Carnegie Vanguard HS Summer Math Packet for students entering Pre-AP Geometry (this covers Algebra 1 objectives)

Carnegie Vanguard HS – Summer Math Packet for students entering Pre-AP Geometry (this covers Algebra 1 objectives) ( 1. Simplify 2(2 2 − 8 2 ) + 3 7...
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Carnegie Vanguard HS – Summer Math Packet for students entering Pre-AP Geometry (this covers Algebra 1 objectives)

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1. Simplify 2(2 2 − 8 2 ) + 3 7(−3) + 20 ÷ 10 + (−3) 2  2 y − 2z 3 2. Evaluate  2 2  16 x − 4 y

)

3

  when x = 2 and y = −5 and z = 3 

3. Is x = −2 a solution to the inequality 4 x 2 + 8 x − 17 ≥ 5 ? 4. Rewrite each algebraic expression, equation, or inequality. a) The difference of five times a number c and 12 is 38. b) The quotient of 100 and 7 times a number w is at most 11. c) The sum of 10 and the square of a number z is less than 26 more than the product of 9 and a number d.

5. Does the following table represent a function? Explain using the definition of a function. x y

0 -2

4 14

8 30

12 46

16 52

6. The points ( 2 , 1 ) , ( 4 , 7 ) and ( 6 , 25 ) are part of the continuous linear function A(n) . a) Rewrite ( 4 , 7 ) using function notation. b) Reorganize the points above into a table of values.

c) Write an expression using function notation that represents the function. (the "rule" for the function)

7. Fred's discount DVD sells DVDs on eBay. Fred computes the shipping costs using a fee of $7.00 plus $2.00 per DVD. Based on packaging restrictions (and a desire by Fred to make more money on shipping), Fred will not ship more than 8 DVDs at a time. You want to buy some DVDs but you will not buy more than will fit in one shipment. a) What are the independent and dependent variables in this situation? Write a sentence to describe the relationship using the phrase "is a function of." b) Represent each variable with a letter. Write an equation using function notation to express this situation. c) What is the domain for this situation? d) What is the range for this situation? e) IF you were to graph this function (DON'T actually graph it), what would the $2.00 in your equation represent? Is this meaningful in the context of the problem? Why or why not?

8. f ( x) = x 2 + 8 x + 6 is defined on the domain { -8 , -6 , -4 , -2 , 0 } a) What is the range of f (x) ? b) Fill in the mapping below based on f (x) .

c) Write f (x) as a list of ordered pairs. d) Graph f (x) on the grid below. Plot your points very darkly so I can see them.

e) A one-to-one function is a function in which each output comes from exactly one input. Is f (x) a one-toone function? Explain. 9. Classify each number as natural, whole, integer, rational, or irrational (some will have multiple answers, give ALL classifications that are appropriate): 7 • − 8 • 13 • 0 5 • 42 • −5 10. Evaluate when x = −4 and y = −5 .

2( 3 x + 4 ) − 5( 6 y + 22 )

11. Write an equation or inequality to represent the following: • Seven times the difference of a number p and fifty-nine is eight more than twice the quotient of a number r and a number t. •

One-third of the sum of ten and a number j is at least quadruple the difference of a number z and fifty.

12. For each of the following, give the full name of the property illustrated. a. r ( st ) = (rs )t 1 b. 7  = 1 7 c. − 10 + 5 = 5 + (−10) d. k + (−k ) = 0 e. a (b + c) = ab + ac 13. Simplify each of the following. a. 5w 3 (6 w 3 + 3w 2 − 2 w − 8) − 10 w 4 (4 w 2 − 10 w − 3) 2 21  b. − 7(4 x − 6 y + 10 ) + 15 y − 27 z +  3 2

14. Edgar buys an mp3 player on sale for 40% off of the regular price. He also used a coupon for an additional 20% off of the sale price. On the receipt it says the coupon saved $24.00. a. What was the sale price of the mp3 player? b. What was the regular price of the mp3 player?

15. Hendrix High's enrollment decreases at an average rate of 50 students per year, while Martinez High's enrollment increases at an average rate of 70 students per year. At the beginning of this school year, Hendrix High has 2450 students and Martinez High has 1850 students. a. Write an expression to represent the number of students enrolled at Hendrix High t years from now. b. Write an expression to represent the number of students enrolled at Martinez High t years from now. c. If enrollments continue to change at the same rate, write an equation for when the two schools will have the same number of students. d. Based on your equation from part c, during what school year will the two schools have the same number of students? (solve your equation from part c)

16. Lyric, Javan, and Gervaisa work together for a furniture company building chairs. Lyric can build 3 chairs per hour, Javan can build 4 chairs per hour, and Gervaisa can build 5 chairs per hour. Lyric starts work at 9AM, Javan starts work at 9:45AM and Gervaisa starts work at 11:00AM. They all finish working at the same time and together they build 71 chairs. a. Let t represent the number of hours since 9AM. Write an equation to model this problem. b. Solve your equation. c. What time did they finish working? d. How many chairs did Javan build? 17. Find the area of a rectangle whose length is five less than four times its width and whose perimeter is thirty more than six times its width.

18. Solve



4 (10 x − 15) = −6 x + 4 5

19. Solve

5k 25 = 2k − 3 7

20. Graph each of the equations below on the grids provided. Give the x-intercept as a coordinate pair, the yintercept as a coordinate pair, and the slope. 4 x −8 3

a) 2 x + 3 y = 18

b) y =

x-intercept:

x-intercept:

y-intercept:

y-intercept:

slope:

slope:

21. The point (d , f ) is located in the third quadrant. All the following questions are based on this point. • In what quadrant(s) is (−d , f ) located? • In what quadrant(s) is ( f , d ) located? • In what quadrant(s) is (2d ,−7 f ) located? • In what quadrant(s) is (d − 12, f − 17) located? • In what quadrant(s) is (d + f , f + d ) located? 22. Which of the following lines are parallel to each other? Explain. (A) y = −3 x + 10 2 (B) y = x −8 5 (C) − 5 x + 2 y = −16 (D) 2 x − 5 y = 30 (E) 6 x + 2 y = −10 23. Which of the following represent direct variation? Give the constant of variation for those that do. b) 9 x − 3 y = 0 c) y = −2 x a) y = 15 x + 10 d)

e)

24. The table below gives the amount of money Eric earns at his job after a certain number of hours. Number of hours Pay ($)

2 $15

4 $25

8 $45

14 $75



What is the rate of change of pay with respect to number of hours?



Does this relationship represent direct variation? Explain.

5 and passes through the point (12,8) . Write the equation of this line in point-slope 6 form, slope-intercept form, and standard form.

25. A line has a slope of −

26. A line passes through the points (−2,5) and (8,35) . Write the equation of this line in point-slope form, slope-intercept form, and standard form.

2 x + 42 . 5 a. Write an equation of the line parallel to line k that passes through the point (20,−15) .

27. Line k has the equation y =

b. Write an equation of the line perpendicular to line k that passes through the point (20,−15) . 28.

a. Give the equations of the lines that bound the figure above. b. Are any lines in the figure above parallel? How do you know? c. Are any lines in the figure above perpendicular? How do you know? 29. 3 a. Graph y = − x 2

b. Translate the line in part (a) to the right 4 units and up 5 units. c. Write the equation for the translated line in the form y = m( x − h) + k d. Rewrite the equation in slope-intercept form.

30. When y =

2 2 x is changed to y = ( x + 12) + 17 , how is the graph translated? 5 5

31. Write the equation in the form y = m( x − h) + k when y = −7 x is translated right 15 and down 42. 32. Write the equation in the form y = m( x − h) + k of a line with a slope of −

4 that passes through the point 7

(−10,25) .

33. For the line y = 5( x − 5) − 25 , give the slope and one point it passes through. 34.

Write two different equations of the line above (using the two points marked) in the form y = m( x − h) + k .

35. Solve each inequality. Graph your solution on a number line. a. 6(3 x − 5) ≤ 2(9 x + 7) + 10 b. − 27 ≤ −5 x − 17 < 38 c. − 3 x + 7 > 34 OR 4 x + 8 ≥ 24

36. Check if each of the following is a solution to the given inequality. a. Is (3 ,− 4) a solution of 4 x − 8 y > 12 ? b. Is (5 , − 5) a solution of 5 x + 2 y ≤ 15 ? 37. Graph each of the following inequalities a. y > −

2 x+3 3

b. 7 x − 4 y ≥ −28

38. Give the inequality of each of the graphs below in point-slope form, slope-intercept form, and standard form. a. b.

39. Sam is selling CDs and DVDs to raise money to buy a new mp3 player. He is charging $3 for each CD and $7 for each DVD. The cheapest mp3 player at the store is $105 and the most expensive one is $210 (and there are many others in between these prices). This means that he needs to make at least $105 and at most $210 from his sales. Let x be the number of CDs Sam sells and let y be the number of DVDs Sam sells. a. Write an inequality in standard form using the minimum Sam needs to raise (at least $105). b. Write an inequality in standard form using the maximum Sam needs to raise (at most $210). c. Combine your answers from parts a and b into a compound inequality. d. Graph your compound inequality from part c on the graph below. Label the axes. (hint: do this by graphing the inequalities from parts a and b and only shading where they overlap)

e. Give 2 possible combinations of CDs and DVDs that Sam could sell that would earn him enough money to buy an mp3 player at this store. Explain how you know these combinations provide enough money based on your graph in part d. 40. Claire is selling tickets to the school musical. The tickets cost $6 for adults and $4 for students. The auditorium can hold at most 550 people. The organizers of the musical must make at least $2500 to cover the costs of the set construction, costumes, and programs. a. Write a system of inequalities to represent this situation. b. If Claire sells 150 adult tickets and 375 student tickets, is that a valid solution to this problem? Explain.

41. Solve the following system by graphing. Give the solution and classify the system.

3 x − 2 y = 12 6 x − 4 y = −6

42. Solve each of the following systems using substitution or elimination. Give the solution and classify the system.

4x + 5 y = 7 6 x − 3 y = −21

43. Graph the following system of inequalities.

x ≥ −4 y −20

44. Write a system of inequalities for the shaded region.

2 7 5 x+ y =− 3 3 3 4 14 x− y =2 5 5



For questions 45-46, set up and solve a system of linear equations. 45. Thalia is running a bake sale to raise money for charity. She is selling cookies for $1.50 each and brownies for $3.50 each. At the end of the day she has raised $210 and sold 100 items. How many of each item did she sell? 46. Two families go to the movies. One family purchases two adult tickets and four youth tickets for $30. Another family purchases three adult tickets and five youth tickets for $41. How much would it cost to purchase six adult tickets and eight youth tickets? 47. Write a polynomial with zeros of 3 and −

4 . 5

48. Find the product.

(3 x − 2)(2 x 2 + 5 x − 4) 49. Factor each of the following. a. x 2 − 9 x + 18 b. 8 x 2 + 18 x − 5 c. 25 x 2 + 70 x + 49 d. 45 x 4 − 80 x 2 50. For what value(s) of k is kx 2 + 60 x + 100 a perfect square trinomial? 51. Solve each of the following. a. 12 x 3 − 10 x = −14 x 2 b. 4 x 3 + 5 x 2 − 36 x − 45 = 0 52. You have two rectangular mirrors that have the same area. One mirror has dimensions of 3 x − 4 and x + 2 . The other mirror has dimensions of x + 6 and 2 x − 3 . All dimensions are in feet. a. Write an equation that relates the areas of the two mirror. b. Find the dimensions of each mirror. 53. Solve x 2 + 3 x − 70 = 0 by factoring. 54. Solve x 2 − 14 x − 15 = 0 by completing the square. 55. Solve 3 x 2 + 10 x + 7 = 0 using the quadratic formula. 56. For the following function, find the y-intercept, x-intercepts, vertex, axis of symmetry, domain, and range. DO NOT sketch the graph.

y = 2 x 2 + 5 x − 25

57. Sketch the graph of y =

1 1 2 x − 4 . Describe how y = x 2 − 4 is a transformation of the quadratic parent function. 3 3

For questions 58-59, find the y-intercept, x-intercepts, and vertex; then sketch the graph. 58. y = x 2 − 8 x + 7

59. y = −2( x − 3) 2 + 9

60. What is the area of a right triangle with legs of x − 4 and x + 3 and a hypotenuse of 17 meters? a) Draw and label this triangle b) Write an equation that relates the sides of the triangle. c) Solve for x. d) Use x to find the legs of the triangle. e) Find the area of the triangle.

For questions 61-65, use the properties of exponents to simplify each of the expressions. 61. (5r 6 ) 2 ⋅ 4r 5 62. (2 x −6 y 3 z −7 ) 5

63.

2 4 ⋅ 43 84

26 x 3 y −6 64. (4 x − 4 y −1 ) 2

 m 4 n12 65.  15 9  4m n

  

 5x 2 y  ⋅  5   y 

0

−3

66. You buy a new HDTV widescreen TV. The ratio of the width of the screen to the height of the screen is 16:9. If the TV is advertised as 60 inches (measured diagonally), what are its dimensions?

67. A cylinder has a radius of 5cm and a height of 30cm. (Round all answers to the nearest hundredth) a) Draw a net for this cylinder. Label all dimensions. b) What is the surface area of the cylinder? c) What is the volume of the cylinder? d) You have a cone with the same radius and height as the cylinder. What is its volume? e) If you multiply all the dimensions of the original cylinder by a scale factor of 2, what would be the surface area and volume of the new cylinder? f) If you multiply only the radius of the original cylinder by a factor of volume of the new cylinder?

1 , what would be the surface area and 3