{"id":983,"date":"2024-05-01T19:54:56","date_gmt":"2024-05-01T19:54:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=983"},"modified":"2025-08-21T23:09:01","modified_gmt":"2025-08-21T23:09:01","slug":"graphing-and-analyzing-linear-equations-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphing-and-analyzing-linear-equations-learn-it-2\/","title":{"raw":"Graphing and Analyzing Linear Equations: Learn It 2","rendered":"Graphing and Analyzing Linear Equations: Learn It 2"},"content":{"raw":"

Graphing Equations by Plotting Points<\/h2>\r\nWe 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\n
How to: Given an equation, graph by plotting points.<\/strong>\r\n
    \r\n \t
  1. Make a table with one column labeled\u00a0[latex]x[\/latex], a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\r\n \t
  2. Enter\u00a0[latex]x[\/latex]-<\/em>values down the first column using positive and negative values. Selecting the\u00a0[latex]x[\/latex]-<\/em>values in numerical order will make the graphing simpler.<\/li>\r\n \t
  3. Select\u00a0[latex]x[\/latex]-<\/em>values that will yield\u00a0[latex]y[\/latex]-<\/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<\/section>
    Graph the equation [latex]y=2x - 1[\/latex].<\/strong>We can begin by substituting a value for [latex]x[\/latex] into the equation and determining the resulting value of [latex]y[\/latex]. Each pair of [latex]x[\/latex] and [latex]y[\/latex]-values is an ordered pair that can be plotted. The table below\u00a0lists values of [latex]x[\/latex] from [latex]\u20133[\/latex] to [latex]3[\/latex] and the resulting values for [latex]y[\/latex].\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 these points from the table. The points for this particular equation form a line, so we can connect them. This is not true for all equations.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"497\"]\"This x,y coordinate plane with a line passing through given points[\/caption]\r\n\r\n<\/section>
    Note that the\u00a0x-<\/em>values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of\u00a0x<\/em>\u00a0to 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.<\/section>
    Graph the equation [latex]y=-x+2[\/latex].[reveal-answer q=\"891289\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"891289\"]First, 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=\"641\"]\"This x,y coordinate plane with a line passing through given points[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    [ohm2_question hide_question_numbers=1]18916[\/ohm2_question]<\/section>","rendered":"

    Graphing Equations by Plotting Points<\/h2>\n

    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

    How to: Given an equation, graph by plotting points.<\/strong><\/p>\n
      \n
    1. Make a table with one column labeled\u00a0[latex]x[\/latex], a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\n
    2. Enter\u00a0[latex]x[\/latex]–<\/em>values down the first column using positive and negative values. Selecting the\u00a0[latex]x[\/latex]–<\/em>values in numerical order will make the graphing simpler.<\/li>\n
    3. Select\u00a0[latex]x[\/latex]–<\/em>values that will yield\u00a0[latex]y[\/latex]–<\/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<\/section>\n
      Graph the equation [latex]y=2x - 1[\/latex].<\/strong>We can begin by substituting a value for [latex]x[\/latex] into the equation and determining the resulting value of [latex]y[\/latex]. Each pair of [latex]x[\/latex] and [latex]y[\/latex]-values is an ordered pair that can be plotted. The table below\u00a0lists values of [latex]x[\/latex] from [latex]\u20133[\/latex] to [latex]3[\/latex] and the resulting values for [latex]y[\/latex].<\/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 these points from the table. The points for this particular equation form a line, so we can connect them. This is not true for all equations.<\/p>\n

      \"This
      x,y coordinate plane with a line passing through given points<\/figcaption><\/figure>\n<\/section>\n
      Note that the\u00a0x-<\/em>values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of\u00a0x<\/em>\u00a0to 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.<\/section>\n
      Graph the equation [latex]y=-x+2[\/latex].<\/p>\n