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Name - __________________________________ HONORS PRECALCULUS SUMMER PACKET Going into Precalculus, there are certain skills that have been taught to you over the previous years that we assume you have. If you do not have these skills, you will find that you will consistently get problems incorrect next year, even though you understand the precalculus concepts. It is frustrating for students when they are tripped up by the algebra and not the precalculus. This summer packet is intended for you to brush up and possibly relearn these topics. We assume that you have basic skills in algebra. Being able to solve equations, work with algebraic expressions, and basic factoring, for example, should now be a part of you. These are skills that are used continually in class. You need to get off to a good start next year so spend some quality time on this packet this summer. It is a mistake to decide to do this now. Let it go until mid summer. We want these techniques to be relatively fresh in your mind in the fall. However, do not wait to do them at the very last minute. These take time. Have a good summer and see you in the fall. DIRECTIONS: Go to www.smsd.org/schools/smnorth to print of the PDF file of the “Precalculus Summer Packet 2012”. Show all work on loose leaf paper, write neatly, and separate each section. The work will be turned in with this packet. This packet will be graded for completion as well as spot- checked for accuracy. A key to the packet will be posted in August. There will be a test on the material covered in this packet. This test will be your first major grade of the semester and it is important that you start off the year strong. This packet is due the day of the test in the fall. DUE DATE: August 15/16, 2012 REAL NUMBERS AND INEQUALITIES Interval Notation Notes Interval Notation is used instead of inequality notation. We use brackets [ or ] to signify that we include the endpoint and we use parentheses ( or ) to signify that we do not include the endpoint. However, we will always use parentheses with the infinity symbol since you can never reach infinity. For example, to express x 5 in interval notation we would write 5, because this interval includes all numbers greater than 5. However, to express x 5 we would write 5, because we can now include the number 5 in our interval. Next, to express x 3 we would write ,3 since we want to include all numbers less than 3. Similarly x 3 would be written as ,3 since we can include 3 now. We can also use interval notation to express compound inequalities. For example the “and” compound inequality 1 x 6 would be written as 1,6 since it includes numbers between 1 and 6 but not including either number. However, if we have 4 x 8 we would write 4,8 since we need to include -4 but not 8. To write the “or” inequality x 2 or x 9 we would write , 2 9, . We use the union symbol to join the “or” inequality. Similarly to express x 4 or x 1 we would write , 4 1, . Use interval notation to describe the interval of real numbers. 1. x 3 7 x 2 4. 2. 5. x is greater than -3 and less than or equal to 4 3. 6. x is positive Find which values of x are solutions of the inequality. 7. 2x 3 7 (a) x 0 (b) x 5 (c) x 6 8. 3x 4 5 (a) x 0 (b) x 3 (c) x 4 9. 1 4x 1 11 (a) x 0 (b) x 2 (c) x 3 10. 3 1 2x 3 (a) x 1 (b) x 0 (c) x 2 Solve the inequality. Write the solution in interval notation. 1 4 1 x 5 1 x 3x 1 x 4 2x 5 3 x 11. 2x 1 4x 3 16. 21. 2 3 4y 2y 3 2 y 3 2x 2 5 12. 3x 1 6x 8 17. 6 8 22. 3x 2 x2 1 3 13. 1 3x 2 7 18. 5 23. 3 5x 7 2y 5 3 4 2 24. 2 x 7 x 15 2 14. 4 19. 3 2 5 3x 3 2x 1 2x 1 15. 2 x 6 9 25. x x 30 x 0 3 2 20. CARTESIAN COORDINATE SYSTEM Find the area and perimeter of the figure determined by the points. 1. 5,3 , 0, 1 , 4, 4 3. 3, 1 , 1,3 , 7,3 , 5, 1 2. 2, 2 , 2, 2 , 2, 2 , 2, 2 4. 2,1 , 2,6 , 4,6 , 4,1 Find the standard form equation for the circle. 5. Center: 1,2 , Radius: 5 7. Center: 1, 4 , Radius: 3 6. Center: 3,2 , Radius: 1 8. Center: 0,0 , Radius: 3 Find the center and radius of the circle. 9. x 32 y 12 36 10. x 2 y 2 5 11. x 42 y 22 121 12. x 22 y 62 25 LINES IN THE PLANE Find the value of x or y so that the line through the pair of points has the given slope. 1. m 2, x,3 , 5,9 3. m 3, 3, 5 , 4, y 1 2. m 3, 2,3 , 4, y 4. m , 8, 1 , x, 2 2 Find a point-slope form equation for the line through the point with given slope. 2 5. m 2, 1, 4 7. m , 4,3 3 6. m 2, 5, 4 8. m 3, 3,4 Find a general (standard) form equation for the line through the pair of points. 9. 7, 2 , 1,6 11. 1, 3 , 5, 3 10. 3, 8 , 4, 1 12. 1, 5 , 4, 2 Find a slope-intercept form equation for the line. 13. m 3, 0,5 15. 4,5 , 4,3 17. 2 x 5 y 12 1 14. m , 1, 2 16. 4, 2 , 3,8 18. 7 x 12 y 96 2 Find the value of x or y for which x,14 and 18, y are points on the graph. 19. y 0.5x 12 21. 3x 4 y 26 20. y 2 x 18 22. 3x 2 y 14 Find an equation for the line passing through the point and (a) parallel to the given line, (b) perpendicular to the given line. 23. 1, 2 , y 3x 2 24. 2,3 , y 2x 4 LINEAR EQUATIONS Find which values of x are solutions of the equation. 1 1 1. 2 x2 5x 3 (a) x 3 (b) x (c) x 2 2 x 1 x 2. 2 6 3 (a) x 1 (b) x 0 (c) x 1 3. 1 x2 2 3 (a) x 2 (b) x 0 (c) x 2 4. x 2 1/3 2 (a) x 6 (b) x 8 (c) x 10 Solve the equation. 1 7 1 1 5. 2x 3 4x 5 7. 2 3 4z 5 2z 3 z 17 9. x 11. x 1 2 8 2 3 2x 3 t 5 t 2 1 6. 4 3 y 2 y 4 8. 3 5z 3 4 2z 1 5z 2 10. 5 3x 12. 4 8 2 3 POLYNOMIAL EQUATIONS AND FACTORING Solve the equation by factoring 1. x x 20 0 3. 4 x 8 x 3 0 5. x 3x 7 6 2 2 2. 2 x 5 x 3 0 4. x 8 x 15 6. x 3x 11 20 2 2 Solve the equation by extracting square roots. 7. 4 x 25 9. 3 x 4 8 2 11. 2 y 2 8 6 2 y 2 2 8. 2 x 5 17 10. 4 u 1 18 2 x 32 169 2 2 12. Solve the equation by completing the square. 5 13. x 6 x 7 15. x 2 7 x 0 17. 2 x2 7 x 9 x 3 x 1 3x 2 4 14. x 5 x 9 0 16. 4 6x x 18. 3x2 6 x 7 x2 3x x x 1 3 2 2 Solve the equation by using the quadratic formula. 19. x 8 x 2 0 21. 3x 4 x 2 2 20. 2 x 3x 1 0 22. x 2 5 3 x 2 Factor completely. 23. x3 y 4 xy 28. 4r 2 25 24. 4 x 2 20 x 25 29. y 3 2 y 2 y 2 25. 3a 2 27ab 54b2 30. 6 x 4 3 x 3 y 2 2 xy 3 y 5 26. 49 x 2 25 y 2 31. 49a 2 7a 6 27. 26 x 4 y 7 34 x 3 y 4 12 x8 y 32. x 2 12 x 32 RATIONAL FUNCTIONS List all numbers (if any) for which each rational expression is undefined. x2 4 2x a 2 2a 3 1. 2. 3. x3 5x 2 x 2 3a 2 11a 6 Divide using long division. 4 x3 4 x 2 7 x 9 2 x 2 3x 8 4. 5. 2x 1 x2 Simplify. 9a 2 25 2a 2 5a 12 a 2 9a 18 6. 10. 3a 2 5a 3ab 5b a 2 10a 24 4a 2 9 2 x 2 5 x 2 xy 2 y 2 16 r 2 r 2 2r 8 7. 11. 2 2 xy y 2 x 3 5 x 2 r 2r 8 4 r2 6 x3 7 x 2 6 x 2 7 x 5 4 8. 12. 12 x 3 36 x 9 y y2 3 5 3y y 1 9. 13. x x 1 4y 2 4y 2 Word Problems 1. Several of the World Cup 1994 soccer matches were played in Stanford University’s stadium in Menlo Park, California. The field if 30 yd longer than it is wide, and the area is 8800 yd2. What the dimensions of this soccer field? 2. John’s paint crew knows from experience that its 18-ft ladder is particularly stable when the distance from the ground to the top of the ladder is 5 ft more than the distance from the building to the base of the ladder. In this position, how far up the building does the ladder reach? 3. A Norman window has the shape of a square with a semicircle mounted on it. Find the width of the window if the total area of the square and the semicircle is to be 200 ft2. 4. Bob Michaels purchased a house 8 years ago for $42,000. This year it was appraised at $67,5000. (a) A linear equation V mt b, 0 t 15 , represents this value V of the house for 15 years after it was purchased. Find m and b. (b) Graph the equation and estimate in how many years after purchase this house will be worth $72,500. (c) Write and solve and equation to determine how many years after purchase this house will be worth $74,000. (d) Determine how many years after purchase this house will be worth $780,250. 3 5. A commercial jet airplane climbs at takeoff with slope m . How far in the horizontal direction will the 8 airplane fly to reach an altitude of 12,000 ft above the takeoff point? 6. Barb wants to drive to a city 105 miles from her home in no more than 2 hours. What is the lowest average speed she must maintain on the drive? 7. Consider the collection of all rectangles that have length 2 inches less than twice their width. (a) Find the possible widths (in inches) of these rectangles if their perimeters are less than 200 inches. (b) Find the possible widths (in inches) of these rectangles if their areas are less than or equal to 1200 in2. 400 8. For a certain gas, P , where P is pressure and V is volume. If 20 V 40 , what is the corresponding V range for P? 9. A company has current assets (cash, property, inventory, and accounts receivable) of $200,000 and current liabilities (taxes, loans, and accounts payable) of $50,000. How much can it borrow if it wants its ratio of assets to liabilities to be no less than 2? Assume the amount borrowed is added to both current assets and current liabilities. Domain and Range Find the domain and range for each function. Express in interval notation. 1. y x 2. y x 2 3. y x 3 4. y x 5. y x