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Section 4 – Topic 1
Arithmetic Sequences
Let’s look at the following sequence of numbers: 3 , 8 , 13 , 18 , 23 ,. ... Ø The “…” at the end means that this sequence goes on forever. Ø 3 , 8 , 13 , 18 , and 23 are the actual terms of this sequence. Ø There are 5 terms in this sequence so far: o 3 is the 1 st^ term o 8 is the 2 nd^ term o 13 is the 𝟑 rd term o 18 is the 𝟒 th term o 23 is the 𝟓 th term This is an example of an arithmetic sequence. Ø This is a sequence where each term is the sum of the previous term and a common difference, 𝑑.
We can represent this sequence in a table: Term Number Sequence Term Term New Notation 1 𝑎. 3 𝑓( 1 ) a formula to find the 1 st^ term 2 𝑎 2 8 𝒇(𝟐) a formula to find the 2 nd^ term 3 𝑎 5 13 𝑓( 3 ) a formula to find the 𝟑 rd^ term 4 𝑎 7 𝟏𝟖 𝑓( 4 ) a formula to find the 𝟒 th^ term 5 𝑎: 𝟐𝟑 𝒇(𝟓) a formula to find the 𝟓 th^ term ⋮ ⋮ ⋮ ⋮ ⋮ 𝑛 𝑎= 𝒂𝒏@𝟏 + 𝒅 𝑓(𝑛) a formula to find the^ 𝒏 th term How can we find the 9 th^ term of this sequence? By adding the common difference until you reach the 9 th term.
Let’s look at another way to find unknown terms: Term Number Sequence Term Term Function Notation 1 𝑎. 3 𝑓( 1 ) 3 2 𝑎 2 8 = 3 + 5 𝑓( 2 ) 3 + 5 ( 1 ) 3 𝑎 5 13 =^8 +^5 =^3 +^5 +^5 𝑓(^3 )^3 +^5 (^2 ) 4 𝑎 7 18 =^13 +^5 =^3 +^5 +^5 +^5 𝑓(^4 )^3 +^5 (^3 ) 5 𝑎: 23 =^18 +^5 =^3 +^5 +^5 +^5 +^5 𝑓(^5 )^3 +^5 (^4 ) 6 𝑎F 28 = 23 + 5 = 3 + 5 + 5 + 5 + 5
- 5 𝑓( 6 ) 3 + 5 ( 5 ) 7 𝑎H 33 = 28 + 5 = 3 + 5 + 5 + 5 + 5 + 5 + 5 𝑓( 7 ) 3 + 5 ( 6 ) 8 𝑎I 38 = 33 + 5 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 𝑓( 8 ) 𝟑 + 𝟓(𝟕) 9 𝑎J 43 = 38 + 5 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 𝑓( 9 ) 𝟑 + 𝟓(𝟖) Write a general equation that we could use to find any term in the sequence. 𝒂𝒏 = 𝟑 + 𝟓 𝒏 − 𝟏 , where 𝒏 is a natural number. 𝒂𝒏 = 𝒂𝟏 + 𝒅 𝒏 − 𝟏 This is an explicit formula. Ø To solve for a term, you need to know the first term of the sequence and the difference by which the sequence is increasing or decreasing.
Let’s Practice!
- Consider the sequence 10 , 4 , − 2 , − 8 , …. a. Write a recursive formula for the sequence. 𝒂𝒏 = 𝒂𝒏@𝟏 − 𝟔 b. Write an explicit formula for the sequence. 𝒂𝒏 = 𝟏𝟎 + (−𝟔)(𝒏 − 𝟏) c. Find the 42 nd^ term of the sequence. 𝒂𝟒𝟐 = 𝟏𝟎 + (−𝟔)(𝟒𝟐 − 𝟏) 𝒂𝟒𝟐 = 𝟏𝟎 + (−𝟔)(𝟒𝟏) 𝒂𝟒𝟐 = 𝟏𝟎 − 𝟐𝟒𝟔 𝒂𝟒𝟐 = −𝟐𝟑𝟔 Try It!
- Consider the sequence 7 , 17 , 27 , 37 , …. a. Find the next three terms of the sequence. 𝟒𝟕, 𝟓𝟕, 𝟔𝟕 b. Write a recursive formula for the sequence. 𝒂𝒏 = 𝒂𝒏@𝟏 + 𝟏𝟎 c. Write an explicit formula for the sequence. 𝒂𝒏 = 𝟕 + (𝟏𝟎)(𝒏 − 𝟏) d. Find the 33 rd^ term of the sequence. 𝒂𝒏 = 𝟕 + (𝟏𝟎)(𝟑𝟑 − 𝟏) 𝒂𝒏 = 𝟕 + (𝟏𝟎)(𝟑𝟐) 𝒂𝒏 = 𝟕 + 𝟑𝟐𝟎 𝒂𝒏 = 𝟑𝟐𝟕
Section 4 – Topic 2
Rate of Change of Linear Functions
Génesis reads 16 pages of The Fault in Our Stars every day. Zully reads 8 pages every day of the same book. Represent both situations on the graphs below using the same scales for both graphs. Graph 1: Génesis’ Reading Speed Graph 2: Zully’s Reading Speed Pages Days Pages Days
Aaron loves Cherry Coke. Each mini-can contains 100 calories. Jacobe likes to munch on carrot snack packs. Each snack pack contains 40 calories. Represent both situations on the graphs below using the same scales for both graphs. Graph 3 : Aaron’s Calorie Intake Graph 4 : Jacobe’s Calorie Intake In each of the graphs, we were finding the rate of change in the given situation. Calories Mini Coke Calories Carrots
Let’s Practice!
- Consider the following graph. a. What is the rate of change of the graph? 𝟑 b. What does the rate of change represent? Souvenirs purchased per day of vacation Days of Vacation Souvenirs Purchased Keisha’s Vacation Souvenirs
Try It!
- Sarah’s parents give her $ 100. 00 allowance at the beginning of each month. Sarah spends her allowance on comic books. The graph below represents the amount of money Sara spent on comic books last month. a. What is the rate of change? −𝟓 b. What does the rate of change represent? $𝟓 spent per comic book Number of Comic Books Purchased Amount of Allowance Left (in Dollars)
BEAT THE TEST!
- A cleaning service cleans many apartments each day. The following table shows the number of hours the cleaners spend cleaning and the number of apartments they clean during that time. Apartment Cleaning Time (Hours) 1 2 3 4 Apartments Cleaned 2 4 6 8 Part A: Represent the situation on the graph below.
Section 4 – Topic 3
Interpreting Rate of Change and 𝒚 - Intercept
in a Real World Context – Part 1
Cab fare includes an initial fee of $ 2. 00 plus $ 3. 00 for every mile traveled. Define the variable and write a function that represents this situation. Let 𝒎 represent number of miles traveled. Let 𝒄(𝒎) represent the cab fare. 𝑪 𝒎 = 𝟐 + 𝟑𝒎 Represent the situation on a graph. Miles driven Total Cost
What is the slope of the line? What does the slope represent? 𝟑 ; cost per mile At what point does the line intersect the 𝑦-axis? What does this point represent? 𝟐 ; initial or starting cost This point is the 𝒚 - intercept of a line. Let’s Practice!
- You saved $ 250. 00 to spend over the summer. You decide to budget $ 25. 00 to spend each week. a. Define the variable and write a function that represents this situation. Let 𝒘 represent the number of weeks. Let 𝑺(𝒘) represent the remaining amount. 𝑺 𝒘 = 𝟐𝟓𝟎 − 𝟐𝟓𝒘
Try It!
- Consider the following graph: a. What is the slope of the line? What does the slope represent? 𝟐 ; Cost per visit b. What is the 𝑦-intercept? What does the 𝑦-intercept represent? (𝟎, 𝟒) ; Membership fee for using the pool c. Define the variables and write a function that represents this situation. Let 𝒙 represent the number of visits. Let 𝒈(𝒙) represent the total cost. 𝒈 𝒙 = 𝟒 + 𝟐𝒙 , where 𝒙 is a whole number. d. What does each point represent? The total cost for that number of visits. Number of Visits to the Community Pool Total Cost (in Dollars)
Consider the three functions that you wrote regarding the cab ride, summer spending habits, and the community pool membership. What do you notice about the constant term and the coefficient of the 𝑥 term? Ø The constant term is the 𝒚 - intercept. Ø The coefficient of the 𝑥 is the slope or rate of change. These functions are written in slope-intercept form. We can use slope-intercept form to graph any linear equation. The coefficient of 𝑥 is the slope and the constant term is the 𝑦-intercept ONLY if the equation is in slope-intercept form, 𝑦 = 𝑚𝑥 + 𝑏.
