fundamentals of 2D and 3D graphs

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fundamentals of 2D and 3D graphs

  1. 1. GRAPHS
  2. 2. 2D and 3Dcoordinate systems
  3. 3. • The Cartesian coordinate system is the most commonly used coordinate system. In two dimensions, this system consists of a pair of lines on a flat surface or plane, that intersect at right angles.
  4. 4. • The lines are called axes and the point at which they intersect is called the origin. The axes are usually drawn horizontally and vertically and are referred to as the x- and y-axes, respectively.
  5. 5. • The Cartesian coordinate system is the most commonly used coordinate system. In two dimensions, this system consists of a pair of lines on a flat surface or plane, that intersect at right angles.
  6. 6. • The lines are called axes and the point at which they intersect is called the origin. The axes are usually drawn horizontally and vertically and are referred to as the x- and y-axes, respectively.
  7. 7. • A point in the plane with coordinates (a, b) is a units to the right of the y axis and b units up from the x axis if a and b are positive numbers.
  8. 8. • If a and b are both negative numbers, the point is a units to the left of the y axis and b units down from the x axis. In the figure above point P1 has coordinates (3, 4), and point P2 has coordinates (-1, -3).
  9. 9. 3D COORDINATE SYSTEM
  10. 10. • In a 3D Cartesian coordinate system, a point P is referred to by three real numbers (coordinates), indicating the positions of the perpendicular projections from the point to three fixed, perpendicular, graduated lines, called the axes which intersect at the origin.
  11. 11. • Often the x-axis is imagined to be horizontal and pointing roughly toward the viewer (out of the page), the y-axis is also horizontal and pointing to the right, and the z-axis is vertical, pointing up.
  12. 12. • The system is called right-handed if it can be rotated so that the three axes are in the position as shown in the figure above. The x-coordinate of of the point P in the figure is a, the y-coordinate is b, and the z-coordinate is c.
  13. 13. Coordinate Systems: Right Hand RulePlace your fingers in the direction of the positive x-axis androtate them in the direction of the y-axis. Your thumb willpoint in the direction of the positive z-axis.
  14. 14. Left or Right-Handed?The systems are right-handed (positive). X Z Y Y Z XThese systems are left-handed (negative). Z X Y Y X Z
  15. 15. Coordinates in 2 Dimensions y (3,2) 2 3 x
  16. 16. The 3rd Dimension z (3,2,4) 4 y (3,2,0) x
  17. 17. Consider the Point A(5,4,2) C B D G E F
  18. 18. A(2,4,0)B(0,4,3)C(2,4,3)D(2,0,3)
  19. 19. B(3,0,0)C(3,4,0)Q(3,0,2)R(3,4,2) (0,2,0) (0,4,1) (1½,4,0) (1½,2,0) (1½,2,2) (1½,4,1)
  20. 20. Centre of Box? (1½,2,1)
  21. 21. Graphical MethodWhen two quantities are so related that achange in one produces a correspondingchange in other, the relation between themcan well be shown by means of a graphicalmethod. The two quantities are said to bevariables.
  22. 22. Variables…• Ex: y=3x-5, y=2x2-6x+10• When x is given a value y will have a definite corresponding value. X and y are called the variables.
  23. 23. Axes of reference• In a suitably chosen position two lines are drawn; one horizontally OX, and one vertically OY, meeting at the point O. the position of the point O is determined by the values of the variables. How to establish this position will be shown later.• These two lines OX,OY, at right angles are called the “Axes of reference” or usually “The axes.”
  24. 24. • The point O is called the “origin of axes” or “the origin.”• The horizontal axis OX is the axis along which x values are plotted and is called the axes of abscissa.
  25. 25. • The vertical axis OY is the axis along which y values are plotted and is called the axis of ordinates.• Along OX the axis from O is divided into equal parts, each part being equal to the same number of x units. Similarly the axis OY is divided into y units.
  26. 26. COORDINATE PLANE Y-axis Parts of a plane 1. X-axis 2. Y-axis2nd QUADRANT 1ST QUADRANT 3. Origin 4. Quadrants I-IV Origin ( 0 , 0 ) X-axis3rd QUADRANT 4th QUADRANT
  27. 27. PLOTTING POINTS Remember when plotting points you always start at the origin. B Next you go left (if x-coordinate C is negative) or right (if x- coordinate is positive. Then you go up (if y-coordinate is positive) or down (if y- coordinate is negative) AD Plot these 4 points A (3, -4), B(5, 6), C (-4, 5) and D (-7, -5)
  28. 28. Example 1.Plot the points A (3, -4), B(5, 6),C (-4, 5) and D (-7, -5) on theCartesian plane.
  29. 29. SLOPESlope is the ratio of the vertical rise to the horizontalrun between any two points on a line. Usuallyreferred to as the rise over run. Run is 6 Slope triangle between because we two points. Notice that the went to the slope triangle can beRise is 10 right drawn two different ways.because we Rise is -10went up because we went down 10 5 The slope in this case is Run is -6 6 3 because we went to the left 10 5 The slope in this case is 6 3 Another way to find slope
  30. 30. FORMULA FOR FINDING SLOPE The formula is used when you know two points of a line.They look like A( X 1 , Y1 ) and B( X 2 , Y2 ) RISE X 2 X1 X1 X 2SLOPE RUN Y2 Y1 Y1 Y2 EXAMPLE
  31. 31. Find the slope of the line between the two points (-4, 8) and (10, -4) If it helps label the points. X 1 Y1 X2 Y2 Then use the formula X 2 X1 (10 ) ( 4) Y2 Y1 SUBSTITUTE INTO FORMULA ( 4) (8) (10 ) ( 4) 10 4 14 7 Then Simplify ( 4) (8) 4 ( 8) 12 6
  32. 32. X AND Y INTERCEPTSThe x-intercept is the x-coordinate of a pointwhere the graph crosses the x-axis.The y-intercept is the y-coordinate of a pointwhere the graph crosses the y-axis. The x-intercept would be 4 and is located at the point (4, 0). The y-intercept is 3 and is located at the point (0, 3).
  33. 33. SLOPE-INTERCEPT FORM OF A LINE The slope intercept form of a line is y = mx + b, where “m” represents the slope of the line and “b” represents the y-intercept. When an equation is in slope-intercept form the “y” is always on one side by itself. It can not be more than one y either. If a line is not in slope-intercept form, then we must solve for “y” to get it there. Examples
  34. 34. IN SLOPE-INTERCEPT NOT IN SLOPE-INTERCEPT y = 3x – 5 y – x = 10 y = -2x + 10 2y – 8 = 6x y = -.5x – 2 y + 4 = 2xPut y – x = 10 into slope-intercept form Add x to both sides and would get y = x + 10Put 2y – 8 = 6x into slope-intercept form. Add 8 to both sides then divide by 2 and would get y = 3x + 4Put y + 4 = 2x into slope-intercept form. Subtract 4 from both sides and would get y = 2x – 4.
  35. 35. GRAPHING LINES BY MAKING A TABLE OR USING THE SLOPE-INTERCEPT FORM I could refer to the table method by input-output table or x-y table. For now I want you to include three values in your table. A negative number, zero, and a positive number. Graph y = 3x + 2 INPUT (X) OUTPUT (Y) -2 -4 0 2 1 5By making a table it gives me three points, in this case (-2, -4) (0, 2) and (1, 5) to plotand draw the line. See the graph.
  36. 36. Plot (-2, -4), (0, 2) and (1, 5)Then draw the line. Make sure yourline covers the graph and hasarrows on both ends. Be sure touse a ruler. Slope-intercept graphing
  37. 37. Slope-intercept graphingSteps1. Make sure the equation is in slope-intercept form.2. Identify the slope and y-intercept.3. Plot the y-intercept.4. From the y-intercept use the slope to get another point to draw the line. 1. y = 3x + 2 2. Slope = 3 (note that this means the fraction or rise over run could be (3/1) or (-3/-1). The y-intercept is 2. 3. Plot (0, 2) 4. From the y-intercept, we are going rise 3 and run 1 since the slope was 3/1.
  38. 38. FIND EQUATION OF A LINE GIVEN 2 POINTS Find the equation of the line between (2, 5) and (-2, -3).1. Find the slope between the two points. 1. Slope is 2.2. Plug in the slope in the slope- 2. y = 2x + b intercept form. 3. Picked (2, 5) so3. Pick one of the given points and plug (5) = 2(2) + b in numbers for x and y. 4. b = 14. Solve and find b. 5. y = 2x + 15. Rewrite final form. Two other ways
  39. 39. Steps if given the slope and If given a graph there are three ways.a point on the line.1. Substitute the slope into One way is to find two points on the slope-intercept the line and use the first method we talked about. form.2. Use the point to plug in Another would be to find the for x and y. slope and pick a point and use the second method.3. Find b.4. Rewrite equation. The third method would be to find the slope and y-intercept and plug it directly into y = mx + b.
  40. 40. Exercise: Plot the following points1.(5,6);(4,2)2.(-1,2);(3,0)3.(-3,-4);(-3,-1)4.(8,-3);(3,-8)5.(0,2);(3,-4)6.(1,-6);(-5,2)7.(5,7);(-3,-8)8.(-4,-5);(-4,6)9.(-1,-6);(3,3)10.(3,-2);(-2,-3)
  41. 41. DRAWING A CURVEGiven a series of values of x and thecorresponding values of y, the relationbetween x and y can be shown by plottingthe given points and then drawing theircurve. The curve is found by joining up thepoints, using the “smoothest” curve whichwill pass through all the points.
  42. 42. When, however, the points are gainedfrom experimental data, say, the curveobtained by joining all the points wouldconsist of irregular angles and sharpbends. In this case the rule is to draw thesmoothest curve which is the bestapproximation to that which would passthrough all the points.
  43. 43. • Some points will be on the curve, some above it and some below it. The curve can then be used to find the error in those values lying off it.
  44. 44. Example:• The following table gives values of x corresponding the values of y. x -3 -2 -1 0 1 2 3 y 9 4.2 1 0 0.7 3.9 9
  45. 45. 10 -3, 9 9 3, 9 8 7 6 5Y -2, 4.2 4 2, 3.9 3 2 -1, 1 1 1, 0.7 0 0, 0-4 -3 -2 -1 0 1 2 3 4 X
  46. 46. • Here the values of x are evenly distributed on either side of zero. Hence the

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