{"id":341,"date":"2015-09-18T22:38:43","date_gmt":"2015-09-18T22:38:43","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=341"},"modified":"2015-11-04T21:52:38","modified_gmt":"2015-11-04T21:52:38","slug":"graphing-equations-by-plotting-points","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/graphing-equations-by-plotting-points\/","title":{"raw":"Graphing Equations by Plotting Points","rendered":"Graphing Equations by Plotting Points"},"content":{"raw":"We can plot a set of points to represent an equation. When such an equation contains both an x <\/em>variable and a y <\/em>variable, it is called an equation in two variables<\/strong>. Its graph is called a graph in two variables<\/strong>. Any graph on a two-dimensional plane is a graph in two variables.\r\n\r\nSuppose we want to graph the equation [latex]y=2x - 1[\/latex]. We can begin by substituting a value for x<\/em> into the equation and determining the resulting value of y<\/em>. Each pair of x<\/em>- and y<\/em>-values is an ordered pair that can be plotted. The table below\u00a0lists values of x<\/em> from \u20133 to 3 and the resulting values for y<\/em>.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/td>\r\n[latex]y=2x - 1[\/latex]<\/td>\r\n[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]-3[\/latex]<\/td>\r\n[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\r\n[latex]\\left(-3,-7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]-2[\/latex]<\/td>\r\n[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\r\n[latex]\\left(-2,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]-1[\/latex]<\/td>\r\n[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\r\n[latex]\\left(-1,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0[\/latex]<\/td>\r\n[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\r\n[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1[\/latex]<\/td>\r\n[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\r\n[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2[\/latex]<\/td>\r\n[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\r\n[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3[\/latex]<\/td>\r\n[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\r\n[latex]\\left(3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can plot the points in the table. The points for this particular equation form a line, so we can connect them.\u00a0<\/strong>This is not true for all equations.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"This Figure 6<\/b>[\/caption]\r\n\r\nNote that the x-<\/em>values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.\r\n
\r\n

How To: Given an equation, graph by plotting points.<\/h3>\r\n
    \r\n\t
  1. Make a table with one column labeled x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\r\n\t
  2. Enter x-<\/em>values down the first column using positive and negative values. Selecting the x-<\/em>values in numerical order will make the graphing simpler.<\/li>\r\n\t
  3. Select x-<\/em>values that will yield y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\r\n\t
  4. Plot the ordered pairs.<\/li>\r\n\t
  5. Connect the points if they form a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Example 2: Graphing an Equation in Two Variables by Plotting Points<\/h3>\r\nGraph the equation [latex]y=-x+2[\/latex] by plotting points.\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nFirst, we construct a table similar to the one below. Choose x<\/em> values and calculate y.<\/em>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    [latex]x[\/latex]<\/td>\r\n[latex]y=-x+2[\/latex]<\/td>\r\n[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]-5[\/latex]<\/td>\r\n[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\r\n[latex]\\left(-5,7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]-3[\/latex]<\/td>\r\n[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\r\n[latex]\\left(-3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]-1[\/latex]<\/td>\r\n[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\r\n[latex]\\left(-1,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]0[\/latex]<\/td>\r\n[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\r\n[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]1[\/latex]<\/td>\r\n[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\r\n[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]3[\/latex]<\/td>\r\n[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\r\n[latex]\\left(3,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]5[\/latex]<\/td>\r\n[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\r\n[latex]\\left(5,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow, plot the points. Connect them if they form a line.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"This Figure 7<\/b>[\/caption]\r\n\r\n<\/div>\r\n
    \r\n\r\nConstruct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].\r\n\r\n<\/div>\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    [latex]x[\/latex]<\/td>\r\n[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\r\n[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]-2[\/latex]<\/td>\r\n[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\r\n[latex]\\left(-2,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]-1[\/latex]<\/td>\r\n[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\r\n[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]0[\/latex]<\/td>\r\n[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\r\n[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]1[\/latex]<\/td>\r\n[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\r\n[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]2[\/latex]<\/td>\r\n[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\r\n[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"This Figure 8<\/b>[\/caption]\r\n\r\n<\/div>","rendered":"

    We can plot a set of points to represent an equation. When such an equation contains both an x <\/em>variable and a y <\/em>variable, it is called an equation in two variables<\/strong>. Its graph is called a graph in two variables<\/strong>. Any graph on a two-dimensional plane is a graph in two variables.<\/p>\n

    Suppose we want to graph the equation [latex]y=2x - 1[\/latex]. We can begin by substituting a value for x<\/em> into the equation and determining the resulting value of y<\/em>. Each pair of x<\/em>– and y<\/em>-values is an ordered pair that can be plotted. The table below\u00a0lists values of x<\/em> from \u20133 to 3 and the resulting values for y<\/em>.<\/p>\n\n\n\n\n\n\n\n\n\n\n
    [latex]x[\/latex]<\/td>\n[latex]y=2x - 1[\/latex]<\/td>\n[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n
    [latex]-3[\/latex]<\/td>\n[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\n[latex]\\left(-3,-7\\right)[\/latex]<\/td>\n<\/tr>\n
    [latex]-2[\/latex]<\/td>\n[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\n[latex]\\left(-2,-5\\right)[\/latex]<\/td>\n<\/tr>\n
    [latex]-1[\/latex]<\/td>\n[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\n[latex]\\left(-1,-3\\right)[\/latex]<\/td>\n<\/tr>\n
    [latex]0[\/latex]<\/td>\n[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\n[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n
    [latex]1[\/latex]<\/td>\n[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\n[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n
    [latex]2[\/latex]<\/td>\n[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\n[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n
    [latex]3[\/latex]<\/td>\n[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\n[latex]\\left(3,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    We can plot the points in the table. The points for this particular equation form a line, so we can connect them.\u00a0<\/strong>This is not true for all equations.<\/p>\n

    \"This<\/p>\n

    Figure 6<\/b><\/p>\n<\/div>\n

    Note that the x-<\/em>values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.<\/p>\n

    \n

    How To: Given an equation, graph by plotting points.<\/h3>\n
      \n
    1. Make a table with one column labeled x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\n
    2. Enter x-<\/em>values down the first column using positive and negative values. Selecting the x-<\/em>values in numerical order will make the graphing simpler.<\/li>\n
    3. Select x-<\/em>values that will yield y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\n
    4. Plot the ordered pairs.<\/li>\n
    5. Connect the points if they form a line.<\/li>\n<\/ol>\n<\/div>\n
      \n

      Example 2: Graphing an Equation in Two Variables by Plotting Points<\/h3>\n

      Graph the equation [latex]y=-x+2[\/latex] by plotting points.<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      First, we construct a table similar to the one below. Choose x<\/em> values and calculate y.<\/em><\/p>\n\n\n\n\n\n\n\n\n\n\n
      [latex]x[\/latex]<\/td>\n[latex]y=-x+2[\/latex]<\/td>\n[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]-5[\/latex]<\/td>\n[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\n[latex]\\left(-5,7\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]-3[\/latex]<\/td>\n[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\n[latex]\\left(-3,5\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]-1[\/latex]<\/td>\n[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\n[latex]\\left(-1,3\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]0[\/latex]<\/td>\n[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\n[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]1[\/latex]<\/td>\n[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\n[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]3[\/latex]<\/td>\n[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\n[latex]\\left(3,-1\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]5[\/latex]<\/td>\n[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\n[latex]\\left(5,-3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Now, plot the points. Connect them if they form a line.<\/p>\n

      \"This<\/p>\n

      Figure 7<\/b><\/p>\n<\/div>\n<\/div>\n

      \n

      Construct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].<\/p>\n<\/div>\n

      \n\n\n\n\n\n\n\n\n
      [latex]x[\/latex]<\/td>\n[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\n[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]-2[\/latex]<\/td>\n[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\n[latex]\\left(-2,1\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]-1[\/latex]<\/td>\n[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\n[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]0[\/latex]<\/td>\n[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\n[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]1[\/latex]<\/td>\n[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\n[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\n<\/tr>\n
      [latex]2[\/latex]<\/td>\n[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\n[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
      \"This<\/p>\n

      Figure 8<\/b><\/p>\n<\/div>\n<\/div>\n\n\t\t\t

      \n\t\t\t

      Candela Citations<\/h3>\n\t\t\t\t\t
      \n\t\t\t\t\t\t
      \n\t\t\t\t\t\t\t
      CC licensed content, Specific attribution<\/div>