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Algebra II Honors Midterm Review **In addition to doing the problems below, also study all quizzes, tests, notes, and homeworks. Chapter 1 Tools of Algebra 1. Simplify 33 43 (9 4) 32 1 2. Evaluate b a (c d )2 if a , b 4, c 2, d 3 . 2 3. df 3g 4h ; solve for f 4. solve 3 x 6 6 3( x 2) 5. Solve x x . Write answer in interval notation. 5 6. Evaluate r 3r 15 ; for r 3 2 5( y 2) ( y 1) 7. Evaluate ; for y 2 2 y 1 8. Solve 4 3x 10 . Write solution in interval notation 9. Solve P 2( AB 1) for A . State any restrictions on the variables. 1 10. Solve ( x 5) k for x . State any restrictions on the variables. 3 11. What is the value of x 3 5x 7 for x 9 ? 7 12. Simplify by combining like terms 3(a 5b) (2b a) 2 3 13. Solve inequality 7 m 5 . Write solution in interval notation 2 14. Solve inequality 2 y 3 3 y 5 . Write solution in interval notation 15. Solve inequality 3x 2 5 . Write solution in interval notation 16. Solve inequality 2 3x 5 8 24 . Write solution in interval notation 17. Two buses leave Philadelphia at the same time and travel in opposite directions. One bus averages 42mi/h, and the other bus averages 50 mi/h. When will they be 230 mi apart? 18. Solve compound inequality 12 3x 3 18 . Write solution in interval notation. 19. Solve compound inequality 5t 1 10 or 2t 5 1 . Write solution in interval notation. 1 20. Solve compound inequality 7a 3 32 and 3(a 1) 9 . Write solution in interval notation. 21a. Find the area of a rectangle with width of 3 inches and length of 7x – 46. b. If the area of the rectangle is 123 square inches find the value of x. 22. Find the area of the shaded region to the right in terms of x. 23. The cost of renting a DVD at a certain store is described by the function f(x) = 3x + 4 in which f(x) is the cost and x is the time in days. If Lupe has $82 to spend, what is the maximum number of days that she can rent a single DVD if tax is not considered? 24. After a ball is dropped the height of the ball decreases with each bounce. The equation y = 8(.6)x shows the relationship between x, the number of bounces, and y the height of the bounce, for a certain ball. What is the height of the ball after the fourth bounce? b b2 4ac 25. Find x given x and a = 5, b = 4, c = -1. 2a 2 26. Find the multiplicative inverse of - . 3 27. Find the additive inverse of -4. 28. Which property does the following statement illustrate? (3x + 5) + 8 = 3x + (5 + 8) 29. Is the set of whole numbers closed under the operation of subtraction? 30. Determine the solution to the inequality -3x + 5 > 12 if the replacement set is the set of positive even integers 31. Johnny jogged between 18 miles and 45 miles this week. represent each situation as an inequality and using interval notation. 32. Mr. Raven is thinking of a number (n). The number is less than 3 or the number is at least 8. represent each situation as an inequality and using interval notation. In problems 33-40, solve for x. Write your answers using interval notation when appropriate. 1 1 2 33. 3(2x – 1) + 4x = 2x – (x – 3) 34. x 35. 4x2 + 2 = 83 5 3 5 36. 5ax – 2b = 4b – (7ax – 6b) 37. 2x – (3x + 2) 5 38. 2a(3x + 4) = 4bx – 5b 39. 0.8x – 0.1 = 2(0.4x + 0.3) 40. 2(4x + 3) > 10x – (2x + 6) 41. The formula for the surface area of a cylinder is S =2πrh+2πr2 solve for h. 2 h b1 b2 42. Solve for h. a 2 43. Solve 3| 2x + 5 | - 3 = 9 44. Solve | x – 4 | + 5 < 12 45. Solve | 2x + 5 | = -3 46. Solve | 3x – 4 | ≥ 12 47. Given f(x) = 2x2, find a. f(-2) b. f(2x – 1) c. find the value of x when f(x) = 50. 48. Solve for x. Write your answers using interval notation when appropriate. a. 3x + c = 4ax b. | 2x – 1 | < 9 Chapter 2 Functions, Equations, and Graphs 1~10 Multiple Choices ___________ 1. Which of the following is an equation of the line that has an x-intercept of -3 and an y-intercept of 2? (A) 4+x + 6y = 18 (B) 4x + 6y = 12 (C) 4x + 6y = -12 (D) -4x + 6y =18 (E) -4x + 6y=12 1 ___________ 2. If the slope of a line that passes through the points (a, 4) and (1, -2) is - , what is 2 the value of a? (A) 13 (B) -13 (C) 11 (D) -11 (E) none of the above ___________ 3. Which of the following relations shows Y varies directly with x? (A) {(1,5), (2, 10), (3,18)} (B) {(7,21), (8,24), (9, 27)} (C) {(1,2), (5,10), (10,25)} (D) {(1,1, 1.21), (1.2, 1.32), (1.3, 1.53)} (E) {(1,4), (2,5),(3,6)} ___________ 4. In the xy-plane, the line with equation y = - 5x - 10 crosses the x-axis at the point (a, 0). What is the value of a? (A) -2 (B) 2 (C) 5 (D) - 10 (E) -5 ___________ 5. Line p has a negative slope and passes through the point (0, 0). If line k is perpendicular to line p, which of the following MUST be true? (A) Line k has a positive slope. (B) Line k has a negative slope. (C) Line k has positive x-intercept. (D) Line k has a negative y-intercept (E) Line k passes through the point (0,0) 3 ___________ 6. Given the function y = -5x - 12, which statement best describes the effect of increasing the y-intercept by 6? (A) The new line is parallel to the original (B) The new line has a greater rate of change (C) The new line has a smaller slope (D) The y-intercept decreases. (E) The x-intercept increases. ___________ 7. What is the slope of the line identified by 2y = -3(2x - 2)? 3 2 (A) 3 (B) -3 (C) (D) (E) 2 2 3 ___________ 8. Given f(x)= - 2x - 3, g(x) = |x-2| +3, what is f(6) – g(2) ? (A) 9 (B) -6 (C) -9 (D) -15 (E) -18 ___________ 9. What graph best represents all the pairs of numbers (x, y) such that x - y > 6? A. B. C. D. 10. a. Graph the function f(x) = -| x - 1 | + 6. Label the 5 points you used to graph. b. Find the vertex c. Find the x-intercept(s), if any. d. Find the y-intercept. 4 11. a. Graph the function f(x) = 2| x + 2 | - 2. Label the 5 points you used to graph. b. Find the vertex c. Find the x-intercept(s), if any. d. Find the y-intercept. 12. Write the piecewise function for the function f(x) graphed below. ___________, x _____ fx ___________, x _____ 13. The graph of y = |x| is translated down 2 units and right 4 units, vertically stretch by a factor of 2, and reflected across the x-axis. Find the equation of the new graph. Graph the new function. 14. Write an equation for the translation so the graph has the given vertex. a. y = - | x |; vertex ( -3 , - 2 ) b. y = 2| x |; vertex ( 2 , - 2 ) c. y = | x |; vertex ( 2 , -5 ) 15. Describe the translation(s) that takes the first function to the second function. a. y = | x - 5 |, y = | x - 2 | b. y = | x + 5 | + 1, y = | x + 2 | - 3 c. y = | x - 5 |, y = 2 | x - 5 | - 1 16. Graph y ≥ | x – 4 | - 2 17. Graph y + | x + 2 | < 6 18. Graph the following piecewise functions. 3x 4, x 1 fx x 2 , x 1 5 19. Graph the following functions. a. 2y < - x + 6 b. 6x - 3y ≤ 18 20. The population, P (in 1000s), of a town can be modeled by P = 17 (t + 56.6) where t = 0 represents 1980. What was the population in 1987? 21. The formula for simple interest is A = P + Prt. Solve the formula for P. 22. The largest egg laid by any bird is that of the ostrich. An ostrich egg can reach 8 inches in length. The smallest egg is that of the vervain hummingbird. Its eggs are approximately 0.4 inches in length. Write an inequality that represents the various lengths of bird eggs. 23. The sales tax on a dress is 8 %. a. Write a function f(x) that expresses the total cost of the dress including the sales tax in terms of the price of the dress (x). b. If the total cost of the dress is $56.16 find the price of the dress? 24. Find the value of y if (3, y) is a solution to the equation 4x - 5y = 9. 25. Represent the function y = 5 – (2x + 3) using a mapping when the replacement set is {-1,0,2}. 26. Given f(x)= 3 – 6x, find: a. f(2) b. f(3a – 4) c. the value of x when f(x) = -8 27. Given f(x) = x2 – 3x, find: a. f(2) b. f(a – 4) 28. Use the graph of f(x) to the right to answer the following questions. a. Find f(6) b. Find the value of x when f(x) = 4 c. Find the domain of the function. d. Find the range of the function. e. Find the average rate of change for the x interval 0 ≤ x ≤ 3. 29. Alex has $5 and plans to save $4 per week. Write a function for the amount of money that Alex has in terms of the number of weeks. Graph this function. 6 30. Determine if each table of values represents a function. If it is a function determine if it is a linear function. If it is linear, write a function for the relationship. a. b. c. 31. Graph each function. a. y = -2x + 5 b. 3x – 2y = 8 c. y – 4 = .5(x + 1) d. y=3 e. x = -2 32. Graph each inequality on a coordinate plane. a. y < -2x + 5 b. 3x – 2y < 8 c. y – 4 ≥ .5(x + 1) d. y≤3 e. x < -2 33. Write the equation for each function graphed. a. b. c. 34. Write the equation of a line that passes through the points (1, 3) and (-2, 9) in the following forms. a. point-slope form b. slope-intercept form c. standard form 35. Write the equation of a line (slope intercept form) parallel to the line in number 34 that passes through the point (6, -2). 36. Write the equation of a line (slope intercept form) perpendicular to the line in number 34 that passes through the point (-6, 1). 7 37. Given f(x)= 5 – 4x find: a. f(-4) b. f(3x – 5) 38. Use the graph of f(x) to the right to answer the following questions. a. Find f(3). b. Find the value(s) of x where f(x) = 1. c. Find the domain of this function. d. Find the range of this function. e. Find the average rate of change over the interval 2 ≤ x ≤ 4. f. Find the intervals of x where f(x) ≥ 4. Do not approximate. You may need to write the equation of a line to find the exact x values. 39. Graph 2x - 3y = 6 40. Given the table of values 41. Write the equation of for a linear function below, the function graphed write the equation for the below. function. x y 0 7 1 10 2 13 3 16 42. Two values of a linear function, f(x), are f(2) = 5 and f(-3) =15, write an equation for this function. 43. The domain of the function f(x) = -3x - 4 is restricted to 4.2 ≤ x ≤ 9.3. Find the range of f(x). 44. Denise is 12 feet from her locker and she is walking towards her locker at a rate of 2 feet per second. a. Write a function for her walking rate, r, (speed) in terms of the time, t. Graph this function. b. Write a function for her distance, d, from her locker in terms of the time, t. Find the reasonable domain and range. Graph this function over the reasonable domain. 45. Graph each equation. a. 3x – 4y = 8 b. x=5 c. y=8 8 46. Write the equation for the linear function f(x): given f(2) = 3 and f(4) = 9. 2 47. Write the equation for the line that is perpendicular to y x 4 and passes through the origin. 3 48. If the line in the answer to problem 47 is shifted up 8 units, write the equation for this new line. Ch 3 Linear Systems 1. Two cans of paint and one roller cost $29. Three cans of paint and two rollers cost $46. Write a system of equations that models this situtation. Remember to define your variables. 2. Solve the system of equations. x + y = 13 2x + 3y = 11 y 2x 3 3. Solve the system of equations. xy 8 4. Solve the system of equations. 2x - 10y = 6 3x - 6y = 0 5. Solve the system of equations. 4x 5y z 6 2x y 2z 11 x 2y 2z 6 6. Graph the system of inequalities. y ≤ - 2| x | y>x-4 9 7. The table below shows the per capita consumption of whole and lowfat milk for the years 1980 and 1990. Over the ten year period, the consumption of whole milk decreased at a rate that was approximately linear. During the same time, the consumption of lowfat milk increased at a rate that was approximately linear. 1980 1990 Whole Milk 17 gal 10 gal Lowfat Milk 11 gal 16 gal a. Write two equations in slope-intercept form that represent the per capital consumption, M, of milk. (One for whole and one for lowfat). Let t = 0 represent 1980. b. Graph this system. Be sure to label each axis and each line clearly. c. Estimate the year that the consumption of lowfat milk surpasses the consumption of whole milk. 8. Find the values of x and y that maximize or minimize the objective function. x+y≤8 2x + y ≤ 10 x ≥ 0, y ≥ 0 Maximum for C = 80x + 20y 9. Solve the system of equations graphed below. 10~15 Multiple Choices 10. Which point gives the minimum value for P = 3x + 2y and lies within the system of restrictions? 1≤x≤6 2≤y≤5 x + y ≤ 10 a. (1, 2) b. (0, 0) c. (5, 5) d. (7, 5) 11. What is the slope of the equation x = a ? a. 0 b. undefined c. a d. 1 12. A system of equations that has exactly one solution is classified as ___________. a. Dependent b. Independent c. Inconsistent d. Undefined 13. The system of x = y + 3 and y = x – 3 is classified as_______________. a. Dependent b. Independent c. Inconsistent d. Undefined 10 14. A system of parallel lines is classified as _______________. a. Dependent b. Independent c. Inconsistent d. Undefined 15. Mary has 15 dimes and quarters worth $2.85. Which equation could not be used in the system representing this situation? a. d + q = 15 b. d + q = 2.85 c. 10d + 25q = 285 d. d = 15 - q 16. Suppose you hire someone to come to your house to make some repairs. One person charges $25 per job plus $45 per hour for labor. Another person charges $60 per job plus $38 per hour for labor. a. Write a system of equations that models this situation. b. How many hours of labor would be necessary before the second one would be less expensive? 17. Suppose you bought eight oranges and one grapefruit for a total of $4.60. Later that day you bought six oranges and three grapefruit for a total of $4.80. Now you want to find the price of each orange and each grapefruit. Write an equation for each purchase. Solve the system of equations. 18. Two sides of a triangle are equal. Each of the two equal sides is 5 inches more than the third side. The perimeter of the triangle is 35 inches. Find the length of each side. 19. Solve the following systems of equations. a. 3x – 2y = -7 b. y = 4x – 9 c. -2x + 3y = 8 x + 4y = 2 15x + 3y = 7 4x – 6y = 9 20. Graph the following inequalities or system of inequalities 2 a. 2x – 3y ≥ 6 b. y x 1 c. 2x + 5y < 10 3 2 y x4 3 Chapter 4 Matrices 1~ 5 Multiple Choice ____ 1. How many elements are in an m n matrix? a. m + n b. c. d. mn ____ 2. A matrix contains 60 elements. Which of the following cannot equal the number of rows of the matrix? a. 30 b. 60 c. 18 d. 10 11 ____ 3. Suppose A and B are 2 x 5 matrices. Which of the following are the dimensions of the matrix A + B? a. 2x5 b. 10 x 10 c. 7x1 d. 7 x 7 Evaluate the determinant. ____ 4. a. 20 b. 4 c. 20 d. 4 3 3 3 3 ____ 5. a. –58 b. 58 c. –102 d. 102 6. The graph shows the populations of two towns. 10 Easton 9 a. Write a 2 x 3 matrix A to represent the data. Weston 8 Population (thousands) 7 6 5 4 3 2 b. Find element a 21 . What does this element represent? 1 1990 1995 2000 Year 12 7~10 Use matrices A, B, and C. Find the sum or difference if you can. 3 5 7 2 0 0 8 6 A B C 9 0 1 4 2 6 2 1 7. B+A 8. A2 9. 3C – B 10. AB 11. Find the product. 12. Find the values of the variables. f= k= w= 13. 0 13 27 0 13 3b 5a 25 3a 2b 7 25 17 7 a= b= 14. The Art Department and the Homecoming Committee at a local school are ordering supplies. The supplies they need are listed in the table. Paint Brushes Paper Glue Sticks Tape (bottles) (reams) (boxes) (rolls) Art Department 11 12 4 11 4 Homecoming Committee 10 14 7 17 7 A bottle of paint costs $4, a paint brush costs $2, a ream of colored paper costs $8, a box of glue sticks costs $3, and a roll of tape costs $2. Find the matrix that represents the total cost of supplies for each group. 13 15. Given quadrilateral A (4, -2), B ( 0, 2), C ( -4, -2), and D( 0, -6). a. Write the vertices of the figure above in a matrix. b. Graph the figure and its image after a reflection in the x-axis. c. Do any of the vertices of the preimage have the same coordinates as the vertices of the reflection? Explain. d. Give the coordinates of a point of the preimage that is also a point of the reflection. 16. What does the expression represent if A is a matrix? If exists, what can you say about the dimensions of A? Explain. 17. The matrix has no inverse. Explain how you can determine the value of x. Then find x. 5 3 18. Find the inverse of the given matrix. 7 2 19. Solve the matrix equation. 4 2 3 2 0 8 3 5 3 X 5 9 1 0 14 20. A triangle has vertices A(1, 1), B(3, 2), and C(-2, 4). a. 0 1 Transform the triangle by multiplying . 1 0 b. Graph the preimage and image. c. Describe the transformation. Use the matrices below to perform the matrix operation in problems 21 – 28. 0 1 1 0 0 1 3 A B C D 2 1 E 2 3 0 1 3 1 4 21. A+B 22. A+E 23. B–C 24. C•A 25. D+E 26. A•D 27. 2E 28. D•B 29. Use matrices to find the coordinates ABC with A(-1, 1); B(2, 4); C(4, -3), if it is translated three units to the left and 4 units up. 30. Use matrices to find the coordinates of the quadrilateral DEFG with D(1, -4); E(0, 2); F(2, 3); G(4, 1) a. rotated 90˚ about the origin. b. rotate 270˚ about the origin c. reflect across the x-axis d. reflect across the y-axis 15 Chapter 5 Quadratic Equations and Functions 1. What is the vertex of y = -3x2 + 12x – 16? 2. Give the standard form of y = 2(x – 3)2 + 5. 3. For the function y= x2 – 6x + 11, find the vertex and axis of symmetry. 4. What are the x-intercepts of y = -2x2 +10x + 28? 5. What is the value of “d” if the y-intercept of y = -2x2 – 3x + 2d -1 is (0, 7)? Graph the following. Find the axis of symmetry, vertex, y-intercept, the maximum/minimum value, domain, and range. On the graph label the axis of symmetry, vertex and 4 additional points. 6. y = x2 -4x +7 7. y = -3(x – 1)2 + 6 8. The graph of y = (x - 3)2 + 5 is translated to the left 3 units and up 4 units, what is the equation of the new parabola in vertex form Factor each expression completely. 9. 3x2 + 9x – 30 10. 3x2 + 9x 11. 4m2 - 20m + 25 12. 6b3 – 24b 13. x2 + 18x + 32 14. -15x2 – 5x 15. 4a2 + 20a + 24 16. 16y2 – 169 17. 2a2 + 13a – 7 18. 12x2 – x – 6 19. 2x2 – 54 Solve each quadratic equation. 20. -3x2 + 7x + 6 = 0 21 (2x - 1)2 = 25 22. 6x2 - 13x + 5 = 0 23. 4x2 – 128 = 0 Write a quadratic equation in standard form with given solutions 24. - 3 and 5 25. - and - ¾ 16 26. A company that makes basketballs has a daily production cost C in hundreds of dollars described by C = b2 – 10b + 37, where b is the number of basketballs made in hundreds. (6 points) a. How many basketballs can be made in order to minimize cost? b. What is the cost when this many items are made? 3 27. Suppose you cut a small square from a square of fabric as shown in the diagram. Write an expression for the remaining shaded area. Factor the expression. 28. Show that is equal to . x Simplify: 29. 45 30. 5 200 31. 3 7i 1 2i 32. (4 5i) (2 3i) 33. (2 3i )(2 3i ) 34. (2 7i) 2 35. (4 5i )(7 3i ) 36. 36 1 1 25 Solve each quadratic equation by completing the square. 37. x 2 6 x 10 38. x 2 7 x 12 0 39. 2 x2 8x 19 1 2 1 1 2 2 40. 3x 2 6 x 9 41. x 2x 42. x x5 0 2 4 3 3 Rewrite each equation in vertex form by completing the square. 43. y x 2 8 x 20 44. y x 2 12 x 9 45. y 2 x 2 16 x 24 46. y 8 x 2 2 x 20 17 Find the value of k that would make the left side of each equation a perfect square trinomial. 4 47. x 2 kx 49 0 48. 4 x 2 kx 25 0 49. x 2 kx 0 81 Solve each equation using the Quadratic Formula. 50. 3x 2 1 5 x 51. 3x 2 11x 4 0 Evaluate the discriminant of each equation. Tell how many solutions each equation has and whether the solutions are real or imaginary. 52. 6 x 2 13x 6 0 53. x( x 9) 5 54. 10 x 3x 1 2 55. 400 x 400 x2 100 Chapter 6 Polynomials and Polynomial Functions 1~6 a. Name the leading coefficient. b. State the degree and name of the polynomial function if possible. c. Describe the end behaviors. d. Find the zeroes and state the behavior at the zeroes. e. Find the y-intercept. f. Sketch the graph. 1. y = (x – 2)2(x + 3) 2. y = −(x + 2)4(x – 1)3 1 2 3 3. y = −x(x − 2)3(x + 1) 4. y x 1 x 5 x 3 2 5. y = −(x + 1)3(x – 2)2 6. y = (x − 3)3(x − 2) Factor completely. 7. x4 – 81 8. 2r3 – 16t3 9. 21 – 7y + 3x – xy 10. m4 – 1 11. 10w2 – 14wv – 15w + 21v 12. x4 – 10x2 + 25 13. 16a4 – 81 Simplify 14. (x – 4)(x2 + 4x + 16) 15. (x + y)(x2 – xy + y2) ** Since we just finished with Ch 6, this is just a few problems from Ch 6. Study all quizzes, notes, and homeworks from Ch 6 as well. 18

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