behaviors
programs that support students to create a positive school environment
implemented in areas including:
all classrooms
non-classroom settings (such as hallways, cafeteria and restrooms)

6. PARKSIDE #RAMFAM
# - To bring immediate attention, unite a group with a common theme
Being a part of Parkside’s Legacy
Respectful – Having high regard for self and others while appreciating differences and commonalities
Being polite and mannerly toward others will create a positive environment
Attendance – To be present and devoted
Being here and on time prepares you to go elsewhere
Management – To take charge and handle until progress is made or accomplished
Being able to devote appropriate time to academics and extracurriculars

7. PARKSIDE #RAMFAM
Focused – The act of directing one’s attention or efforts.
Be focused on your potential
Achievement – The act of reaching an objective, accomplishment, and completing something successfully
Being able to say “I did it!”
Meaningful – To find purpose, value, and significance
Being able to make your time here count

8. Incentives and Supports
“RAM”dom Pride Awards
Staff will nominate students who are showing #RAMRAM within the school environment.
Students will be recognized on Channel 27, receive a certificate, and fun prizes!
Students of the Month
Each department will nominate an Academic SOM and Character SOM who are showing #RAMFAM.
Students will be recognized on Channel 27, PBIS bulletin board, receive a certificate, and eligible for an end of the year event.

9. Staff Incentives RAMtastic Teacher Award
Students, Staff, and parents can nominate staff who are showing #RAMFAM within the school
Staff will be recognized on Channel 27, bulletin board, receive a certificate and fun prizes!

10. #RAMFAM in the Classroom
School
Expectations
Respect
Attendance
Management
Focused
Achievement
Meaningful
Expected Student Behavior
- Be respectful and cooperative with others
- Take pride in my appearance and follow the dress code
- Follow classroom expectations
- Use appropriate language
- Arrive to class on time
- Be in my seat before bell sounds and ready to learn
- Honor the 10/10 policy
- Complete work on time
- Be prepared for class with appropriate materials
- Use agenda for assignments
- Plan ahead for tests/projects/ papers/etc.
- Work quietly
- Strive to do the best job you can do
- Review class material often
- Study hard
- Set obtainable goals and strive to accomplish them
- Actively Participate
- Complete all assignments honestly
- Work to the best of my ability
- Ask questions when appropriate
- Participate in class discussions

11. Who’s Who in our Class Name Role Model(s) Hobbies Summer Fun
Challenges (in school or personal)
Goal for this year

12. Creating a SMART Goal
Specific: the goal should be well defined so that you know exactly the intended outcome for the goal.
Measurable: the goal should have a way to measure progress to provide evidence of it being met or not met.
Achievable: the goal should challenge you, so that it is worthwhile to work on, but that it is also a realistic task for you to accomplish.
Rewarding/Relevant: the goal should be important to you, your work and focus on results.
Time-bound: the goal should always indicate a timed deadline for its outcome.

13. Videos Mathmaticious https://www.youtube.com/watch?v=6cAs1YBELmA
Logic puzzles

16. How many tiles did he add?
INTRO TO SEQUENCES AND SERIES
Guido wants to create a tile mosaic around the Ram-Fountain. In the first week he begins his work by placing red tiles around the fountain as shown:
How many tiles did he add?

17. How many tiles did he add?
In the second week, he adds to his work by placing purple tiles around the fountain as shown:
How many tiles did he add?

18. How many tiles did he add?
In the third week, he adds to his work by placing green tiles around the fountain as shown:
How many tiles did he add?

19. INTRO TO SEQUENCES AND SERIES
If he continues this pattern, how many blue tiles will he need to complete his fourth week of work?

20. INTRO TO SEQUENCES AND SERIES
In the 10th week, how many tiles would you expect him to add. How many total are around the fountain? Explain how you arrived at this answer.

21. INTRO TO SEQUENCES AND SERIES
What is a “Sequence”?
A list of things (usually numbers) that are in order.

22. INTRO TO SEQUENCES AND SERIES
What is an “Infinite Sequence”?
An infinite sequence is a function whose domain is the set of positive integers. The function values
a1, a2, a3, a4, a5, a6, a7. . .
Are the terms of the sequence. If the domain of a function consists of the first n positive integers only, the sequence is a finite sequence
A list of numbers separated by commas:
1, 2, 4, 8...., 128………
INTRO TO SEQUENCES AND SERIES
What is a “Sequence”?
A list of numbers separated by commas:
1, 2, 4, 8...., 128.

23. INTRO TO SEQUENCES AND SERIES
Types of a “Sequence”?

24. INTRO TO SEQUENCES AND SERIES
Types of a “Sequence”?
Arithmetic: a sequence of numbers that has a common difference (d). EX: 1, 3, 5, 7 the common difference is 2. (each term is arrived at through addition)

25. INTRO TO SEQUENCES AND SERIES
Types of a “Sequence”?
Arithmetic: a sequence of numbers that has a common difference (d). EX: 1, 3, 5, 7 the common difference is 2. (each term is arrived at through addition)
Geometric: a sequence of numbers that has a common ratio (r). EX: 3, 12, 48, the common ratio is 4. (each term is arrived at through multiplication)

27. How to find the Common Ratio

28. Warm-Up Honors Algebra 2 9/7/18
Decide whether each sequence is arithmetic or geometric. State the common difference or common ratio.
4, 10, 16, 22, 28, …
5, 8, 11, 14, …
4, 12, 36, 108, …
2, -4, 8, -16, …

29. INTRO TO SEQUENCES AND SERIES
What is a “term”?

30. INTRO TO SEQUENCES AND SERIES
What is a “term”?
A specific number in a sequence or series.
a1= first term
a2= second term
an=nth term (or last term)

31. ex1.
The nth term of a sequence is given by:
an = n2 + 2
a) Write out the first 5 terms.

32. ex 1 (continued)
The nth term of a sequence is given by:
an = n2 + 2
b) What is the value of the 7th term?

33. ex1. (continued)
The nth term of a sequence is given by:
an = n2 + 2
c) Find a9.

34. ex2.
The nth term of a sequence is given by:
an = 4(n + 2)(n – 1)
Use the table function of the graphing utility on your calculator to write out the first 5 terms.

35. INTRO TO SEQUENCES AND SERIES
What is a “Recursively defined Sequence”?
A sequence in which calculating each term is based on the value of the term before.

36. INTRO TO SEQUENCES AND SERIES
Recursively defined Sequence
Find the first six terms of the “famous” sequence described below

37. Important Formulas for an Arithmetic Sequence:
Recursive Formula
Explicit Formula
𝑎 1 = ?
an = (an – 1 ) +d
𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑
Where:
an is the nth term in the sequence
a1 is the first term
n is the number of the term
d is the common difference

38. Important Formulas for a Geometric Sequence:
Recursive Formula
Explicit Formula
𝑎 1 = ?
an = (an – 1 ) r
𝑎 𝑛 = 𝑎 1 ∙ 𝑟 𝑛−1
Where:
an is the nth term in the sequence
a1 is the first term
n is the number of the term
r is the common ratio

39. In the sequence 10, 40, 70, 100, ….
The constant difference between the terms is 30
The first term of the sequence is 10
The recursive formula for this sequence would be:
**I substituted in 10 for the first term and 30 for the constant difference**

40. Write the recursive formula of the sequence 4, 7, 10, 13, ….

41. In the sequence 4, 7, 10, 13, ….
To find the 5th term recursively, I substitute it into the formula I just made:
an = an-1 + 3
a5 = a in words: 5th term equals the 4th term plus 3
a5 =
a5 = 16

42. The constant difference between the terms is 30
In the sequence 10, 40, 70, 100, ….
The constant difference between the terms is 30
The first term of the sequence is 10
The explicit formula for this sequence would be:
an = ( n - 1) which simplifies to: an = n
**I substituted in 10 for the first term and 30 for the constant difference and distributed**

43. Write the explicit formula of the sequence 4, 7, 10, 13, ….

44. In the sequence 4, 7, 10, 13, ….
To find the 11th term explicitly, I substitute in the nth term into the formula I just made:
an = 1 + 3n
a11 = 1 + 3(11)
a11 = 34

45. Find the 15th term of the sequence using the formula: an = 1 + 3n

46. Geometric Sequence Ex: Write the explicit formula for the sequence
9, 3, 1, ….
Write the recursive formula for the sequence
9, 3, 1, ….

48. Go over Hw

49. Determine if the following are Explicit or Recursive Formulas

50. Write the recursive formula for the following
5, 15, 45, ….
Write the explicit formula for the following
5, 15, 45, ….

51. #1. 𝑎 1 =−4 𝑎 𝑛 =−3( 𝑎 𝑛−1 ) #2. 𝑎 1 =5 𝑎 𝑛 = 𝑎 𝑛−1 +𝑛
For each problem, find the next four terms.
#1. 𝑎 1 =−4
𝑎 𝑛 =−3( 𝑎 𝑛−1 )
#2. 𝑎 1 =5
𝑎 𝑛 = 𝑎 𝑛−1 +𝑛

54. HW

64. The indicated sum of the terms of a geometric sequence is called a geometric series. You can derive a formula for the partial sum of a geometric series by subtracting the product of Sn and r from Sn as shown.

65. 𝑷𝑨𝑹𝑪𝑪 𝑹𝒆𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝑺𝒉𝒆𝒆𝒕:
𝑆 𝑛 = 𝑎 1 − 𝑎 1 𝑟 𝑛 1−𝑟 , where r≠1

66. Example 5A: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
S8 for
Step 1 Find the common ratio.

67. Step 2 Find S8 with a1 = 1, r = 2, and n = 8.
Example 5A Continued
Step 2 Find S8 with a1 = 1, r = 2, and n = 8.
Sum
formula
Check Use a graphing calculator.
Substitute.

68. 9/18/18 Homework
Find the indicated sum for each geometric series.
S6 for

70. Step 1 Find the common ratio.
9/18/18 Homework
Find the indicated sum for each geometric series.
S6 for
Step 1 Find the common ratio.

71. Check It Out! Example 5a Continued
Step 2 Find S6 with a1 = 2, r = , and n = 6.
Sum formula
Substitute.

77. Example
A 6-year lease states that the annual rent for an office space is $84,000 the first year and will increase by 8% each additional year of the lease. What will the total rent expense be for the 6-year lease?
 $616,218.04

79. Objectives
Add, subtract, multiply, and divide functions.

81. Example 1A: Adding and Subtracting Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2, find each function.
(f + g)(x)
(f + g)(x) = f(x) + g(x)
= (4x2 + 3x – 1) + (6x + 2)
Substitute function rules.
= 4x2 + 9x + 1
Combine like terms.

82. Example 1B: Adding and Subtracting Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2, find each function.
(f – g)(x)
(f – g)(x) = f(x) – g(x)
= (4x2 + 3x – 1) – (6x + 2)
Substitute function rules.
= 4x2 + 3x – 1 – 6x – 2
Distributive Property
= 4x2 – 3x – 3
Combine like terms.

83. Check It Out! Example 1a
Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function.
(f + g)(x)
(f + g)(x) = f(x) + g(x)
= (5x – 6) + (x2 – 5x + 6)
Substitute function rules.
= x2
Combine like terms.

84. Check It Out! Example 1b
Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6, find each function.
(f – g)(x)
(f – g)(x) = f(x) – g(x)
= (5x – 6) – (x2 – 5x + 6)
Substitute function rules.
= 5x – 6 – x2 + 5x – 6
Distributive Property
= –x2 + 10x – 12
Combine like terms.

85. When you divide functions, be sure to note any domain restrictions that may arise.

86. Example 2A: Multiplying and Dividing Functions
Given f(x) = 6x2 – x – 12 and g(x) = 2x – 3, find each function.
(fg)(x)
(fg)(x) = f(x) ● g(x)
Substitute function rules.
= (6x2 – x – 12) (2x – 3)
= 6x2 (2x – 3) – x(2x – 3) – 12(2x – 3)
Distributive Property
= 12x3 – 18x2 – 2x2 + 3x – 24x + 36
Multiply.
= 12x3 – 20x2 – 21x + 36
Combine like terms.

87. (fg)(x) Given f(x) = x + 2 and g(x) = x2 – 4, find each function.
Check It Out! Example 2a
Given f(x) = x + 2 and g(x) = x2 – 4, find each function.
(fg)(x)
(fg)(x) = f(x) ● g(x)
= (x + 2)(x2 – 4)
Substitute function rules.
= x3 + 2x2 – 4x – 8
Multiply.

91. Warm-Up Honors Algebra 2 9/25/18

97. #1a #1

98. Objectives Graph and recognize inverses of relations and functions.
Find inverses of functions.

99. Vocabulary
inverse relation
inverse function

100. You have seen the word inverse used in various ways.
The additive inverse of 3 is –3.
The multiplicative inverse of 5 is

101. You can also find and apply inverses to relations and functions
You can also find and apply inverses to relations and functions. To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y-values in each ordered pair of the relation.
A relation is a set of ordered pairs. A function is a relation in which each x-value has, at most, one y-value paired with it.
Remember!

102. Example 1: Graphing Inverse Relations
Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. x 1 5 8 y 2 6 9
Graph each ordered pair and connect them. ●
Switch the x- and y-values in each ordered pair. ● ● x 2 5 6 9 y 1 8 ●

103. Example 1 Continued • •
Reflect each point across y = x, and connect them. Make sure the points match those in the table. • • • • • •
Domain:{x|0 ≤ x ≤ 8} Range :{y|2 ≤ y ≤ 9}
Domain:{x|2 ≤ x ≤ 9} Range :{y|0 ≤ y ≤ 8}

104. 1 3 4 5 6 2 • • • • • Check It Out! Example 1
Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. x 1 3 4 5 6 y 2
Graph each ordered pair and connect them.
Switch the x- and y-values in each ordered pair. • • x 1 2 3 5 y 4 6 • • •

105. Check It Out! Example 1 Continued
Reflect each point across y = x, and connect them. Make sure the points match those in the table. • • • • • • • • • •
Domain:{x| 1 ≤ x ≤ 6} Range :{y| 0 ≤ y ≤ 5}
Domain:{x| 0 ≤ x ≤5} Range :{y| 1 ≤ y ≤ 6}

106. When the relation is also a function, you can write the inverse of the function f(x) as f–1(x). This notation does not indicate a reciprocal.
Functions that undo each other are inverse functions.
To find the inverse function, use the inverse operation. In the example above, 6 is added to x in f(x), so 6 is subtracted to find f–1(x).

107. Example 2: Writing Inverses of by Using Inverse Functions
Use inverse operations to write the inverse of f(x) = x – if possible. 1 2
f(x) = x – 1 2
is subtracted from the variable, x. 1 2 1 2
f–1(x) = x +
Add to x to write the inverse. 1 2

108. Check Use the input x = 1 in f(x).
Example 2 Continued
Check Use the input x = 1 in f(x).
f(x) = x – 1 2
f(1) = 1 – 1 2
Substitute 1 for x. = 1 2
Substitute the result into f–1(x) 1 2
f–1(x) = x + 1 2
f–1( ) = + 1 2
Substitute for x.
= 1
The inverse function does undo the original function. 

109. Use inverse operations to write the inverse of f(x) = . x 3
Check It Out! Example 2a
Use inverse operations to write the inverse of f(x) = . x 3 x 3
f(x) =
The variable x, is divided by 3.
f–1(x) = 3x
Multiply by 3 to write the inverse.

110. Check It Out! Example 2a Continued
Check Use the input x = 1 in f(x). x 3
f(x) = 1 3
f(1) =
Substitute 1 for x. = 1 3
Substitute the result into f–1(x)
f–1(x) = 3x
f–1( ) = 3( ) 1 3 1 3
Substitute for x.
= 1
The inverse function does undo the original function. 

111. Use inverse operations to write the inverse of f(x) = x + .
Check It Out! Example 2b
Use inverse operations to write the inverse of f(x) = x 2 3
f(x) = x + 2 3
is added to the variable, x. 2 3 2 3
f–1(x) = x –
Subtract from x to write the inverse. 2 3

112. Check It Out! Example 2b Continued
Check Use the input x = 1 in f(x).
f(x) = x + 2 3
f(1) = 1 + 2 3
Substitute 1 for x. = 5 3
Substitute the result into f–1(x) 2 3
f–1(x) = x – 2 3
f–1( ) = – 5 5 3
Substitute for x.
= 1
The inverse function does undo the original function. 

113. Undo operations in the opposite order of the order of operations.
The reverse order of operations: Addition or Subtraction Multiplication or Division Exponents Parentheses
Helpful Hint

114. Example 3: Writing Inverses of Multi-Step Functions
Use inverse operations to write the inverse of f(x) = 3(x – 7).
The variable x is subtracted by 7, then is multiplied by 3.
f(x) = 3(x – 7) 1 3
f–1(x) = x + 7
First, undo the multiplication by dividing by 3. Then, undo the subtraction by adding 7.
Check Use a sample input.  1 3
f–1(6) = (6) + 7= 2 + 7= 9
f(9) = 3(9 – 7) = 3(2) = 6

115. Use inverse operations to write the inverse of f(x) = 5x – 7.
Check It Out! Example 3
Use inverse operations to write the inverse of f(x) = 5x – 7.
The variable x is multiplied by 5, then 7 is subtracted.
f(x) = 5x – 7.
f–1(x) =
x + 7 5
First, undo the subtraction by adding by 7. Then, undo the multiplication by dividing by 5.
Check Use a sample input.
f–1(3) = = = 2 
f(2) = 5(2) – 7 = 3 10 5
3 + 7

116. You can also find the inverse function by writing the original function with x and y switched and then solving for y.

117. Example 4: Writing and Graphing Inverse Functions
Graph f(x) = – x – 5. Then write the inverse and graph. 1 2 1 2
y = – x – 5
Set y = f(x) and graph f. 1 2
x = – y – 5
Switch x and y.
x + 5 = – y 1 2
Solve for y.
–2x – 10 = y
y = –2(x + 5)
Write in y = format.

118. Example 4 Continued f–1(x) = –2(x + 5) Set y = f(x). f–1(x) = –2x – 10
Simplify. Then graph f–1.
f –1 f

119. Graph f(x) = x + 2. Then write the inverse and graph.
Check It Out! Example 4
Graph f(x) = x + 2. Then write the inverse and graph. 2 3 2 3
y = x + 2
Set y = f(x) and graph f. 2 3
x = y + 2
Switch x and y.
x – 2 = y 2 3
Solve for y.
3x – 6 = 2y
Write in y = format.
x – 3 = y 3 2

120. Set y = f(x). Then graph f–1.
Check It Out! Example 4
f–1(x) =
x – 3 3 2
Set y = f(x). Then graph f–1. f
f –1

121. Anytime you need to undo an operation or work backward from a result to the original input, you can apply inverse functions.
In a real-world situation, don’t switch the variables, because they are named for specific quantities.
Remember!