Mathematics Department Columbia High School. Advanced Algebra 2 Summer Packet

Transcription

1 Mathematics Department Columbia High School Advanced Algebra Summer Packet This summer packet is for students entering Advanced Algebra (10-5) for the Fall. The material contained in this packet represents Algebra 1 skills, procedures and concepts that we expect students to know as they enter Advanced Algebra. Students are expected to complete this packet by the first day of school. We strongly suggest that students not wait until the last minute to do the packet. Completed packets will be collected and graded based on effort and completeness. The packet is worth 5% of the student s first marking period grade. No credit will be given unless all work is shown. Completion of this packet will therefore ensure a smooth transition into new Algebra topics that we need to cover right away. Teachers teaching Advanced Algebra will use this summer packet to conduct an abbreviated review of prerequisite Algebra 1 content during the first days of school, followed by an exam on the prerequisite material in the second week of school. That exam will be worth 15% of the first marking period grade. This summer packet is specifically designed to review the following topics: Working with fractions and ratios when performing algebraic operations. Solving linear equations and linear inequalities. Solving absolute value equations and inequalities. Evaluating functions and identifying the key characteristics of functions (e.g. domain and range). Computing slope or rate of change Graphing linear equations given an equation either in slope-intercept form, standard form and point-slope form. Writing the equation of a line given two points, or given a point and the line s slope. Graphing linear inequalities in two variables. Solving systems of linear equations by graphing, by substitution and/or by elimination. Graphing a system of linear inequalities and identify the solution region.

2 Name Advanced Algebra Summer Packet Determine whether each statement is true or false. If it is true, explain why. If it is false, explain why and make the appropriate correction. No Calculator Allowed a. = 8 b. 3 1 = 8 Write a rule for 3 each of the 4 following operations: c = x y d. x y = = 1 e. 1 x x f. a + b a =1+ b = Each of the following solutions contains an algebraic error. Identify the error and make the appropriate correction. No Calculator Allowed.. a. b. c. d. Adv Alg (10-5), pg

3 Solve the following inequalities and graph the solution on a number line. No Calculator Allowed. 3. a. 5x 7 b. 16 3x 4 (recall that this is an and type compound inequality) What is the difference between solutions to and type inequalities and solutions to or type inequalities? c. x 5 or x 4 (this is an or type compound inequality) I.D Absolute Value Equations and Inequalities Solve the following absolute value equations. No Calculator Allowed. 4. a. 11+ x = 5 Critical thinking: Note that 11 is considered the center of the equation. What do you notice about the relationship of the solutions to the center? b. 11+ x = 5 Critical thinking: What is the center of this absolute value equation? How are the solutions related to the center? What does the solution to an absolute value equation represent? Adv Alg (10-5), pg 3

4 Solve the following absolute value inequalities and graph the solution on a number line. 5. a. 3x 15 b. 34x 15 c. x 1 1 d. 3x ACCURACY OF MEASUREMENTS: Your woodshop instructor requires that you cut several pieces of wood to within th of an inch of his specifications. Let p represent the specification and let x represent the actual length of a cut piece of wood. One piece of wood is specified to be p = 9 1 inches. Write an absolute value inequality that describes the 8 acceptable values of x. Describe the acceptable lengths for the piece of wood. Adv Alg (10-5), pg 4

5 Important Definitions: A relation is a mapping, or pairing, of input values with output values. The set of input values is the domain. The set of output values is the range. A relation is a function provided there is exactly one output for each input. 7. Give the definition of a function, but this time in your own words. a. Give a real-life example of a relation that is a function. b. Give a real-life example of a relation that is not a function. 8. Why does y = 3 represent a function, while x = 3 does not? 9. CAP SIZES: Your cap size is based on your head circumference (in inches). For head circumferences from 0 7 to 5 inches, cap size s can be modeled as a function of head 8 circumference c by this equation: c 1 s 3 a. Identify the domain and range of the function. Then graph the function. b. If you wear a size 7 cap, what is your head circumference? Adv Alg (10-5), pg 5

6 10. Evaluate the function for the given value of x. a. f x x x f 3 ( ) 9 ; () b. f ( x) ; f ( 4) c. f x x x f 3 ( ) 5; (6) y y1 slope = m = x x1 Parallel lines have the same slope. The slopes of perpendicular lines are opposite reciprocals. Horizontal lines have zero slope. Vertical lines have undefined slope. Calculator Allowed. 11. Determine which line is steeper. Line 1: through (, 4) and (1, 7) Line : through (5, ) and (3, 1) 1. CRITICAL THINKING: Does it make a difference what two points on a line you choose when finding slope? Does it make a difference which point is (x 1, y 1 ) and which point is (x, y ) in the formula for slope? Draw a line and calculate its slope using several pairs of points to support your answer. Adv Alg (10-5), pg 6

7 13. WHEELCHAIR RAMP DESIGN: You are in charge of building a wheelchair ramp for a doctor s office. Federal regulations require that the ramp must extend 1 inches for every 1 inch of rise. The ramp needs to rise to a height of 18 inches. a. How far should the end of the ramp be from the base of the building? b. Use the Pythagorean theorem to determine the length of the ramp. c. Some northern states require that outdoor ramps extend 0 inches for every 1 inch of rise because of the added problems of winter weather. Under this regulation, what should be the length of the ramp? d. How does changing the slope of the ramp affect the required length of the ramp? 14. MISSING COORDINATES: Find the value of k so that the line through the given points has the given slope. Check your solution. a. (5, k) and (k, 7); m = 1 b. (, k) and (k, 4); m = Graph the following equations. Find the slope, y-intercept, and x-intercept of each equation. No Calculator Allowed. a. y = 4 x +1 b x y Adv Alg (10-5), pg 7

8 c. y5 3( x 4) d. 3 y x CAR WASH: A car wash charges $8 per wash and $1 per wash-and-wax. After a busy day, sales totaled $100. Use the verbal model shown below to write an equation that shows the different numbers of washes (call that x) and wash-and-waxes (call that y) that could have been done. Then graph the equation. What are the x- and y-intercepts of the equation? What do the x- and y-intercepts mean in the context of this situation? What is the slope of the equation? Interpret the slope in the context of this situation. (Calculator Allowed) Price per wash * Number of washes + Price per wash-and-wax * Number of wash-and-waxes = Total sales Adv Alg (10-5), pg 8

9 17. SAILING: The owner of a sailboat takes passengers to an island 5 miles away to go snorkeling. A sailboat averages about 9 miles per hour when using its sails and about 14 miles per hour when using its motor. Write an equation that shows the numbers of minutes the sailboat can use its sails and its motor to get to the island. Then graph the equation. (Hints: How is this problem similar to the previous problem. How can you relate distance traveled to speed and time? Also, you will need to convert miles per hour to miles per minute.) What is the slope of this equation and what is the significance of the slope in the context of this situation? (Calculator Allowed). Slope-Intercept Form: y = mx + b Standard Form: Ax + By = C Point-Slope Form: y y 1 = m(x x 1 ) y y1 slope = m = x x No Calculator Allowed. 18. Write the equation of the line that passes through the given point and has the given slope. a. (9, 3); m = /3 1 b. ( 6, 5); m = Write the equation of the line that passes through (6, 10) and has the same slope as a line that passes through (4, 6) and (3, 4). Adv Alg (10-5), pg 9

10 0. Write an equation of the line that passes through the given points: ( 5, 9) and (, 7). Write your equation in all three forms: slope-intercept, point-slope and standard forms. 1. Find k so that the line through (7,k) and (4, 3) is parallel to the line through (1,k 1) and (3,5).. Find k so that the line through ( k1, k) is perpendicular to the line through ( 3,) and (,5). 3. Suppose a line goes through the points ( x1, y1) and ( x, y ). Express the y-intercept of the line in terms of these four coordinates. Adv Alg (10-5), pg 10

11 No Calculator Allowed. 4. Graph the following inequalities: a. xy4 b. y 3 x c. 5x 0 d. 1 x y 3 5. Write the inequality whose graph is shown. Adv Alg (10-5), pg 11

12 6. You have $00 to spend on DVDs and Blu-ray Videos. DVDs cost $10 each and Blu-rays cost $15. a. Write a linear inequality that represents the number of DVDS and Blu-rays that you can buy. b. Graph the inequality. c. Choose three possible solutions and interpret their meaning in the context of the problem. 7. You earn $3 per hour when you babysit the Thompson children. You earn $3.50 per hour when you babysit the Stewart children. You would like to buy a $47.50 ticket for a concert that is coming to town in 5 weeks. a. Write and graph an inequality that represents the number of hours you need to babysit for the Thompson s and Stewart s to earn enough money to buy your concert ticket. b. Choose three possible combinations of babysitting hours that satisfy the inequality and interpret them in the context of the situation. Adv Alg (10-5), pg 1

13 8. Graph the linear system and state how many solutions it has based on the graph. If there is exactly one solution, estimate the solution using the graph and check it algebraically. No Calculator Allowed. a. x 4y10 3x y 3 b. 4y 4x 4 y 6x1 9. Solve the following systems of equations using the best method of either the substitution method or the elimination method. Explain the reasoning for your choice of methods. (Calculator Allowed). a. x3y 5 x5y 9 b. x y 6 4xy 5 c. 3xy6 6x 3y 6 9x 6y 0 d. 1x 8y 0 Adv Alg (10-5), pg 13

14 30. Solve by elimination. c. 1x 8y 1 9x5y8 x y 3z 7 d. 4x 5y z 7 x y z The band boosters are organixing a trip to a national competition for the 6-member marching band. A bus will hold 70 students and their instruments. A van will hold 8 students and their instruments. A bus costs $80 to rent for the trip. A van costs $70 to rent for the trip. The boosters have $980 to use for transportation. a. Define your variables. b. Write a system of linear equations whose solution is how many buses and vans should be rented. c. Solve the system by whichever method you choose. Adv Alg (10-5), pg 14

15 3. A sporting goods store receives a shipment of 14 golf bags. The shipment includes two types of bags, full-size and collapsible. The full-size bags cost $38.50 each. The collapsible bags cost $.50 each. The bill for the shipment is $3430. a. Define your variables. b. Write a system of linear equations to represent the situation. c. Solve the system and determine how many of each type of golf bag are in the shipment. 33. Mrs. Thompson has one-dollar, five-dollar, and ten-dollar bills that total $171. She has the same number of five-dollar bills as one-dollar bills and ten-dollar bills combined. If she has 30 bills in all how many bills of each denomination does she have? 34. Last Tuesday, Bow-Tie Cinemas sold a total of 8500 movie tickets. Proceeds totaled $64,600. Tickets can be bought in one of three ways: a matinee admission costs $5, student admission is $6 all day, and regular admissions are $8.50. Twice as many student tickets were sold as matinee tickets. How many of each type of ticket was sold? Adv Alg (10-5), pg 15

16 35. Graph the system of linear inequalities (Calculator Allowed). a. x 5 x 4 b. x y 5x y 3 c. x4y 0 x y1 3x y DAILY DIET OF AN ADULT MOOSE: Each day, an average adult moose can process about 3 kilograms of terrestrial vegetation (twigs and leaves) and aquatic vegetation. From this food, it needs to obtain about 1.9 grams of sodium and 11,000 Calories of energy. Aquatic vegetation has about 0.15 gram of sodium per kilogram and about 193 Calories of energy per kilogram, while terrestrial vegetation has minimal sodium and about four times more energy than aquatic vegetation. Write and graph a system of inequalities describing the amounts t and a of terrestrial and aquatic vegetation, respectively, for the daily diet of an average adult moose. Adv Alg (10-5), pg 16

17 37. Graph the following quadratic functions. Describe your strategy for graphing each function. Label the vertex and axis of symmetry. No Calculator. a. y x x 6 5 b. y x x 3 y x c. 38. Factor each expression into a product of two binomials. a. x x 1 b. c. d. x x x 4x 1 x 4 1 6x 5 e. x 3x 70 f. x 1x 13 Adv Alg (10-5), pg 17

18 IV.C Solving Quadratic Equations 39. Solve the equation (Calculator Allowed). a. 9x 4 0 b. x 4x1 0 c. 4x d. x 6x 7 0 Adv Alg (10-5), pg 18