Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics

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1 Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics 1 DISTANCE BETWEEN TWO POINTS - REVIEW To find the distance between two points, use the Pythagorean theorem. The difference between x 1 and x 2 is one leg and the difference between y 1 and y 2 is the other leg. distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 1. Point M (-1, 3) and point N (6, 5) are points on the coordinate plane. What is the length of the segment MN? 2. Points A and B lie in the standard (x,y) coordinate plane. The (x,y) coordinates of A are (7, 1), and the (x,y) coordinates of B are (-2,-4). What is the distance from A to B? USING TWO POINTS TO FIND THE SLOPE - REVIEW change in y slope = m = = rise = y 2 y 1 change in x run x 2 x 1 3. What is the slope of the line that contains the points J (2,3) and K (0,-2)? 4. What is the slope of the line that contains the points H (-9, 1) and G (2, -7)? SLOPE-INTERCEPT FORM - REVIEW y = mx + b (where m represents the slope and b represents the y-intercept) 5. What is the slope of the line y = 1 4 X + 3? 6. At what point does the line y = 2 X - 1 cross the y axis? 9 FINDING SLOPE AND INTERCEPT FROM LINEAR EQUATIONS - REVIEW To find the slope or y-intercept of a line that is not in slope-intercept form, re-write the equation in slopeintercept form so it looks like y = mx + b. 7. At which y-coordinate does the line described by the equation 6y - 3x = 18 intersect the y-axis? 8. What is the slope of the line described by the equation y - 2x = 8?

2 CREATE AN EQUATION FOR A LINE FROM 2 POINTS - REVIEW Finding an equation for a line when we are only given 2 points takes 2 steps. We know that y = mx + b, where m is the slope and b is the y -intercept. So in order to create the equation we need m and we need b. 2 Step 1: find m (slope) We already know how to find the slope from 2 points. slope = m = change in y = rise = y 2 y 1 change in x run x 2 x 1 Step 2: find b (y-intercept) We're going to use the formula y = mx + b to solve for b. Plug in the slope you found in step 1 for m, then choose one of the points that was given (it doesn't matter which one), plug in the x value for x in the formula and the y value in for y in the formula. Now the only variable that doesn't have a known value is b. Solve for b. 9. What is the equation of the line that passes through points A (-8,3) and B (0,-1)? 10. What is the equation of the line that passes through the points (0,0) and (6,3)? 11. What is the equation of the line that passes through the points (4,5) and (-1, 0)? PARALLEL LINES ON THE COORDINATE PLANE - REVIEW Parallel lines always have the same slope. Therefore, if two lines have the same slope then they are parallel. 12. Write the equation of a line which is parallel to the line y = 2x + 3 and passes through the point (0, -1).

3 3 PERPENDICULAR LINES ON THE COORDINATE PLANE When two lines are perpendicular, the slope of one is the negative reciprocal of the other. If the slope of one line is m, the slope of the other is 1. m For example, in the image above, the slope of one line is -4. The slope of the other line is 1 4. negative reciprocals of each other. These lines are perpendicular. -4 and 1 4 are 13. Are these 2 lines perpendicular? y = 2x + 3 y = 1 2 x Are these 2 lines perpendicular? y = 1 2 x + 5 y = 2 3 x Are these 2 lines perpendicular? y = 2 5 x + 3 y = 2 5 x Are these 2 lines perpendicular? x - 2y = 3 2y = x - 7

4 17. Which of these is an equation of a line with a y-intercept of 3 and is perpendicular to the line 3y - x = 6? A. y = 1 3 x B. y = 1 3 x + 3 C. y = 3x + 3 D. y = 2x + 6 E. y = -3x Which of these is an equation of a line perpendicular to the line 2y = x - 7 which goes through the point (0, 5)? A. y = 5x - 7 B. y = -2x + 5 C. y = 1 2 x - 7 D. y = 1 2 x + 5 E. y = 1 2 x Line m passes through the point (4,3) in the standard (x,y) coordinate plane and is perpendicular to the line described by the equation y = 4 x + 6. Which of the following equations describes line m? (Hint: to 5 make a line we need a slope and a y-intercept. A perpendicular slope is easy to figure out. To find the y- intercept, use the generic formula y = mx + b and plug in the perpendicular slope for m, and use the point (4,3) for the x and y value, then solve for b.) A. y = 5 4 x - 2 B. y = 5 4 x + 2 C. y = 4 5 x - 2 D. y = 4 5 x +2 E. y = x - 2

5 20. Line q passes through the point (-2, -3) in the standard (x,y) coordinate plane and is perpendicular to the line described by the equation y = 2 x + 4. What is the equation that describes line q? Line z passes through the point (3, 5) in the standard (x,y) coordinate plane and is perpendicular to the line described by the equation y = 3 x - 2. What is the equation that describes line z? In the standard (x,y) coordinate plane, what is the equation of the line perpendicular to the line y = -2x + 2 and that passes through the point (0,-3)? 23. In the standard (x,y) coordinate plane, line m is perpendicular to the line containing the points (5,6) and (6,10). What is the slope of line m? (Hint: how do you find the slope of a line using 2 points? Then how do you find the perpendicular slope?) A. -4 B C. 4 D. 4 E In the standard (x,y) coordinate plane, line P is perpendicular to the line containing the points (8,5) and (4, 9). What is the slope of line P? 25. In the standard (x,y) coordinate plane, line P is perpendicular to the line containing the points (3,1) and (-2, 5). What is the slope of line P?

6 SYSTEMS OF EQUATIONS Remember that when we have an equation with two variables, there is more than one possible answer. A line shows all the possible answers for a linear equation. For example, the line in the graph to the right shows all the possible answers to the equation y = 2x + 2. However, if there are 2 linear equations there is only one possible answer; it is the point where the two lines meet. It is only at this point where the variables work for both equations (see graph at far right). 6 SOLVING SYSTEMS OF EQUATIONS BY SUBSTITUTION - REVIEW Solve the system of linear equations y = 3x - 7 and 4x - 2y = 5. y = 3x - 7 Start with the original equation 4x - 2y = 5 Start with the original equation 4x - 2(3x - 7) = 5 Substitute (3x -7) for y since (3x-7) = y 4x - 6x + 14 = 5 Distributive property -2x + 14 = 5 Add like terms -2x = - 9 Subtract 14 from both sides x = 9 2 Divide both sides by -2. x = 9 2 y = 3x - 7 y = 3( 9 2 ) -7 y = 27 2 y = y = 13 2 To solve for y, start with the original equation (either one will work) Plug in 9 2 for x - 7 Distributive property Re-write 7 as 14 (common denominator) 2 The point where these lines meet is at ( 9, 13 ) 2 2 SOLVING SYSTEMS OF EQUATIONS BY ELIMINATION (ADDING AND SUBTRACTING) - REVIEW We will solve the same system of linear equations using elimination. y = 3x - 7 Start with the original equation 4x - 2y = 5 Start with the original equation y = 3x - 7-2y = -4x + 5 Re-write the equation so that like terms are above each other 2(y) = 2(3x - 7) Multiply both sides by 2 (we're trying to get rid of the y value and I see that 2y - 2y = 0) -2y = -4x + 5 2y = 6x - 14 Distributive property -2y = -4x = 2x - 9 Add the two equations 9 = 2x Add 9 to both sides. 9 = x To solve for y, plug in 9 for x in either equation, just the same as before. 2 2

7 7 Here's another example. What is the x-coordinate of the intersection point, in the (x,y) coordinate system, of the lines 2x + 3y = 8 and 5x + y = 7? We can use either substitution or elimination to solve the equation. I choose elimination. 2x + 3y = 8 Start with the original equation. -3(5x + y) = -3(7) Multiply the other equation by -3. I choose this because 3y -3y = 0. 2x + 3y = 8 Original equation. -15x -3y = -21 Distributive property. -13 x = -13 Add the two equations; the y variable has been eliminated. x = 1 Divide both sides by -13. The question only asked for the x value so we're done. Sample Questions: 26. What are the (x,y) coordinates of the intersection point, in the coordinate system, of the lines 3x - 4y = 12 and 2x + y = 3? 27. What is the x-coordinate of the intersection point, in the (x,y) coordinate system, of the lines 2x + y = -1 and 4x + 3y = 1? LINEAR INEQUALITIES Sometimes we're asked to graph linear inequalities like this y 2x -4. To do this we first graph the line y = 2x -4. Then we shade everything under the line. Everything on the shaded side of the graph will work in the equation. You can choose any point and test it. For example, the point (3, -3) is in the shaded area. I will plug in those values and see if it works in the equation. y 2x -4 Original equation -3 2(3) -4 Plug in the point (3, -3) Distributive -3 2 This is true, so the point (3, -3) works. Let's try a point that is not on the shaded side and see if it works. I will choose the point (0,0). y 2x -4 Original equation -3 2(0) -4 Plug in the point (0, 0) Distributive -3-4 This is not true, so the point (0, 0) doesn't work.

8 Sometimes people get confused which side to shade. If you think of the line as a division between a mountain and the sky, then the when y is < or shade the mountain (or ground), regardless of how steep the mountain is or which direction it is sloped. If they're asking for y > or, shade the sky regardless of how steep the mountain is or which direction it is sloped. The way to distinguish between < or is with a solid or dashed line. If it's then the line is solid. If it's <, then the line is dashed to indicate that the line itself is not included. The same is true for > or. If it's, then the line will be solid. If they want > then the line will be dashed. The graph at the right shows the equation y > 3. 8 Sample Questions: Graph these inequalities on the coordinate graphs: 28. y 5 4 x y < -2x + 1 Sometimes we have to find an area bound by 2 or even 3 lines. To do that, graph each line and choose to shade the mountain or sky side depending on what they ask for. The part where the shading overlaps is the part that satisfies all the equations. Sample Questions: 30. Which of the following systems in inequalities is represented by the shaded region on the coordinate plane below? a. y < 6 and y > 3x 5 b. x < 6 and y > 3x 5 c. y < 6 and y < 3x 5 d. x < 6 and y < -3x 5 e. y < 6 and y > (1/3)x Jamie drew a triangle bounded by the lines y = -x, x = -2, and y = 8 and shaded the interior, as shown in the figure below. Then Jamie decided to reflect this triangle across the y-axis and shade the interior of the new triangle. Which of the following would describe the shaded region of Jamie's new triangle? a. x 2, y 8, y x b. x 2, y 8, y x c. x 2, y -8, y x d. x -2, y 8, y x e. x -2, y -8, y x

9 9 MIDPOINT BETWEEN 2 POINTS ON THE COORDINATE PLANE Sometimes you need to find the point that is exactly between two other points. For instance, you might need to find a line that bisects (divides into equal halves) a given line segment. This middle point is called the "midpoint". If you need to find the point that is exactly halfway between two given points, just average the x- values and the y-values. Think about it this way: If you are given two numbers, you can find the number exactly between them by averaging them, by adding them together and dividing by two. For example, the number exactly halfway between 5 and 10 is = 15 2 = 7.5. The Midpoint Formula works exactly the same way. We find the average of the x values and the average of the y values. Midpoint formula: ( x 1+ x 2 2 For example: find the midpoint between ( 1, 2) and (3, 6). ( x 1+ x 2 y, 1 + y 2 ) = ( , ) = ( 2, 4 ) = (1, -2) , y 1 + y 2 ) 2 Sample Questions: 32. In the standard (x,y) coordinate plane, points P and Q have coordinates (2, 3) and (12, -15), respectively. What are the coordinates of the midpoint of PQ? 33. The endpoints of the diameter of a circle O are A and C. in the standard (x, y) coordinate plane, A is at (4, 3) and C is at (-9, -2). What is the y-coordinate of the center of the circle? (Hint: the center will be at the midpoint) 34. In the standard (x,y) coordinate plane, the endpoints of line segment QR lie at (7, -8) and (-2,1). What is the x-coordinate of the midpoint of line segment QR? 35. The midpoint of line segment MN in the standard (x,y) coordinate plane has coordinates (1,-1). The (x,y) coordinates of M and N are (-3,2) and (B, -4), respectively. What is the value of B?

10 The midpoint of line segment AC in the standard (x,y) coordinate plane has coordinates (4,8). The (x,y) coordinates of A and C are (4,2) and (4, Z), respectively. What is the value of Z? (Hint: you already have the midpoint, in this case you need to work backwards to find Z). 37. In the standard (x, y) coordinate plane, the midpoint of AB is (4, -3) and A is located at (1, -5). If (x, y) are the coordinates of B, what is the value of x + y? 38. The diagonals of a rectangle connect the opposite corners of the rectangle. The vertices of the rectangle are located at the points (2, 6), (12, 6), (12, -2), and (2, -2). What are the coordinates of the point where they meet? MORE ON FUNCTIONS, DOMAIN AND RANGE Remember that the domain is the set of all possible x-values which will make the function "work", and the range is the set of all possible y-values which will make the function "work." When we're asked to find the domain of a function, we're looking to see if there are any places where the equation would be undefined. Specifically, we're looking to see if there is a possibility of a 0 in the denominator. If there are, then the domain is everywhere except there. Sample Questions: 39. What is the domain of the function f(x) = x 1 x 2? 40. What is the domain of the function f(x) = 41. For what values of t is A. 0 and 5 B. 0 only C. -4 and 0 D. -4 and 5 E. -4, 0, and 5 t+4 t(t 5) undefined? x+3 x(x 2)(x+7)?

11 11 VERTICAL LINE TEST Not everything on a graph fits the definition of a "function." Part of the definition of a function is that for every question (x value) there is only one answer (y value). Since y represents the "answers" we can use a vertical line to test if a graph is a function or not. If a vertical line touches the graph in two places anywhere then it is not a function, because that means that at least once a single question (x value) has more than one answer (y value). Sample Questions: 42. Which of the graphs at the right are functions?

12 12 BEYOND LINES - CIRCLES, PARABOLAS AND ELLIPSES Thus far, we have focused on linear equations on the coordinate plane, which, of course, means equations that make straight lines. There are equations that make other shapes. We will briefly introduce equations of circles, parabolas, and ellipses. Circles and ellipses are not functions, but we can still graph them on the coordinate plane. Parabolas may or may not be functions, depending on which direction they face. EQUATION FOR A CIRCLE The equation for a circle of radius r and centered at (h,k) is: This means that if we see an equation that looks like this, we can recognize that it will be a circle and we can figure out where the center of the circle is and what the radius is. Using the example of the equation (x 2) 2 + ( y + 1) 2 = 25. Since the equation of a circle: (x h) 2 + ( y k) 2 = r 2 we can compare this to the given equation. (x h) 2 + ( y k) 2 = r 2 (x 2) 2 + ( y + 1) 2 = 25 I notice that h (or the x value of the center of the circle) must be 2 since I see (x 2) 2 I notice that k (or the y value of the center of the circle) must be -1 since I see ( y + 1) 2. This one is a little trickier, since I need to remember that it is y - k and since I see y + 1 that means I need a negative value for k since subtracting a negative value is the same as adding. (y [ 1]) = ( y + 1). I also notice that the radius must be 5 since r 2 = 25. Putting all this together I know that the center of the circle is at the point (2, -1) and the radius is 5. With this information I can also figure out other things about the circle like the diameter, circumference, and area of the circle. The diagram below shows the graph of the circle with center at point (2,-1) and a radius of 5.

13 43. The equation of a circle in the standard (x,y) coordinate plane is given by the equation (x + 5) 2 + ( y 5) 2 = 5. What is the center of the circle? A. ( 5, 5) B. (-5, 5) C. ( 5, 5) D. (5, -5) E. (5, 5) The equation of a circle in the standard (x,y) coordinate plane is given by the equation (x 2) 2 + ( y + 4) 2 = 16. What is the center of the circle? 45. The equation of a circle in the standard (x,y) coordinate plane is given by the equation (x) 2 + ( y) 2 = 12. What is the center of the circle? 46. The equation of a circle in the standard (x,y) coordinate plane is given by the equation (x + 1) 2 + ( y + 3) 2 = 49. What is the center of the circle? 47. The equation of a circle in the standard (x,y) coordinate plane is given by the equation (x 8) 2 + ( y + 1) 2 = 20. What is the radius of the circle? A. 20 B. 8 C. 9 D. 2 5 E The equation of a circle in the standard (x,y) coordinate plane is given by the equation (x 3) 2 + ( y 2) 2 = 36. What is the radius of the circle?

14 49. The equation of a circle in the standard (x,y) coordinate plane is given by the equation (x 4) 2 + ( y + 7) 2 = 12. What is the radius of the circle? A circle with the equation x 2 + y 2 = 49 is graphed in the standard (x,y) coordinate plane. At which 2 points does this circle intersect the x-axis? (Hint: Where is the center of the circle? What is the radius of the circle? Would it help to draw a graph?) A. (-1, 0) and (1, 0) B. (-7, 0) and (7, 0) C. (-14, 0) and (14, 0) D. (-21, 0) and (21, 0) E. (-49, 0) and (49, 0) 51. In the standard (x,y) coordinate plane, what is the area of the circle (x 3) 2 + ( y + 2) 2 = 25? A. 5π B. 10π C. 25π D. 125π E. 225π 52. In the standard (x,y) coordinate plane, what is the area of the circle (x + 1) 2 + ( y + 6) 2 = 36? 53. In the standard (x,y) coordinate plane, what is the circumference of the circle (x 7) 2 + ( y 4) 2 = 81? 54. Which of the following is an equation for the circle in the standard (x, y) coordinate plane that has its center at (-1, -1) and passes through the point (7,5)? (Hint: can you eliminate any answers? The radius is the distance from the center to the point.) A. (x 1) 2 + (y 1) 2 = 10 B. (x + 1) 2 + (y + 1) 2 = 10 C. (x 1) 2 + (y 1) 2 = 12 D. (x 1) 2 + (y 1) 2 = 100 D. (x + 1) 2 + (y + 1) 2 = 100

15 EQUATION FOR A PARABOLA The graph of an equation in the form y = ax 2 + bx + c is a parabola. A parabola has a line of symmetry. The point on the parabola that is on its line of symmetry is called the vertex. We can find the x value of the vertex with the formula: b, then by plugging the x value into the equation, we can find the y value of the 2a vertex. 15 For example to find the vertex of the parabola y = 2x 2 + 8x + 5 we compare it to the formula y = ax 2 + bx + c y = 2x 2 + 8x + 5 comparing the two formulas I can see that a = 2, b = 8 and c = 5. Then use the formula b 2a and plug in the values for a and b. 8 2(2) = 8 4 = -2. So the x value of the vertex is -2. Then I plug that into the original equation y = 2x 2 + 8x + 5 y = 2( 2) 2 + 8( 2) + 5 = -3 so the y value of the vertex is -3. So the vertex of the equation y = 2x 2 + 8x + 5 is (-2, -3) Sample Questions: 55. What is the vertex of the parabola y = 3x 2 + 9x + 2? 56. What is the vertex of the parabola y = x 2 + 6x 5

16 16 Since the vertex is the most important point on the parabola there is another form for writing an equation of a parabola called the vertex form. It looks like this: y = a(x h) 2 + k. Where (h,k) is the vertex. For example to find the vertex of the parabola described by the equation y = 2(x 1) we can compare it to the vertex form of a parabola. y = a(x h) 2 + k y = 2(x 1) from this I can see that a = -2, h = 1, and k = 2. Since the vertex is just (h,k) the vertex of this parabola must be the point (1, 2). Done. Sample Questions: 57. What is the vertex of the parabola y = 4(x + 1) 2 + 3? 58. What is the vertex of the parabola y = 1(x 3) 2 + 7? 59. What is the vertex of the parabola y = 2(x 2) 2 + 5? 60. What is the vertex of the parabola y = 3(x + 4) 2 2? This form has the added benefit of showing any transformations performed on the parabola. The "a" shows if the parabola has been stretched or reflected. The "h" indicated any horizontal shift and the "k" shows any vertical shift. For example the equation y = (x + 3) 2 shows that the "h" value is -3. That means that the graph is shifted to the left 3 units. The equation y = x 2 shows that the "a" value is -1. A negative value means that the graph is reflected around the x-axis (it is upside down). The equation y = 2x 2 shows that the "a" value is 2. This means that the y-coordinates of the parabola have been multiplied by 2 and creates a vertical stretch. The equation y = x 2 2 shows that the "k" value is -2. This means that the parabola has been shifted down 2 units.

17 17 Sample Questions: 61. Figure 1 below shows the graph of y = x 2 in the standard (x, y) coordinate plane. Which of the following is the equation for the graph in Figure 2? a. y = (x 4) 2 1 b. y = (x 4) c. y = (x + 1) 2 4 d. y = (x + 4) 2 1 e. y = (x + 4) The table below gives the values of f(x) for selected values of x in the function f(x) = (x + 4) 2 1, where x and y are both real numbers. For the equation above, which of the following gives the greatest value for f(x)? (Hint: is this a line or a parabola? Graph the points and see.) A. -4 B. -5 C. -6 D. -7 E A basketball team made breakfast for the school community as a fundraiser. The team president used a parabolic function to predict the amount of profit the team would make depending on what price they charged for breakfast. The prices at which profit would be $0 are $3.25 and $7.75. Which statement, according to the model, is true? A. There are not any prices at which the team will lose money. B. The maximum profit occurs when the ticket price is approximately $5.50. C. A ticket price of $3.25 yields a profit of $400. D. The maximum profit occurs when the ticket price is $7.75. E. Points G and H represent the points at which the team will make the maximum profit. Notice that in question 63, the points where the graph crosses the x-axis are called zeros on the graph. The highest point on the graph is called the maximum. These points: the zeros and maximum, are very important and useful. Here they indicate where the profit is zero and where the profit is maximum.

18 64. A parabola with vertex (-3, -2) and axis of symmetry y = -2 crosses the y-axis at (0, ). At what other point does the parabola cross the y-axis? (Hint: graph the parabola and see which, if any, point described comes closest). A. No other point B. (0, ) C. (0, 2-3 3) D. (0, ) E. Cannot be determined from the given information Which of the following is an equation of the parabola graphed in the following (x,y) coordinate plane? (Hint: would it help to try a few x values like, 3 or -3, and solve for y and see which equation works?) A. y = x2 3-3 B. y = x2 3 3 C. y = x D. y = x E. y = 3x The equation y = x 2 is graphed in the standard (x, y) coordinate plane, then reflected across the x-axis. Which of the following is the equation of this reflection? (Hint: would it help to graph the parabola and its reflection?) A. y = x 2 B. y = -x 2 C. y = ( x) 2 D. y = x E. y = x 67. The graph of y 2 = x is shown in the standard (x, y) coordinate plane below for values of x such that 0 x 4. The e-coordinates of points D and E are both 4. What is the area of DEO, in square coordinate units? A. 5 2 B. 4 C. 8 D. 12 E. 16

19 ELLIPSES The equation for an ellipse looks like this: x2 + y2 a 2 b2 = 1. Where a is the distance from the center to farthest horitonal point on the ellipse and b is the distance form the center to the farthest vertical point on the ellipse. For example, the following figure shows the graph of x2 + y is the major axis and QS is the minor axis. = 1 which can also be written x2 + y = 1. PR 19 Sample Questions: 68. What is the equation of the ellipse graphed in the standard (x, y) coordinate plane below? A. x 2 + y = 1 B. x = 1 C. x = 1 D. x = 1 E. x = 1

20 20 MINIMUMS, MAXIMUMS AND ZEROS A minimum is the lowest point on a graph and a maximum is the highest point on a graph. In a parabola the minimum or maximum will be the y value of the vertex. The zeros are the x values where the graph crosses the x-axis (or the x-intercepts). 69. What is the minimum of the parabola in the following graph? 70. What are the zeros of the graph? 71. What is the maximum of the parabola in the following graph? 72. What are the zeros of the graph? 73. What is the minimum of the parabola in the following graph? 74. What are the zeros of the graph? 75. Referring to the above question, what does the maximum mean? 76. Referring to the above question, what do the zeros mean?

21 Answers or or y = 1 2 x y = 1 2 x 11. y = x y = 2x yes 14. no 15. no 16. no 17. E 18. B 19. A 20. y = 3 2 x y = 4 3 x y = 1 2 x B ( 24 11, ) see graph below 29. see graph below 30. A 31. B 32. (7, -6) (7, 2) 39. all real numbers except x = all real numbers except x = 0, 2, A 42. B, D 43. B 44. (2, -4) 45. (0, 0) 46. (-1, -3) 47. D B 51. C π π 54. D 55. ( 3 2, 19 4 ) 56. (-3, -14) 57. (-1, 3) 58. (3, 7) 59. (2, 5) 60. (-4, -2) 61. K 62. E 63. B 64. D 65. A 66. B 67. C 68. A , , ,4 75. maximum profit 76. profit is zero