Chapter Review
Chapter Review Things to Know Formulas Distance formula (p. 5)
d = 41x2 - x122 + 1y2 - y122 1x, y2 = ¢
Midpoint formula (p. 7) Slope (p. 27) Parallel lines (p. 37) Perpendicular lines (p. 38)
x1 + x2 y1 + y2 , ≤ 2 2 y2 - y1 m = , if x1 Z x2 ; undefined if x1 = x2 x2 - x1 Equal slopes 1m1 = m22 with different y-intercepts Product of slopes is -1 1m1 # m2 = -12
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CHAPTER 1
Graphs
Equations x = a
Vertical line (p. 31)
y - y1 = m1x - x12; m is the slope of the line, 1x1 , y12 is a point on the line
Point–slope form of the equation of a line (p. 32)
y = b y = mx + b; m is the slope of the line, b is the y-intercept
Horizontal line (p. 33) Slope–intercept form of the equation of a line (p. 34) General form of the equation of a line (p. 35) Standard form of the equation of a circle (p. 45) Equation of the unit circle (p. 45)
1x - h22 + 1y - k22 = r2; r is the radius of the circle, 1h, k2 is the center of the circle x2 + y2 = 1
General form of the equation of a circle (p. 48)
x2 + y2 + ax + by + c = 0
Ax + By = C; A, B not both 0
Objectives Section 1.1 1.2
1 ✓ 2 ✓ 1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓ 6 ✓ 7 ✓
✓ 1 ✓ 1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓ 6 ✓ 7 ✓ 8 ✓ 9 ✓ 10 ✓ 1 ✓ 2 ✓ 3 ✓ 8
1.3 1.4
1.5
You should be able to
Review Exercises
Á
Use the distance formula (p. 4) Use the midpoint formula (p. 7) Graph equations by hand by plotting points (p. 11) Graph equations using a graphing utility (p. 13) Use a graphing utility to create tables (p. 14) Find intercepts from a graph (p. 15) Find intercepts from an equation (p. 16) Use a graphing utility to approximate intercepts (p. 16) Test an equation for symmetry with respect to the x-axis, the y-axis, and the origin (p. 17) Know how to graph key equations (p. 19) Solve equations in one variable using a graphing utility (p. 24) Calculate and interpret the slope of a line (p. 27) Graph lines given a point and the slope (p. 30) Find the equation of a vertical line (p. 31) Use the point–slope form of a line; identify horizontal lines (p. 32) Find the equation of a line given two points (p. 33) Write the equation of a line in slope–intercept form (p. 34) Identify the slope and y-intercept of a line from its equation (p. 34) Graph lines written in general form using intercepts (p. 35) Find equations of parallel lines (p. 36) Find equations of perpendicular lines (p. 38) Write the standard form of the equation of a circle (p. 44) Graph a circle by hand and by using a graphing utility (p. 46) Work with the general form of the equation of a circle (p. 47)
1(a)–6(a), 51, 52(a), 54–56 1(b)–6(b), 55 9–14 8 8 7 9–14, 47–50 8 15–22 23, 24 25–28 1(c)–6(c); 1(d)–6(d), 53 57 31 29, 30 32–34 29–38 39–42 9, 10 35, 36 37, 38 43–46 47–50 47–50, 55
Review Exercises In Problems 1–6, find the following for each pair of points: (a) The distance between the points. (b) The midpoint of the line segment connecting the points. (c) The slope of the line containing the points. (d) Then interpret the slope found in part (c). 1. 10, 02; 14, 22
2. 10, 02; 1-4, 62
3. 11, -12; 1-2, 32
4. 1-2, 22; 11, 42
5. 14, -42; 14, 82
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6. 1-3, 42; 12, 42
Chapter Review
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8. Graph y = -x2 + 15 using a graphing utility. Create a table of values to determine a good initial viewing window. Use a graphing utility to approximate the intercepts.
7. List the intercepts of the following graph. y
2 4
4
x
2
In Problems 9–14, determine the intercepts and graph each equation by hand by plotting points. Verify your results using a graphing utility. Label the intercepts on the graph. 9. 2x - 3y = 6 10. x + 2y = 4 11. y = x2 - 9 12. y = x2 + 4 13. x2 + 2y = 16 14. 2x2 - 4y = 24 In Problems 15–22, test each equation for symmetry with respect to the x-axis, the y-axis, and the origin. 15. 2x = 3y2 16. y = 5x 17. x2 + 4y2 = 16 4 2 3 19. y = x + 2x + 1 20. y = x - x 21. x2 + x + y2 + 2y = 0
18. 9x2 - y2 = 9 22. x2 + 4x + y2 - 2y = 0
24. Sketch a graph of y = 1x.
23. Sketch a graph of y = x3.
In Problems 25–28, use a graphing utility to approximate the solutions of each equation rounded to two decimal places. All solutions lie between -10 and 10. 25. x3 - 5x + 3 = 0 26. -x3 + 3x + 1 = 0 27. x4 - 3 = 2x + 1 28. -x4 + 7 = x2 - 2 In Problems 29–38, find an equation of the line having the given characteristics. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer. Graph the line. 29. Slope = -2; containing the point 13, -12 30. Slope = 0; containing the point 1-5, 42 31. Slope undefined; containing the point 1 -3, 42 32. x-intercept = 2; containing the point 14, -52 33. y-intercept = -2; containing the point 15, -32 34. Containing the points 13, -42 and 12, 12 35. Parallel to the line 2x - 3y = -4; containing the point 1-5, 32 36. Parallel to the line x + y = 2; containing the point 11, -32 37. Perpendicular to the line x + y = 2; containing the point 14, -32 38. Perpendicular to the line 3x - y = -4; containing the point 1-2, 42 In Problems 39–42, find the slope and y-intercept of each line. 1 5 1 2 42. - x + y = 2 x + y = 10 2 2 3 2 In Problems 43–46, find the standard form of the equation of the circle whose center and radius are given. 43. 1h, k2 = 1-2, 32; r = 4 44. 1h, k2 = 13, 42; r = 4 45. 1h, k2 = 1-1, -22; r = 1 46. 1h, k2 = 12, -42; r = 3 39. 4x + 6y = 36
40. 7x - 3y = 42
41.
In Problems 47–50, find the center and radius of each circle. Graph each circle by hand. Determine the intercepts of the graph of each circle. 47. x2 + y2 - 2x + 4y - 4 = 0 48. x2 + y2 + 4x - 4y - 1 = 0
49. 3x2 + 3y2 - 6x + 12y = 0 50. 2x2 + 2y2 - 4x = 0
51. Show that the points A = 13, 42, B = 11, 12, C = 1-2, 32 are the vertices of an isosceles triangle.
and
53. Show that the points A = 12, 52, B = 16, 12, C = 18, -12 lie on a straight line by using slopes.
and
56. Find two numbers y such that the distance from 1-3, 22 to 15, y2 is 10. 2 57. Graph the line with slope containing the point 11, 22. 3 58. Create four problems that you might be asked to do given the two points 1-3, 42 and 16, 12. Each problem should involve a different concept. Be sure that your directions are clearly stated. 59. Describe each of the following graphs in the xy-plane. Give justification. (a) x = 0 (b) y = 0 (c) x + y = 0 (d) xy = 0 (e) x2 + y2 = 0
52. Show that the points A = 1-2, 02, B = 1-4, 42, and C = 18, 52 are the vertices of a right triangle in two ways: (a) By using the converse of the Pythagorean Theorem (b) By using the slopes of the lines joining the vertices
54. Show that the points A = 11, 52, B = 12, 42, and C = 1-3, 52 lie on a circle with center 1-1, 22. What is the radius of this circle?
55. The endpoints of the diameter of a circle are 1-3, 22 and 15, -62. Find the center and radius of the circle. Write the general equation of this circle.
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