Review how to find the distance between two points To find the distance between two points, use the Pythagorean theorem. The difference between is one leg and the difference between and is the other leg. and If you prefer, rather than work through the Pythagorean theorem, you may memorize the distance formula which is just the points plugged into the Pythagorean theorem. 1. What is the distance between the points (-1,6) and (5,-2)? A. 68 B. 10 C. 8 D. 2 E. 2 2. How many units apart are the points P (1, -2) and Q (-2, 2) in the standard (x,y) coordinate plane? 3. Point M (-1, 3) and point N (6, 5) are points on the coordinate plane. What is the length of the segment MN? 4. 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? 1
Review using two points to find the slope slope = m = = = 5. What is the slope of the line that contains the points J (2,3) and K (0,-2)? 6. What is the slope of the line that contains the points H (-9, 1) and G (2, -7)? Review slope-intercept form y = mx + b (where m represents the slope and b represents the y-intercept) 7. What is the slope of the line y = X + 3? 8. At what point does the line y = X - 1 cross the y axis? Review finding slope and intercept from linear equations - writing equations in slope-intercept form To find the slope or y-intercept of a line that is not in slope-intercept form, re-write the equation in slope-intercept form so it looks like y = mx + b. 9. At which y-coordinate does the line described by the equation 6y - 3x = 18 intersect the y-axis? 10. What is the slope of the line described by the equation y - 2x = 8? Review how to create an equation for a line from 2 points 2
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. Step 1: find m (slope) We already know how to find the slope from 2 points. slope = m = = = 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. 11. What is the equation of the line that passes through points A (-8,3) and B (0,-1)? 12. What is the equation of the line that passes through the points (0,0) and (6,3)? 13. What is the equation of the line that passes through the points (4,5) and (-1, 0)? Review Parallel lines on the coordinate plane 3
Parallel lines always have the same slope. Therefore, if two lines have the same slope then they are parallel. 14. Write the equation of a line which is parallel to the line y = 2x + 3 and passes through the point (0, -1). 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, the slope of the other is. For example, in the image above, the slope of one line is -4. The slope of the other line is negative reciprocals of each other. These lines are perpendicular. -4 and are 4
15. These 2 lines are perpendicular: y = 2x + 3 y = x - 4 True False 16. These 2 lines are perpendicular: y = x + 3 y = x True False 17. These 2 lines are perpendicular: y = x + 5 y = x - 3 True False 18. These 2 lines are perpendicular: y = x + 3 y = x - 2 True False 19. These 2 lines are perpendicular: x - 2y = 3 2y = x - 7 True False 5
20. These 2 lines are perpendicular: 3y + x = 6 3y = 9x +9 True False 21. 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 = x + 2 B. y = x + 3 C. y = 3x + 3 D. y = 2x + 6 E. y = -3x + 3 22. 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 = x - 7 D. y = x + 5 E. y = x + 5 23. Write the equation of a line which is perpendicular to the line y = x + 3 and passes through the point (0, -1) 6
24. 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 = x + 6. Which of the following equations describes line m? (Hint: to 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 = x - 2 B. y = x + 2 C. y = x - 2 D. y = x +2 E. y = - x - 2 25. 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 = x + 4. What is the equation that describes line q? 26. 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 = x - 2. What is the equation that describes line z? 27. 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)? 28. 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. D. 4 E. 8 7
29. 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? 30. 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? 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 = = 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: For example: find the midpoint between ( 1, 2) and (3, 6). = = (1, -2) 31. In the standard (x,y) coordinate plane, points P and Q have coordinates (2,3) and (12,-15), respectively. What are the coordinate of the midpoint of PQ? 32. Aubrey decides to graph her office and the nearest smoothie shop in the standard (x,y) plane. If her office is at point (-1, -5) and the smoothie shop is at point (3,3), what are the coordinates of the point exactly halfway between those of her office and the shop? (You may assume Aubrey is able to walk a straight line between them). 8
33. In the standard (x,y) coordinate plane, the endpoints of line segment EF lie at (-5, 9) and (3,-3). What is the y-coordinate of the midpoint of line segment EF? (Hint: they only asked for the y coordinate) 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 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). 36. 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? Beyond lines - circles and parabolas 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 and equations of parabolas. 9
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 + = 25. Since the equation of a circle: + = we can compare this to the given equation. + = + = 25 I notice that h (or the x value of the center of the circle) must be 2 since I see I notice that k (or the y value of the center of the circle) must be -1 since I see. 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. = I also notice that the radius must be 5 since = 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. 10
37. The equation of a circle in the standard (x,y) coordinate plane is given by the equation + = 5. What is the center of the circle? A. (, ) B. (-5, 5) C. (, ) D. (5, -5) E. (5, 5) 38. The equation of a circle in the standard (x,y) coordinate plane is given by the equation + = 16. What is the center of the circle? 39. The equation of a circle in the standard (x,y) coordinate plane is given by the equation + = 12. What is the center of the circle? 40. The equation of a circle in the standard (x,y) coordinate plane is given by the equation + = 49. What is the center of the circle? 41. The equation of a circle in the standard (x,y) coordinate plane is given by the equation + = 20. What is the radius of the circle? A. 20 B. 8 C. 9 D. 2 E. 5 11
42. The equation of a circle in the standard (x,y) coordinate plane is given by the equation + = 36. What is the radius of the circle? 43. The equation of a circle in the standard (x,y) coordinate plane is given by the equation + = 12. What is the radius of the circle? 44. A circle with the equation + = 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) 45. In the standard (x,y) coordinate plane, what is the area of the circle + = 25? A. 5 B. 10 C. 25 D. 125 E. 225 46. In the standard (x,y) coordinate plane, what is the area of the circle + = 36? 47. In the standard (x,y) coordinate plane, what is the circumference of the circle + = 81? 12
Equation for a parabola The graph of an equation in the form 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:, then by plugging the x value into the equation, we can find the y value of the vertex. For example to find the vertex of the parabola we compare it to the formula comparing the two formulas I can see that a = 2, b = 8 and c = 5. Then use the formula and plug in the values for a and b. = = -2. So the x value of the vertex is -2. Then I plug that into the original equation = -3 so the y value of the vertex is -3. So the vertex of the equation is (-2, -3) 48. What is the vertex of the parabola? 49. What is the vertex of the parabola 13
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: Where (h,k) is the vertex. For example to find the vertex of the parabola described by the equation we can compare it to the vertex form of a parabola. vertex form of a parabola 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. 50. What is the vertex of the parabola? 51. What is the vertex of the parabola? 52. What is the vertex of the parabola? 53. Graph the parabola. (Hint: find the vertex, then select a few values for x and find the corresponding y value. Graph enough points until you know the basic shape of the parabola.) 14
Answers: 1. B 2. 5 3. or 7.28 4. or 10.296 5. 6. 7. 8. -1 9. 3 10. 2 11. y = - x - 1 12. y = x 13. y = x + 1 14. y = 2x - 1 15. True 16. True 17. False 18. False 19. False 20. True 21. E 22. B 23. y = x - 1 24. A 25. y = x - 6 28. B 29. 1 30. 31. (7, -6) 32. (1, -1) 33. 3 34. or 2.5 35. 14 36. 5 37. B 38. (2, -4) 39. (0, 0) 40. (-1, -3) 41. D 42. 6 43. 2 or 3.46 44. B 45. C 46. 36 47. 18 48. (-1, -1) 49. (-3, -14) 50. (4, 5) 51. (-1, 3) 52. (3, 7) 53. 26. y = x + 9 27. y = x - 3 15