MATH NATION SECTION 4 H.M.H. RESOURCES

Transcription

1 MATH NATION SECTION 4 H.M.H. RESOURCES

2 SPECIAL NOTE: These resources were assembled to assist in student readiness for their upcoming Algebra 1 EOC. Although these resources have been compiled for your convenience from the recently adopted textbook materials from Houghton Mifflin Harcourt, digital versions of these materials can also be accessed via the textbook link found in the employee portal. Please be reminded that these materials are copyrighted and should not be posted on school or private websites without prior written permission from the publisher.

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4 4-1 Identifying and Graphing Sequences Reteach A list of numbers in a specific order, or pattern, is called a sequence. Each number, or term, in the sequence corresponds with the position number that locates it in the list. You can write a sequence as a function, where the domain is {1, 2, 3, 4, } or the set of position numbers. The range is the set of the numbers, or terms, in the list. Domain or position number: n Range or term: f(n) This sequence can be described by an explicit rule that defines each f(n) in terms of n. The explicit rule is f(n) 2n. A sequence can be shown on a graph. Use the domain and range to make ordered pairs, (n, f(n)); then plot on a graph. Example Domain or position number: n Range or term: f(n) Ordered pairs: (1, 2) (2, 4), (3, 6), and (4, 8) Complete each table for the given sequence. Then write the ordered pair. 1. f(n) 3n 2 2. f(n) 1 n 1 3. f(n) n 1 2 n f(n) n f(n) n f(n) ordered pairs: ordered pairs: ordered pairs: 69

5 4-1 Identifying and Graphing Sequences Practice and Problem Solving: Modified Complete the table and state the domain and range for the sequence. The first one is done for you. 1. n f(n) , 2, 3, 4, 5 Domain: 3, 6, 9, 12,15 Range: 2. n f(n) Domain: Range: A taxi charges $4 per ride plus $2 for each mile driven. For 3 4, use the explicit rule f(n) 2n 4. The first one in each is done for you. 3. Complete the table. 4. Graph the sequence using the ordered pairs. n f(n) 2n 4 f(n) 1 f(1) 2(1) f( ) 2( ) 4 3 f( ) 2( ) 4 4 f( ) 2( ) 4 5 f( ) 2( ) 4 Use the table to create ordered pairs. The ordered pairs are (n, f(n)). (1, 6), (2, ), (3, ), (4, ), (5, ) 70

6 4-2 Constructing Arithmetic Sequences Reteach An arithmetic sequence is a list of numbers (or terms) with a common difference between each number. 0, 6, 12, 18, Find how much you add or subtract to move from term to term. The difference between terms is constant. In this example, f(1) 0, f(2) 6, f(3) 12, f(4) 18,. The common difference is 6. Use the common difference, d, to write rules for an arithmetic sequence. A recursive rule has this general form: Given f(1), f(n) f(n 1) d for n 2 Substitute d 6: f(n) f(n 1) 6 for n 2 An explicit rule has this general form: f(n) f(1) d(n 1) Substitute d 6 from the example: f(n) f(1) 6(n 1) Indicate whether each sequence is arithmetic. If so, find the common difference, and write an explicit rule for the sequence. 1. 1, 2, 3, 4, 2. 14, 12, 10, 8, 3. 3, 6, 9, 27, Write a recursive rule and an explicit rule for each sequence. 4. 5, 0, 5, 10, 5. 7, 4, 1, 2, 6. 4, 7, 10, 13, Use the explicit rule given to write the first three terms for each sequence. 7. f(n) 6 3(n 1) 8. f(n) 68 2(n 1) 9. f(n) f(n 1) 7 71

7 4-2 Constructing Arithmetic Sequences Practice and Problem Solving: Modified Find the common difference for each arithmetic sequence. The first one is done for you. 1. 8, 13, 18, 23, 2. 9, 23, 37, 51, 3. 28, 22, 16, 10, 5 Find the next three terms for each arithmetic sequence. The first one is done for you , 13, 15, 17, 5. 8, 5, 2, 1, 6. 4, 7, 18, 29, 19, 21, 23 Write an explicit rule and a recursive rule for each sequence. The first one is done for you n f(n) n f(n) f(n) 1 2(n 1) f(1) 1, f(n) f(n 1) 2 for n 2 9. n f(n) n f(n) Solve. 11. The first term of an arithmetic sequence is 20 and the common difference is 15. Find the fifth term of the sequence. 12. Renata does 30 sit-ups every day from Monday to Friday. The graph shows the sequence. Write an explicit rule for the sequence. 72

8 4-3 Modeling with Arithmetic Sequences Reteach You can graph a function and use it to solve real-world problems. A carnival game awards a prize if Karen can shoot a basket. The charge is $5.00 for the first shot, then $2.00 for each additional shot. Karen needed 6 shots to win a prize. What is the total amount Karen spent to win a prize? Table Ordered Pairs Graph Number of Shots Cost ($) (1, 5) (2, 7) (3, 9) (4, 11) (5, 13) (6, 15) 1. Anna buys 1 raffle ticket for $4. Each ticket after that costs $2. How many raffle tickets can she buy with $12? Complete the table and graph to solve. Table Ordered Pairs Graph Number of Tickets Cost ($) 1 (1, ) 2 (2, ) (3, ) (4, ) (5, ) 73

9 4-3 Modeling with Arithmetic Sequences Practice and Problem Solving: Modified Use the table to find the common difference. Then find the value of f(7) for each. The first one is done for you Number of Weeks n Membership Fees f(n) Number of Months n Toys Collected f(n) Common Difference: 6 f(7) 48 Common Difference: f(7) Number of Kilometers n Number of Pounds n Hours Driving f(n) Boxes of Fruit f(n) Common Difference: Common Difference: f(7) f(7) Each student is training for a race. How many miles did each student run after 6 days? The first one is done for you f(6)

10 7-1 Modeling Linear Relationships Reteach Linear equations and their graphs can sometimes be used to model real-world situations. The school store sells a binder for $5 and a notebook for $4. The store needs to sell $80 worth of these two items each week. Write a linear equation that describes the problem. Graph the linear equation, making sure to label both axes with appropriate titles. Use the graph to approximate the number of notebooks the store must sell if 12 binders are sold. Step 1 Analyze the data. binder $5, notebook $4, need to sell $80 Step 2 Make a plan. Let b represent number of binders and n represent number of notebooks. Sales from binders 5b and sales from notebooks 4n Step 3 Write a linear equation to model the problem. 5b4n 80 Step 4 Calculate three sets of values for binders and notebooks. Binders Notebooks Step 5 Plot the points on a coordinate grid. Connect the points to graph the equation and label the axes. Step 6 Find the point on the line for 12 binders to find the number of notebooks needed to meet the goal. Use the graph to answer the questions. 1. What does the point (0, 20) represent? _ 2. What does the point (16, 0) represent? _ 3. What is the approximate number of notebooks that need to be sold if 12 binders are sold? 75

11 7-1 Modeling Linear Relationships Practice and Problem Solving: Modified Solve. The first one is started for you. 1. Van s Deli sells hot dogs for $2 and hamburgers for $5. His daily sales goal is $200. a. Complete the chart. Hamburgers Hot Dogs b. Write a linear equation that describes the problem. 2d 5b 200 c. Graph the linear equation. d. If Van sells 14 hamburgers, how many hot dogs must he sell to reach his goal? 2. The Good Fruit stand sells baskets of cherries for $4 and baskets of blueberries for $3. Its daily sales goal is $720. a. Complete the chart. Cherries Blueberries 0 0 b. Write a linear equation that describes the problem. c. Graph the linear equation. d. If the fruit stand sells 60 baskets of cherries, how many baskets of blueberries must it sell to meet the goal? 76

12 11-1 Solving Linear Systems by Graphing Reteach The solution to a system of linear equations can be found by graphing. Write both equations so that they are in slope-intercept form and draw their lines on a coordinate graph. The point of intersection is the solution. If the lines have the same slope but different y-intercepts they won t intersect and there is no solution. If the graphs are the same line then there are an infinite number of solutions. Example Solve the system. y 2x 2 y 2x 6 4x 2y 4 6x 3y 18 Rewrite each equation in slope-intercept form. Graph the lines and look for the point of intersection. The lines intersect at (2, 2). The solution to the system is (2, 2). Solve each linear system of equations by graphing. x y x 2y x y 10 2x 4y x y 4 x y x 3y 12 2x 2y 10 77

13 11-1 Solving Linear Systems by Graphing Practice and Problem Solving: Modified Tell the number of solutions for each system of two linear equations and if the system is consistent or inconsistent. The first one is done for you One solution, consistent Solve each system of linear equations by graphing. The first one is done for you. 4. x y 9, x-int 9, y-int x y 8, x-int y-int x y 1, x-int 1, y-int 1 x y 7, x-int y-int (5, 4) 6. 6x 2y 12, x-int y-int 7. 3x y 6, x-int y-int x 2y 4, x-int y-int x 2y 6, x-int y-int 78

14 11-2 Solving Linear Systems by Substitution Reteach You can use substitution to solve a system of equations if one of the equations is already solved for a variable. Solve y x 2 3x y 10 Step 1: Choose the equation to use as the substitute. Use the first equation y x 2 because it is already solved for a variable. Step 2: Solve by substitution. x 2 3x y 10 3x (x 2) 10 Substitute x 2 for y. 4x 2 10 Combine like terms x 8 4x x 2 Step 3: Now substitute x 2 back into one of the original equations to find the value of y. y x 2 y 2 2 y 4 The solution is (2, 4). Check: Substitute (2, 4) into both equations. y x2 3xy10 4? 22 3(2)4? 10 4? 4 64? 10 10? 10 You may need to solve one of the equations for a variable before solving with substitution. Solve each system by substitution. y x 2 1. y 2x 5 2. x y 10 x 2y 3 3. x y 3 2x y y x 8 5x 2y 9 79

15 11-2 Solving Linear Systems by Substitution Practice and Problem Solving: Modified For each linear system, tell whether it is more efficient to solve for x and then substitute for x or to solve for y and then substitute for y. The first one is done for you. 1. 2x 3y 8 x 4y x y 6 3x 2y x 4y 3 5x y 6 x For each linear system, write the expression you could substitute for x from the first equation to solve the second equation. The first one is done for you. 4. x 2y 17 3x 5y x 5y 5 2x y x 6y 16 3x 10y 8 2y 17 Solve each system by substitution and check your answer. The first one is done for you. 7. y x 6 3x y x 2y 3 2x 5y y x 7 3x 2y 3 (3, 9) 10. x 4y 1 2x y x y 17 5x 2y x 3y 2 7x y 5 Write a system of equations to solve. The first one is done for you. 13. Jan is five years older than her brother Dan. The sum of their ages is 27. How old are Jan and Dan? Jan is 16 years old and Dan is 11 years old. 14. Mariko has 30 nickels and dimes. She has 12 more nickels than dimes. How many dimes does Mariko have? 80

16 11-3 Solving Linear Systems by Adding or Subtracting Reteach To use the elimination method to solve a system of linear equations: 1. Add or subtract the equations to eliminate one variable. 2. Solve the resulting equation for the other variable. 3. Substitute the value for the known variable into one of the original equations. 4. Solve for the other variable. 5. Check the values in both equations. Use the elimination method when the coefficients of one of the variables are the same or opposite. 3x 2y 7 5x 2y 1 3x 2y 7 5x 2y 1 Substitute x 1 into 3x 2y 7 and 3x 2y 7 solve for y: 3(1) 2y 7 2y 4 y 2 The solution to the system is the ordered pair (1, 2). 8x 8 Solve for x. x 1 Check using both equations: 3x 2y 7 5x 2y 1 3(1) 2(2)? 7 5(1) 2(2)? The y-terms have opposite coefficients, so add. Add the equations. Solve each system by adding or subtracting. 2x y x y x 2y 10 3x 2y x y 12 2x y x y 1 2x 3y 5 81

17 11-3 Solving Linear Systems by Adding or Subtracting Practice and Problem Solving: Modified Which method is easier to use to solve the system of equations: substitution or addition/subtraction? The first one is done for you. 1. y 4y x 3y x 2y 11 2x 5y 11 2x 3y 18 3x y 0 substitution Solve each system of linear equations by adding or subtracting. Check your answer. The first one is done for you. 4. 2x 5y 4 5. x 4y 4 2x 8y 8 3x 4y 4 (12, 4) 6. 6x y x 4y 2 3x y 4 9x 4y x y 9 9. x 2y 8 7x 2y 24 x 2y 13 Write a system of equations to solve. The first one is started for you. 10. The sum of two numbers is 70. When the smaller number is subtracted from the bigger number, the result is 24. Find the numbers. x y 70; x y Two pairs of socks and a pair of slippers cost $30. Five pairs of socks and a pair of slippers cost $42. How much does a pair of socks cost? 82

18 11-4 Solving Linear Systems by Multiplying First Reteach To solve a system by elimination, you may first need to multiply one of the equations to make the coefficients match. 2x 5y 9 x 3y 10 Multiply bottom equation by 2. 2x 5y 9 2( x 3 y) 2(10) 2x 5y 9 2x 6y y 11 Solve for y: 11y 11 Substitute 1 for y in x 3y x 3(1) 10 y 1 x x 7 The solution to the system is the ordered pair (7, 1). You may need to multiply both of the equations to make the coefficients match. 5x 3y 2 4x 2y 10 Multiply the top by 2 and the bottom by 3. The solution to this system is the ordered pair (13, 21). - 2(5x + 3y = 2) 3(4x + 2y = 10) 10 x ( 6 y) 4 12x 6y 30 2x 0 26 x 13 After you multiply, add or subtract the two equations. Solve for the variable that is left. Substitute to find the value of the other variable. Check in both equations. Solve each system by multiplying first. Check your answer. 2x 3y x 2y 1 3x y 2 8x 2y x 5y 22 10x 3y x 2y 14 7x 3y 8 83

19 11-4 Solving Linear Systems by Multiplying First Practice and Problem Solving: Modified For each linear system, tell whether you would multiply the terms in the first or second equation in order to eliminate one of the variables. Then write the number by which you could multiply. The first one is done for you. 1. 3x 2y x y 16 x 5y 17 3x 2y 22 Multiply the equation equation 3 or 3 2 nd by. Multiply the by. 3. 4x 3y x 7y 8 5x 12y 32 x 2y 2 Multiply the equation by. Multiply the equation by. Solve each system of equations from 1 4 and check your answer. The first one is done for you. 5. 3x 2y 12 x 5y x y 16 3x 2y x 3y 19 5x 12y 32 (2, 3) 8. 4x 7y 8 x 2y 2 Write a system of equations to solve. 9. A newspaper and three hot chocolates cost $7. Two newspapers and two hot chocolates cost $6. How much does one hot chocolate cost? 84

20 12-1 Creating Systems of Linear Equations Reteach There are three important points you can use to write a system of linear equations using a graph of the equations. y-intercept of line a: y-intercept of line b: (0, 3) (0, 2) intersection point line a m = 3 - (-1) 0-2 = b 3 line b = -2 m = -1-(-1) 0-2 = -1 b 2 y = -2x +3 y = 1 2 x = 1 2 Find the y-intercepts and intersection point for each graph. Then write a system of equations for each graph y-intercept of line a: y-intercept of line a: y-intercept of line a: y-intercept of line b: y-intercept of line b: y-intercept of line b: intersection: intersection: intersection: 85

21 12-1 Creating Systems of Linear Equations Practice and Problem Solving: Modified Use the situation below to complete 1 2. Gym Rats Health Club has a starting membership fee of $25 and charges $12 per month. Greens and Soy Health Club has a starting membership fee of $35 and charges $10 per month. After how many months would the cost for the two health clubs be the same? What is that cost? Write an equation for the cost of each health club, using the slope and the y-intercept. The first one is done for you. 1. Gym Rats: slope: 12 y-intercept: 25 y 12x 25 equation: 2. Greens and Soy: slope: y-intercept: equation: Solve the system of equations by filling in the blanks. The first one is done for you. 10x x x x 6. 12x 10x 7. 2x 8. 2 x 2 9. x 10. The cost for Gym Rats and Greens and Soy are the same after months. 11. Using your answer from Exercise 10, what is the cost for the number of months that each health club charges the same price for? 86

22 12-2 Graphing Systems of Linear Inequalities Reteach You can graph a system of linear inequalities by combining the graphs of the inequalities. Graph of y 2x 3 Graph of y x 6 Graph of the system y 2x3 y x 6 All solutions are in this double shaded area. Two ordered pairs that are solutions: (3, 4) and (5, 2) Solve each system of linear inequalities by graphing. Check your answer by testing an ordered pair from each region of your graph. 1. yx3 y x 6 2. y x y 2x 1 3. y2x2 y 2x 3 87

23 12-2 Graphing Systems of Linear Inequalities Practice and Problem Solving: Modified For each inequality, write the equation of the corresponding line in slope-intercept form. Then state whether you shade above or below the line to graph the inequality. The first one is done for you. 1. 2x y 4 2. y 3x x y 7 y 2x 4; below Tell whether the ordered pair (3, 2) is a solution of the given system. The first one is done for you. 4. y y 2x 5 x 2 5. x y 5 3x 2y x y 3y 2 3x 7 no Graph the system of linear inequalities. a. Give two ordered pairs that are solutions. b. Give two ordered pairs that are not solutions. The first one is started for you. 7. y y x 1 2x 8. y y 2x 4 x1 Solve. a. (1, 0) and (3, 2) a. b. (0, 3) and (4, 0) b. 9. Coach Jules bought more than five bats. Some were wood and some were composite. The wood bats cost $49 each and the composite bats cost $100 each. Coach Jules spent less than $400. Write the system of equations that could be used to represent this situation. Let w stand for wood bats and c stand for composite bats. 88