Section 4 – Topic 1 Arithmetic Sequences, Study notes of Algebra

Download Section 4 – Topic 1 Arithmetic Sequences and more Study notes Algebra in PDF only on Docsity! 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 1st term o 8 is the 2nd term o 13 is the 𝟑rd term o 18 is the 𝟒th term o 23 is the 𝟓th 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 1st term 2 𝑎2 8 𝒇(𝟐) a formula to find the 2nd term 3 𝑎5 13 𝑓(3) a formula to find the 𝟑rd term 4 𝑎7 𝟏𝟖 𝑓(4) a formula to find the 𝟒th term 5 𝑎: 𝟐𝟑 𝒇(𝟓) a formula to find the 𝟓th term ⋮ ⋮ ⋮ ⋮ ⋮ 𝑛 𝑎= 𝒂𝒏@𝟏 + 𝒅 𝑓(𝑛) a formula to find the 𝒏th term How can we find the 9th term of this sequence? By adding the common difference until you reach the 9th term. Let’s Practice! 1. 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 42nd term of the sequence. 𝒂𝟒𝟐 = 𝟏𝟎 + (−𝟔)(𝟒𝟐 − 𝟏) 𝒂𝟒𝟐 = 𝟏𝟎 + (−𝟔)(𝟒𝟏) 𝒂𝟒𝟐 = 𝟏𝟎 − 𝟐𝟒𝟔 𝒂𝟒𝟐 = −𝟐𝟑𝟔 Try It! 2. 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 33rd term of the sequence. 𝒂𝒏 = 𝟕 + (𝟏𝟎)(𝟑𝟑 − 𝟏) 𝒂𝒏 = 𝟕 + (𝟏𝟎)(𝟑𝟐) 𝒂𝒏 = 𝟕 + 𝟑𝟐𝟎 𝒂𝒏 = 𝟑𝟐𝟕 BEAT THE TEST! 1. Yohanna is conditioning all summer to prepare for her high school’s varsity soccer team tryouts. She is incorporating walking planks into her daily workout training plan. Every day, she will complete four more walking planks than the day before. Part A: If she starts with five walking planks on the first day, write an explicit formula that can be used to find the number of walking planks Yohanna completes on any given day. 𝒂𝒏 = 𝟓 + (𝟒)(𝒏 − 𝟏) Part B: How many walking planks will Yohanna do on the 12th day? A 49 B 53 C 59 D 64 Answer: A 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 Pa ge s Days Pa ge s Days Let’s Practice! 1. 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 So uv en irs P ur ch as ed Keisha’s Vacation Souvenirs 2. Freedom High School collected data on the GPA of various students and the number of hours they spend studying each week. A scatterplot of the data is shown below with the line of best fit. a. What is the slope of the line of best fit? 𝟎. 𝟐 b. What does the slope represent? Change in GPA per hour spent studying each week Hours Spent Studying Each Week G PA Part B: The data suggest a linear relationship between the number of hours spent cleaning and the number of apartments cleaned. Assuming the relationship is linear, what does the rate of change represent in the context of this relationship? A The number of apartments cleaned after one hour. B The number of hours it took to clean one apartment. C The number of apartments cleaned each hour. D The number of apartments cleaned before the company started cleaning. Answer: C Part C: Which equation describes the relationship between the time elapsed and the number of apartments cleaned? A 𝑦 = 𝑥 B 𝑦 = 𝑥 + 2 C 𝑦 = 2𝑥 D 𝑦 = 2𝑥 + 2 Answer: C 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 To ta l C os t 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! 1. 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 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. Section 4 – Topic 4 Interpreting Rate of Change and 𝒚-Intercept in a Real World Context – Part 2 Let’s Practice! 3. Graph 𝑦 = 2𝑥 + 3. 4. Consider the equation 2𝑥 + 5𝑦 = 10. a. How does this equation look different from slope-intercept form of an equation? It is not solved for 𝒚. b. Rewrite the equation in slope-intercept form. 𝟐𝒙 − 𝟐𝒙 + 𝟓𝒚 = 𝟏𝟎 − 𝟐𝒙 𝟓𝒚 𝟓 = 𝟏𝟎 𝟓 − 𝟐𝒙 𝟓 𝒚 = −𝟐𝟓𝒙 + 𝟐 c. Identify the slope and 𝑦-intercept. Slope = −𝟐𝟓, 𝒚-intercept = 𝟐 d. Graph the equation. 2. The senior class at Elizabeth High School was selling tickets to raise money for prom. The graph below represents the situation. Part A: How much does one ticket cost? $𝟐𝟓 Part B: How much money did the senior class have at the start of the fundraiser? $𝟏𝟎𝟎 Number of Tickets Sold A m ou nt o f M on ey R ai se d (in d ol la rs ) Section 4 – Topic 5 Introduction to Systems of Equations A system of equations is a set of 2 or more equations. Consider the following systems of equations. Line 1: 2𝑥 − 𝑦 = −5 Line 2: 2𝑥 + 𝑦 = 1 Graph the system of equations on the coordinate plane below. Recall that a solution to a linear equation is any ordered pair that makes that equation a true statement. What do you notice about the point (−2,5)? It falls on the Line 𝟐. What do you notice about the point (1,7)? It falls on the Line 𝟏. What do you notice about the point (−1,3)? It falls on both lines. What do you notice about the point (1,1)? It does not fall on either line. 4. Consider the following system of equations. 4𝑥 + 3𝑦 = 3 2𝑥 − 5𝑦 = −5 a. Graph the system of equations. b. What is the solution to the system? (𝟎, 𝟏) BEAT THE TEST! 1. Consider the following system of equations. 𝑥 + 𝑦 = 5 2𝑥 − 𝑦 = −2 Part A: Sketch the graph of the system of equations. Part B: Determine the solution to the system of equations. (𝟏, 𝟒) Part C: Create a third equation that could be added to the system so that the solution does not change. Graph the line on the coordinate plane above. Answers vary. Sample Answer: Add the two equations together to get a new equation 𝟑𝒙 = 𝟑 Section 4 – Topic 6 Finding Solution Sets to Systems of Equations Using Substitution and Graphing There are many times that we are able to use systems of equations to solve real world problems. One method of solving systems of equations is by graphing like we did in the previous video. Let’s Practice! 1. Brianna’s lacrosse coach suggested that she practices yoga to improve her flexibility. “Yoga-ta Try This!” Yoga Studio has two membership plans. Plan A costs $20.00 per month plus $10.00 per class. Plan B costs $100.00 per month for unlimited classes. a. Define a variable and write two functions to represent the monthly cost of each plan. Let 𝒄 represent the number of monthly classes attended and 𝒇(𝒄) represent monthly cost. Plan A: 𝒇 𝒄 = 𝟐𝟎 + 𝟏𝟎𝒄 Plan B: 𝒇 𝒄 = 𝟏𝟎𝟎 Try It! 3. Vespa Scooter Rental rents scooters for $45.00 and $0.25 per mile. Scottie’s Scooter Rental rents scooters for $35.00 and $0.30 per mile. a. Define a variable and write two functions to represent the situation. Let 𝒎 represent the number of miles driven. Vespa: 𝒈 𝒎 = 𝟒𝟓 + 𝟎. 𝟐𝟓𝒎 Scotties: 𝒈 𝒎 = 𝟑𝟓 + 𝟎. 𝟑𝒎 b. Represent the two situations on the graph below. Miles To ta l C os t c. What is the rate of change of each line? What do they represent? Vespa: 𝟎. 𝟐𝟓 Scotties: 𝟎. 𝟑 They represent the cost per mile driven. d. What do the 𝑦-intercepts of each line represent? The initial cost It’s difficult to find the solution by looking at the graph. In such cases, it’s better to use substitution to solve the problem. 4. Use the substitution method to help the renter determine when the two scooter rentals will cost the same amount. a. When will renting a scooter from Vespa Scooter Rental cost the same as renting a scooter from Scottie’s Scooter Rental? 𝟒𝟓 + 𝟎. 𝟐𝟓𝒎 = 𝟑𝟓 + 𝟎. 𝟑𝒎 𝟒𝟓 − 𝟒𝟓 + 𝟎. 𝟐𝟓𝒎 = 𝟑𝟓 − 𝟒𝟓 + 𝟎. 𝟑𝒎 𝟎. 𝟐𝟓𝒎 = −𝟏𝟎 + 𝟎. 𝟑𝒎 𝟎. 𝟐𝟓𝒎− 𝟎. 𝟑𝒎 = −𝟏𝟎 + 𝟎. 𝟑𝒎 − 𝟎. 𝟑𝒎 @𝟎.𝟎𝟓𝒎 @𝟎.𝟎𝟓 = @𝟏𝟎 @𝟎.𝟎𝟓 𝒎 = 𝟐𝟎0 Driving 𝟐𝟎𝟎 miles the cost will be the same. b. Describe a situation when renting from Vespa Scooter Rental would be a better deal than renting from Scottie’s Scooter Rental. Vespa will be a better deal if you drive more than 𝟐𝟎𝟎 miles. BEAT THE TEST! 1. Lyle and Shaun open a savings account at the same time. Lyle deposits $100 initially and adds $20 per week. Shaun deposits $500 initially and adds $10 per week. Lyle wants to know when he will have the same amount in his savings account as Shaun. Part A: Write two equations to represent the amount of money Lyle and Shaun have in their accounts. Let 𝒙 represent the number of weeks they make deposits. Lyle: 𝒚 = 𝟏𝟎𝟎 + 𝟐𝟎𝒙 Shaun: 𝒚 = 𝟓𝟎𝟎 + 𝟏𝟎𝒙 Part B: Which method would you use to solve the problem, substitution or graphing? Explain your answer. Answers vary. Sample answer: I would use substitution since graphing large numbers might be more difficult. Part C: After how many weeks of making the additional deposits will Lyle have the same amount of money as Shaun? 𝟏𝟎𝟎 + 𝟐𝟎𝒙 = 𝟓𝟎𝟎 + 𝟏𝟎𝒙 𝟏𝟎𝟎 − 𝟏𝟎𝟎 + 𝟐𝟎𝒙 = 𝟓𝟎𝟎 − 𝟏𝟎𝟎 + 𝟏𝟎𝒙 𝟐𝟎𝒙 − 𝟏𝟎𝒙 = 𝟒𝟎𝟎 + 𝟏𝟎𝒙 − 𝟏𝟎𝒙 𝟏𝟎𝒙 𝟏𝟎 = 𝟒𝟎𝟎 𝟏𝟎 𝒙 = 𝟒𝟎 At 𝟒𝟎 weeks of deposits, they will have the same amount of money. Consider the original system of equations again. 𝑥 + 𝑦 = 4 𝑥 − 𝑦 = 6 What is the resulting equation when we add the two equations in the system together? 𝟐𝒙 = 𝟏𝟎 or 𝒙 = 𝟓 Graph the new equation on the same coordinate plane with our original system. Algebraically, show that (5, −1) is also a solution to the sum of the two lines. 𝟐𝒙 = 𝟏𝟎, 𝟐 𝟓 = 𝟏𝟎 What is the resulting equation when we subtract the second equation from the first equation? 𝟐𝒚 = −𝟐 or 𝒚 = −𝟏 Graph the new equation on the same coordinate plane with our original system. Algebraically, show that (5, −1) is also a solution to the difference of the two lines. 𝟐𝒚 = −𝟐 or 𝟐 −𝟏 = −𝟐 Let’s revisit the original system: Equation 1: 𝑥 + 𝑦 = 4 Equation 2: 𝑥 − 𝑦 = 6 Complete the following steps to show that replacing one equation by the sum of that equation and a multiple of the other equation produces a system with the same solutions. Create a third equation by multiplying Equation 1 by two. Equation 𝟑: 𝟐𝒙 + 𝟐𝒚 = 𝟖 Create a fourth equation by finding the sum of the third equation and Equation 2. 𝟐𝒙 + 𝟐𝒚 = 𝟖 𝒙 − 𝒚 = 𝟔 𝟑𝒙 + 𝒚 = 𝟏𝟒 BEAT THE TEST! 1. The system 𝐴𝑥 + 𝐵𝑦 = 𝐶 𝐷𝑥 + 𝐸𝑦 = 𝐹 has the solution (1, −3), where 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and 𝐹 are non-zero real numbers. Select all the systems of equations with the same solution. ý 𝐴 − 𝐷 𝑥 + 𝐵 − 𝐸 𝑦 = 𝐶 − 𝐹 𝐷𝑥 + 𝐸𝑦 = 𝐹 o (2𝐴 + 𝐷)𝑥 + (2𝐵 + 𝐸)𝑦 = 𝐶 + 2𝐹 𝐷𝑥 + 𝐸𝑦 = 𝐹 ý 𝐴𝑥 + 𝐵𝑦 = 𝐶 −3𝐷𝑥 − 3𝐸𝑦 = −3𝐹 ý 𝐴 − 5𝐷 𝑥 + 𝐵 − 5𝐸 𝑦 = 𝐶 − 5𝐹 𝐷𝑥 + 𝐸𝑦 = 𝐹 o 𝐴𝑥 + (𝐵 + 𝐸)𝑦 = 𝐶 𝐴 + 𝐷 𝑥 + 𝐸𝑦 = 𝐶 + 𝐹 Section 4 – Topic 8 Finding Solution Sets to Systems of Equations Using Elimination Consider the following system of equations: 2𝑥 + 𝑦 = 8 𝑥 − 2𝑦 = −1 Write an equivalent system that will eliminate one of the variables when you add the equations. Answers vary. Sample answer: Multiply the first equation by 𝟐. 𝟒𝒙 + 𝟐𝒚 = 𝟏𝟔 𝒙 − 𝟐𝒚 = −𝟏 Determine the solution to the system of equations. 𝟒𝒙 + 𝟐𝒚 = 𝟏𝟔 𝒙 − 𝟐𝒚 = −𝟏 𝟓𝒙 = 𝟏𝟓 𝒙 = 𝟑 𝟐 𝟑 + 𝒚 = 𝟖 𝟔 + 𝒚 = 𝟖 𝟔 − 𝟔 + 𝒚 = 𝟖 − 𝟔 𝒚 = 𝟐 The solution to the system is (𝟑, 𝟐). Describe what the graph of the two systems would look like. The lines intersect at the point 𝟑, 𝟐 . This method of solving a system is called elimination. Let’s Practice! 1. Ruxin and Andre were invited to a Super Bowl party. They were asked to bring pizzas and sodas. Ruxin brought three pizzas and four bottles of soda and spent $48.05. Andre brought five pizzas and two bottles of soda and spent $67.25. a. Write a system of equations to represent the situation. Let 𝒙 represent the cost of one pizza. Let 𝒚 represent the cost of one soda. 𝟑𝒙 + 𝟒𝒚 = 𝟒𝟖. 𝟎𝟓 𝟓𝒙 + 𝟐𝒚 = 𝟔𝟕. 𝟐𝟓 b. Write an equivalent system that will eliminate one of the variables when you add the equations. 𝟑𝒙 + 𝟒𝒚 = 𝟒𝟖. 𝟎𝟓 −𝟏𝟎𝒙 − 𝟒𝒚 = −𝟏𝟑𝟒. 𝟓𝟎 BEAT THE TEST! 1. Complete the following table. Solve by Elimination: 2𝑥 − 3𝑦 = 8 3𝑥 + 4𝑦 = 46 Operations Equations Labels 2𝑥 − 3𝑦 = 8 3𝑥 + 4𝑦 = 46 Equation 1 Equation 2 −6𝑥 + 9𝑦 = −24 New equation 1 Multiply equation 2 by 2. New equation 2 −6𝑥 + 9𝑦 = −24 6𝑥 + 8𝑦 = 92 17𝑦 = 68 Divide by 17. Solve for 𝑥. Write 𝑥 and 𝑦 as coordinates. ( , ) Solution to the system Multiply Equation 𝟏 by −𝟑 Add the equations together. 𝟏𝟕𝒚 𝟏𝟕 = 𝟔𝟖 𝟏𝟕 𝒚 = 𝟒 𝟐𝒙 − 𝟑(𝟒) = 𝟖 𝒙 = 𝟏𝟎 𝟒 𝟏𝟎 𝟔𝒙 + 𝟖𝒚 = 𝟗𝟐 2. Which of the systems of equations below could not be used to solve the following system for 𝑥 and 𝑦? 6𝑥 + 4𝑦 = 24 −2𝑥 + 4𝑦 = −10 A 6𝑥 + 4𝑦 = 24 2𝑥 − 4𝑦 = 10 B 6𝑥 + 4𝑦 = 24 −4𝑥 + 8𝑦 = −20 C 18𝑥 + 12𝑦 = 72 −6𝑥 + 12𝑦 = −30 D 12𝑥 + 8𝑦 = 48 −4𝑥 + 8𝑦 = −10 Answer: D Section 4 – Topic 9 Solution Sets to Inequalities with Two Variables Consider the following linear inequality. 𝑦 ≥ 2𝑥 − 1 Underline each ordered pair (𝑥, 𝑦) that is a solution to the above inequality. (𝟎, 𝟓) (4, 1) (−𝟏,−𝟏) (𝟏, 𝟏) (3, 0) (−𝟐, 𝟑) (4, 3) (−𝟏,−𝟑) Plot each solution as a point (𝑥, 𝑦) in the coordinate plane. Graph the line 𝑦 = 2𝑥 − 1 in the same coordinate plane. What do you notice about the solutions to the inequality 𝑦 ≥ 2𝑥 − 1 and the graph of the line 𝑦 = 2𝑥 − 1? The solutions to 𝒚 = 𝟐𝒙 − 𝟏 are also solutions to 𝒚 ≥ 𝟐𝒙 − 𝟏. Try It! 2. The freshman class wants to include at least 120 people in the pep rally. Each skit will have 15 people, and the dance routines will feature 12 people. a. List two possible combinations of skits and dance routines. Answers vary. Sample answer: 𝟒 skits and 𝟓 dance routines 𝟐 skits and 𝟖 dance routines b. Write an inequality to represent the situation. Let 𝒙 represent the number of skits. Let 𝒚 represent the number of dance routines. 𝟏𝟓𝒙 + 𝟏𝟐𝒚 ≥ 𝟏𝟐𝟎 c. Graph the region where the solutions to the inequality would lie. d. What does the 𝑦-intercept represent? The 𝒚 − intercept would represent having only dance routines and no skits. Number of Skits Nu m be r of D an ce R ou ti ne s BEAT THE TEST! 1. Coach De Leon purchases sports equipment. Basketballs cost $20.00 each, and soccer balls cost $18.00 each. He had a budget of $150.00. The graph shown below represents the number of basketballs and soccer balls he can buy given his budget constraint. Part A: Write an inequality to represent the situation. Let 𝒔 represent the number of soccer balls. Let 𝒃 represent the number of basketballs. 𝟏𝟖𝒔 + 𝟐𝟎𝒃 ≤ 𝟏𝟓𝟎 N um be r o f B as ke tb al ls 𝑠 𝑏 Number of Soccer Balls Hours Mowing Lawns Ho ur s W as hi ng C ar s Let’s Practice! 1. Bristol is having a party and has invited 24 friends. She plans to purchase sodas that cost $5.00 for a 12-pack and chips that cost $3.00 per bag. She wants each friend to have at least two sodas. Bristol’s budget is $35.00. a. Write a system of inequalities to represent the situation. Let 𝒙 represent the number of 𝟏𝟐-packs of sodas. Let 𝒚 represent the number of bags of chips. 𝟓𝒙 + 𝟑𝒚 ≤ 𝟑𝟓 𝒙 ≥ 𝟒 b. Graph the region where the solutions to the inequality would lie. Ba gs o f C hi ps Soda Packs c. Name two difference solutions for Bristols’s situation. Answers Vary. Sample Answer: 𝟓 cases of soda and 𝟏 bag of chips. 𝟒 cases of soda and 𝟒 bags of chips. Try It! 2. Anna is an avid reader. Her generous grandparents gave her money for her birthday, and she decided to spend at most $150.00 on books. Reading Spot is running a special: all paperback books are $8.00 and hardback books are $12.00. Anna wants to purchase at least 12 books. a. Write a system of inequalities to represent the situation. Let 𝒙 represent the number of hardback books Let 𝒚 represent the number of paperback books 𝟏𝟐𝒙 + 𝟖𝒚 ≤ 𝟏𝟓𝟎 𝒙 + 𝒚 ≥ 𝟏𝟐