{"id":6646,"date":"2020-10-03T15:44:49","date_gmt":"2020-10-03T15:44:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6646"},"modified":"2026-02-05T07:38:04","modified_gmt":"2026-02-05T07:38:04","slug":"3-3-graphing-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-3-graphing-linear-functions\/","title":{"raw":"3.3: Graphing Linear Functions","rendered":"3.3: Graphing Linear Functions"},"content":{"raw":"
\r\n

section 3.3 Learning Objectives<\/h3>\r\n3.3: Graphing Linear Functions<\/strong>\r\n
    \r\n \t
  • Identify if an ordered pair is a solution to a linear equation<\/li>\r\n \t
  • Graph a linear equation by plotting points<\/li>\r\n \t
  • Identify the x-intercept and y-intercept of a linear equation<\/li>\r\n \t
  • Use the intercepts to graph a linear equation<\/li>\r\n \t
  • Graph horizontal and vertical lines<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

    Identifying ordered pairs as solutions to linear equations<\/h2>\r\nA line is a visual representation of a linear equation, and the line itself is made up of an infinite number of points (or ordered pairs). The picture below shows the line of the linear equation [latex]y=2x\u20135[\/latex]\u00a0with some of the specific points on the line.\r\n\r\n\"Line\r\n\r\nEvery point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. You can try plugging in any of the points that are labeled, like the ordered pair, [latex](1,\u22123)[\/latex].\r\n

    [latex]\\begin{array}{l}\\,\\,\\,\\,y=2x-5\\\\-3=2\\left(1\\right)-5\\\\-3=2-5\\\\-3=-3\\\\\\text{This is true.}\\end{array}[\/latex]<\/p>\r\nYou can also try ANY of the other points on the line. Every point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. All this means is that determining whether an ordered pair is a solution of an equation is pretty straightforward. If the ordered pair is on the line created by the linear equation, then it is a solution to the equation. But if the ordered pair is not on the line\u2014no matter how close it may look\u2014then it is not a solution to the equation.\r\n

    \r\n

    Identifying Solutions<\/h3>\r\nGraphing a linear equations (using techniques learned later in this and upcoming sections) produces a line.\u00a0 Every point on the line is a solution to the linear equation.\u00a0 The line continues forever in both directions and has an infinite number of solutions.\r\n\r\nTo find out whether a specific ordered pair (x,y)<\/em>\u00a0is a solution of a linear equation, you can do the following:\r\n
      \r\n \t
    • Substitute the x<\/em>\u00a0and\u00a0y\u00a0<\/em>values into the equation. If the equation yields a true statement, then the ordered pair is a solution of the linear equation. If the ordered pair does not yield a true statement then it is not a solution.<\/li>\r\n<\/ul>\r\n<\/div>\r\n

      <\/h3>\r\n
      \r\n

      Example 1<\/h3>\r\nDetermine whether [latex](\u22122,4)[\/latex] is a solution to the equation [latex]4y+5x=3[\/latex].\r\n\r\n[reveal-answer q=\"980260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"980260\"]For this problem, you will use the substitution method. Substitute [latex]x=\u22122[\/latex]\u00a0and [latex]y=4[\/latex]\u00a0into the equation.\r\n

      [latex]\\begin{array}{r}4y+5x=3\\\\4\\left(4\\right)+5\\left(\u22122\\right)=3\\end{array}[\/latex]<\/p>\r\nEvaluate.\r\n

      [latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\r\nThe statement is not true, so [latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].\r\n

      Answer<\/h4>\r\n[latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\n

      <\/h3>\r\nhttps:\/\/youtu.be\/9aWGxt7OnB8\r\n

      Graphing lines using ordered pairs<\/span><\/h2>\r\nGraphing ordered pairs helps us make sense of all kinds of mathematical relationships.\r\n
      \r\n\r\nYou can use a coordinate plane<\/strong><\/b> to plot points and to map various relationships, such as the relationship between an object\u2019s distance and the elapsed time. Many mathematical relationships are linear relationships<\/strong><\/b>. Let\u2019s look at what a linear relationship is.\r\n\r\n<\/div>\r\nA linear relationship is a relationship between variables such that when plotted on a coordinate plane, the points lie on a line. Let\u2019s start by looking at a series of points in Quadrant I on the coordinate plane.\r\n\r\nLook at the five ordered pairs<\/strong><\/b> (and their x<\/em>- and y<\/em>-coordinates) below. Do you see any pattern to the location of the points? If this pattern continued, what other points could be included?\r\n

      (0,0) , (1,2) , (2,4) , (3,6) , (4, 8) , (?, ?)<\/p>\r\nYou may have identified that if this pattern continued the next ordered pair would be at (5, 10).\u00a0 Applying the same logic, you may identify that the ordered pairs (6, 12) and (7, 14) would also belong (if this coordinate plane were larger).\r\n\r\nThese series of points can also be represented in a table. In the table below, the x-<\/em>\u00a0and y<\/em>-coordinates of each ordered pair on the graph is recorded.\r\n

      \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      x<\/em><\/strong><\/b>-coordinate<\/strong><\/b><\/td>\r\ny<\/em><\/strong><\/b>-coordinate<\/strong><\/b><\/td>\r\n<\/tr>\r\n
      0<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
      1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
      2<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
      3<\/td>\r\n6<\/td>\r\n<\/tr>\r\n
      4<\/td>\r\n8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that each y<\/em>-coordinate is twice the corresponding x<\/em>-value. All of these x-<\/em>\u00a0and y<\/em>-values follow the same pattern, and, when placed on a coordinate plane, they all line up!\r\n\r\n\"Graph\r\n\r\nOnce you know the pattern that relates the x-<\/em> and y-<\/em>values, you can find a y<\/em>-value for any x<\/em>-value that lies on the line. So if the rule of this pattern is that each y<\/em>-value is twice<\/em> the corresponding x<\/em>-value, then the ordered pairs (1.5, 3), (2.5, 5), and (3.5, 7) should all appear on the line too, correct? Look to see what happens.\r\n\r\n\"Coordinate\r\n\r\nIf you were to keep adding ordered pairs (x<\/em>, y<\/em>) where the y<\/em>-value was twice the x<\/em>-value, you would end up with a graph like this.\r\n\r\n\"A\r\n\r\nLook at how all of the points blend together to create a line. You can think of a line, then, as a collection of an infinite number of individual points that share the same mathematical relationship. In this case, the relationship is that the y<\/em>-value is twice the x<\/em>-value.\r\n\r\nThere are multiple ways to represent a linear relationship\u2014a table, a linear graph, and there is also a linear equation<\/strong><\/b>. A linear equation is an equation with two variables whose ordered pairs graph as a straight line.\r\n\r\nThere are several ways to create a graph from a linear equation. One way is to create a table of values for x<\/em> and y<\/em>, and then plot these ordered pairs on the coordinate plane. Two points are enough to determine a line. However, it\u2019s always a good idea to plot more than two points to avoid possible errors.\r\n\r\nThen you draw a line through the points to show all of the points that are on the line. The line continues endlessly in both directions. Every point on this line is a solution to the linear equation.\r\n
      \r\n

      Example 2<\/h3>\r\nGraph the linear equation [latex]y=\u2212\\frac{3}{2}x[\/latex].\r\n\r\n[reveal-answer q=\"983342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"983342\"]Evaluate [latex]y=\u2212\\frac{3}{2}x[\/latex]\u00a0for different values of x<\/em>, and create a table of corresponding x<\/em> and y<\/em> values.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      x<\/em> values<\/strong><\/td>\r\n[latex]\u2212\\frac{3}{2}x[\/latex]<\/strong><\/td>\r\ny<\/em> values<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]0[\/latex]<\/td>\r\n[latex]\u2212\\frac{3}{2}(0)[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]2[\/latex]<\/td>\r\n[latex]\u2212\\frac{3}{2}(2)[\/latex]<\/td>\r\n[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]4[\/latex]<\/td>\r\n[latex]\u2212\\frac{3}{2}(4)[\/latex]<\/td>\r\n[latex]\u22126[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]6[\/latex]<\/td>\r\n[latex]\u2212\\frac{3}{2}(6)[\/latex]<\/td>\r\n[latex]\u22129[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince the coefficient of x<\/em> is [latex]\u2212\\frac{3}{2}[\/latex], it is convenient to choose multiples of 2 for x<\/em>. This ensures that y<\/em> is an integer, and makes the line easier to graph.\r\n\r\nConvert the table to ordered pairs. Then plot the ordered pairs.\r\n

      [latex](0,0)[\/latex]<\/p>\r\n

      [latex](2,\u22123)[\/latex]<\/p>\r\n

      [latex](4,\u22126)[\/latex]<\/p>\r\n

      [latex](6,\u22129)[\/latex]<\/p>\r\nDraw a line through the points to indicate all of the points on the line.\r\n

      Answer\r\n\"A[\/hidden-answer]<\/h4>\r\n<\/div>\r\n

      <\/h3>\r\nhttps:\/\/youtu.be\/f5yvGPEWpvE\r\n
      \r\n

      Example 3<\/h3>\r\nGraph the linear equation [latex]y=2x+3[\/latex].\r\n\r\n[reveal-answer q=\"834421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834421\"]Evaluate [latex]y=2x+3[\/latex]\u00a0for different values of x<\/em>, and create a table of corresponding x<\/em> and y<\/em> values.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      x<\/em> values<\/strong><\/td>\r\n[latex]2x+3[\/latex]<\/strong><\/td>\r\ny<\/em> values<\/strong><\/td>\r\n<\/tr>\r\n
      0<\/td>\r\n\u00a0[latex]2(0) + 3[\/latex]<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
      1<\/td>\r\n[latex]2(1) + 3[\/latex]<\/td>\r\n5<\/td>\r\n<\/tr>\r\n
      2<\/td>\r\n[latex]2(2) + 3[\/latex]<\/td>\r\n7<\/td>\r\n<\/tr>\r\n
      3<\/td>\r\n[latex]2(3) + 3[\/latex]<\/td>\r\n9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConvert the table to ordered pairs.\u00a0Plot the ordered pairs.\r\n

      [latex](0, 3)[\/latex]<\/p>\r\n

      [latex](1, 5)[\/latex]<\/p>\r\n

      [latex](2, 7)[\/latex]<\/p>\r\n

      [latex](3, 9)[\/latex]<\/p>\r\n \r\n\r\n\"Graph\r\n\r\nDraw a line through the points to indicate all of the points on the line.\r\n

      Answer<\/h4>\r\n\"Line[\/hidden-answer]\r\n\r\n<\/div>\r\n

      Identify the intercepts of a linear equation<\/h2>\r\nThe intercepts of a line are the points where the line intersects, or crosses, the horizontal and vertical axes. To help you remember what \u201cintercept\u201d means, think about the word \u201cintersect.\u201d The two words sound alike and in this case mean the same thing.\r\n\r\nThe straight line on the graph below intersects the two coordinate axes. The point where the line crosses the x<\/i>-axis is called the x<\/i>-intercept<\/b>. The y<\/i>-intercept<\/b> is the point where the line crosses the y<\/i>-axis.\r\n\r\n\"A\r\n\r\nThe x<\/i>-intercept above is the point [latex](\u22122,0)[\/latex]. The y<\/i>-intercept above is the point (0, 2).\r\n\r\nNotice that the y<\/i>-intercept always occurs where [latex]x=0[\/latex], and the x<\/i>-intercept always occurs where [latex]y=0[\/latex].\r\n\r\nTo find the x<\/em>- and y<\/em>-intercepts of a linear equation, you can substitute 0 for y<\/i> and for x,<\/i> respectively.\r\n\r\nFor example, the linear equation [latex]3y+2x=6[\/latex]\u00a0has an x<\/i> intercept when [latex]y=0[\/latex], so [latex]3\\left(0\\right)+2x=6\\\\[\/latex].\r\n

      [latex]\\begin{array}{r}2x=6\\\\x=3\\end{array}[\/latex]<\/p>\r\nThe x<\/em>-intercept is [latex](3,0)[\/latex].\r\n\r\nLikewise the y<\/i>-intercept occurs when [latex]x=0[\/latex].\r\n

      [latex]\\begin{array}{r}3y+2\\left(0\\right)=6\\\\3y=6\\\\y=2\\end{array}[\/latex]<\/p>\r\nThe y<\/i>-intercept is [latex](0,2)[\/latex].\r\n

      Use the intercepts to graph a linear equation<\/h2>\r\nYou can use intercepts to graph some linear equations. Once you have found the two intercepts, draw a line through them.\u00a0 This method is often used when the equation is written in Standard Form.\r\n

      Standard Form of a Line<\/h3>\r\nOne way that we can represent the equation of a line is in standard form<\/strong>. Standard form is given as\r\n

      [latex]Ax+By=C[\/latex]<\/p>\r\nwhere [latex]A[\/latex], [latex]B[\/latex], and [latex]C[\/latex] are integers. The x\u00a0<\/em>and y\u00a0<\/em>terms are on one side of the equal sign, and the constant term is on the other side.\r\n\r\nLet\u2019s use the intercepts to graph the equation [latex]3y+2x=6[\/latex]. You figured out that the intercepts of the line this equation represents are [latex](0,2)[\/latex] and [latex](3,0)[\/latex]. That\u2019s all you need to know.\r\n\r\n\"A\r\n\r\n \r\n

      \r\n

      Example 4<\/h3>\r\nGraph [latex]3x+5y=30[\/latex]\u00a0using the x<\/em> and y<\/em>-intercepts.\r\n\r\n[reveal-answer q=\"153435\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"153435\"]When an equation is in [latex]Ax+By=C[\/latex]\u00a0form, you can easily find the x<\/i>- and y<\/i>-intercepts and then graph.\r\n

      [latex]\\begin{array}{r}3x+5y=30\\\\3\\left(0\\right)+5y=30\\\\0+5y=30\\\\5y=30\\\\y=\\,\\,\\,6\\\\y\\text{-intercept}\\,\\left(0,6\\right)\\end{array}[\/latex]<\/p>\r\nTo find the y<\/i>-intercept, set [latex]x=0[\/latex]\u00a0and solve for y<\/i>.\r\n

      [latex]\\begin{array}{r}3x+5y=30\\\\3x+5\\left(0\\right)=30\\\\3x+0=30\\\\3x=30\\\\x=10\\\\x\\text{-intercept}\\left(10,0\\right)\\end{array}[\/latex]<\/p>\r\nTo find the x<\/i>-intercept, set [latex]y=0[\/latex] and solve for x<\/i>.\r\n

      Answer<\/h4>\r\n\"Decreasing\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\r\nAs we saw in the previous example, when given an equation in the form [latex]Ax+By=C[\/latex], it is often easy to find the x and y-intercepts. Note that because of the communitive property of addition,\u00a0[latex]Ax+By=C[\/latex] is equivalent to [latex]By+Ax=C[\/latex].\u00a0<\/strong>In the example below you will see an example where the equation is in the form\u00a0[latex]By+Ax=C[\/latex].<\/strong>\r\n\r\nhttps:\/\/youtu.be\/k8r-q_T6UFk\r\n\r\nIn the example below, the equation wasn't given in the form\u00a0[latex]Ax + By=C[\/latex],\u00a0<\/strong>but we can still find the x and y-intercepts in the form it is in.\r\n
      \r\n

      Example 5<\/h3>\r\nGraph [latex]y=2x-4[\/latex] using the x<\/em> and y<\/em>-intercepts.\r\n\r\n[reveal-answer q=\"476848\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"476848\"]First, find the y<\/em>-intercept. Set x<\/em> equal to zero and solve for y<\/em>.\r\n

      [latex]\\begin{array}{l}y=2x-4\\\\y=2\\left(0\\right)-4\\\\y=0-4\\\\y=-4\\\\y\\text{-intercept}\\left(0,-4\\right)\\end{array}[\/latex]<\/p>\r\nTo find the x<\/i>-intercept, set [latex]y=0[\/latex]\u00a0and solve for x<\/i>.\r\n

      [latex]\\begin{array}{l}y=2x-4\\\\0=2x-4\\\\4=2x\\\\x=2\\\\x\\text{-intercept}\\left(2,0\\right)\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n\"Line\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe mentioned that we can graph\u00a0some<\/em> linear equations using intercepts.\u00a0 So, what is the exception.\u00a0 In the next example, there is only one intercept; yet, we need two points to construct a graph!\r\n
      \r\n

      Example 6<\/h3>\r\nFind the [latex]x[\/latex] and [latex]y[\/latex]-intercepts and sketch the graph.\r\n\r\n[latex]2x-y=0[\/latex]\r\n\r\n[reveal-answer q=\"252153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"252153\"]\r\n\r\nFirst we find the [latex]y[\/latex]-intercept by plugging in [latex]x=0[\/latex].\r\n

      [latex]2(0)-y=0[\/latex]<\/p>\r\n

      [latex]\\frac{-y}{-1}=\\frac{0}{-1}[\/latex]<\/p>\r\n

      [latex]y=0[\/latex]<\/p>\r\nTherefore, the [latex]y[\/latex]-intercept is the origin, [latex](0,0)[\/latex]\r\n\r\nNext, we find the [latex]x[\/latex]-intercept by plugging in [latex]y=0[\/latex].\r\n

      [latex]2x-0=0[\/latex]<\/p>\r\n

      [latex]\\frac{2x}{2}=\\frac{0}{2}[\/latex]<\/p>\r\n

      [latex]x=0[\/latex]<\/p>\r\nHence, the [latex]x[\/latex]-intercept is also [latex](0,0)[\/latex].\r\n\r\nBut since this only produces one point on the graph, we will be unable to graph it using intercepts alone.\u00a0 Instead, we can revert back to plugging in a different value for [latex]x[\/latex] (or for [latex]y[\/latex]) to find a second solution.\r\n\r\nSuppose we plug in [latex]x=1[\/latex]. We get\r\n

      [latex]2(1)-y=0[\/latex]<\/p>\r\n

      [latex]2-y=0[\/latex]<\/p>\r\n

      [latex]\\underline{-2\\hspace{.35in}-2}\\hspace{.1in}[\/latex]<\/p>\r\n

      [latex]\\frac{-y}{-1}=\\frac{-2}{-1}[\/latex]<\/p>\r\n

      [latex]y=2[\/latex]<\/p>\r\nSo, a second point on the graph is [latex](1,2)[\/latex].\u00a0 Of course, we can continue to find more solutions if we want, but two will be sufficient to produce the graph shown below.\r\n

      \"Increasing<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      Solve for y<\/em>, then graph a linear equation<\/h2>\r\nOften times, it is easier to make a table of values if the equation is in the form [latex]y=mx+b[\/latex] where m<\/em> and b<\/em> are real numbers. But to take advantage of this, we often must first solve for [latex]y[\/latex] in the equation.\r\n
      \r\n

      Example 7<\/h3>\r\nSolve for [latex]y[\/latex] and graph using a table of values.\r\n\r\n[latex]3x+y=5[\/latex].\r\n\r\n[reveal-answer q=\"61530\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"61530\"]\r\n\r\nFirst, isolate [latex]y[\/latex] by subtracting [latex]3x[\/latex] on both sides of the equation.\r\n

      [latex]3x+y=5[\/latex]<\/p>\r\n

      [latex]\\hspace{.02in}\\underline{-3x\\hspace{.35in}-3x}[\/latex]<\/p>\r\n

      [latex]y=-3x+5[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      x <\/em>values<\/strong><\/td>\r\n[latex]\u20133x+5[\/latex]<\/strong><\/td>\r\ny<\/em> values<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]0[\/latex]<\/td>\r\n[latex]\u20133(0)+5[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]1[\/latex]<\/td>\r\n[latex]\u20133(1)+5[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]2[\/latex]<\/td>\r\n[latex]\u20133(2)+5[\/latex]<\/td>\r\n[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]3[\/latex]<\/td>\r\n[latex]\u20133(3)+5[\/latex]<\/td>\r\n[latex]\u22124[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the ordered pairs (shown below).\r\n

      [latex](0,5)[\/latex]<\/p>\r\n

      [latex](1,2)[\/latex]<\/p>\r\n

      [latex](2,\u22121)[\/latex]<\/p>\r\n

      [latex](3,\u22124)[\/latex]<\/p>\r\n\"Graph\r\n\r\nDraw a line through the points to indicate all of the points on the line.\r\n

      Answer<\/h4>\r\n\"A[\/hidden-answer]\r\n\r\n<\/div>\r\nHowever, we must first sometimes solve for [latex]y[\/latex] ourselves to put it into this form, as demonstrated in the next example.\r\n
      \r\n

      Example 8<\/h3>\r\nSolve for [latex]y[\/latex], then graph the equation using a table of values.\r\n\r\n[latex]4x-3y=3[\/latex]\r\n\r\n[reveal-answer q=\"789355\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"789355\"]\r\n\r\nWe first isolate [latex]y[\/latex] in the equation.\r\n

      [latex]4x-3y=3[\/latex]<\/p>\r\n

      [latex]\\underline{-4x\\hspace{.42in}-4x}[\/latex]<\/p>\r\n

      [latex]\\hspace{.75in}-3y=-4x+3[\/latex]<\/p>\r\n

      [latex]\\frac{-3y}{-3}=\\frac{-4x}{-3}+\\frac{3}{-3}[\/latex]<\/p>\r\n

      [latex]y=\\frac{4}{3}x-1[\/latex]<\/p>\r\nAs we select values for [latex]x[\/latex], we note that it would be wise to choose multiples of 3 in order to produce integer answers.\u00a0 We will use -3, 0, and 3.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      x <\/em>values<\/strong><\/td>\r\n[latex]\\frac{4}{3}x-1[\/latex]<\/strong><\/td>\r\ny<\/em> values<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]-3[\/latex]<\/td>\r\n[latex]\\frac{4}{3}(-3)-1=-4-1=-5[\/latex]<\/td>\r\n[latex]-5[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]0[\/latex]<\/td>\r\n[latex]\\frac{4}{3}(0)-1=0-1=-1[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]3[\/latex]<\/td>\r\n[latex]\\frac{4}{3}(3)-1=4-1=3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlotting the points leads to the graph below.\r\n

      \"Increasing<\/p>\r\n\r\n

      Why not use intercepts?<\/span><\/h4>\r\nYou may have noticed that the original equation was in standard form.\u00a0 We noted earlier that it is quick to find intercepts when dealing with this form.\u00a0 In Example 6, we discussed one potential problem with graphing using intercepts (when there is only one intercept at the origin).\u00a0 This example reveals another potential problem you may encounter.\r\n\r\nThe [latex]y[\/latex]-intercept turns out nice here.\u00a0 If we plug in [latex]x=0[\/latex], we get\r\n

      [latex]4(0)-3y=3[\/latex]<\/p>\r\n

      [latex]\\frac{-3y}{-3}=\\frac{3}{-3}[\/latex]<\/p>\r\n

      [latex]y=-1[\/latex]<\/p>\r\nSo, the [latex]y[\/latex]-intercept is [latex](0,-1)[\/latex].\r\n\r\nHowever, if we plug in [latex]y=0[\/latex] to find the [latex]x[\/latex]-intercept, we get\r\n

      [latex]4x-3(0)=3[\/latex]<\/p>\r\n

      [latex]\\frac{4x}{4}=\\frac{3}{4}[\/latex]<\/p>\r\n

      [latex]x=\\frac{3}{4}[\/latex]<\/p>\r\nTherefore, the [latex]x[\/latex]-intercept is [latex]\\left(\\frac{3}{4},0\\right)[\/latex], which include a fractional value.\u00a0 While there is nothing inherently wrong with this, it can be extremely difficult to accurately plot fraction on a graph, especially if you are using an online mathematics program.\u00a0 So, it can be extremely helpful to have alternative techniques.\u00a0 By solving for [latex]y[\/latex] and carefully selecting [latex]x[\/latex]-values to plug in, we were able to avoid having to plot any fractions on our graph.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides another example of solving for [latex]y[\/latex] and then graphing using a table of values.\r\n\r\nhttps:\/\/youtu.be\/6yL3gfPbOt8\r\n

      Graph horizontal and vertical lines<\/h2>\r\nThe linear equations [latex]x=2[\/latex]\u00a0and [latex]y=\u22123[\/latex]\u00a0only have one variable in each of them. However, because these are linear equations, then should graph on a coordinate plane as lines just as the linear equations above do. Just think of the equation [latex]x=2[\/latex]\u00a0as [latex]x=0y+2[\/latex]\u00a0and think of [latex]y=\u22123[\/latex]\u00a0as [latex]y=0x\u20133[\/latex].\r\n
      \r\n

      Example 9<\/h3>\r\nGraph [latex]y=\u22123[\/latex].\r\n\r\n[reveal-answer q=\"140758\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"140758\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      x<\/em> values<\/strong><\/td>\r\n[latex]0x\u20133[\/latex]<\/strong><\/td>\r\ny<\/em> values<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]0[\/latex]<\/td>\r\n[latex]0(0)\u20133[\/latex]<\/td>\r\n[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]1[\/latex]<\/td>\r\n[latex]0(1)\u20133[\/latex]<\/td>\r\n[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]2[\/latex]<\/td>\r\n[latex]0(2)\u20133[\/latex]<\/td>\r\n[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]3[\/latex]<\/td>\r\n[latex]0(3)\u20133[\/latex]<\/td>\r\n[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWrite [latex]y=\u22123[\/latex]\u00a0as [latex]y=0x\u20133[\/latex], and evaluate y<\/em> when x<\/em> has several values. Or just realize that [latex]y=\u22123[\/latex]\u00a0means every y-<\/em>value will be [latex]\u22123[\/latex], no matter what x<\/em> is.\r\n

      [latex](0,\u22123)[\/latex]<\/p>\r\n

      [latex](1,\u22123)[\/latex]<\/p>\r\n

      [latex](2,\u22123)[\/latex]<\/p>\r\n

      [latex](3,\u22123)[\/latex]<\/p>\r\nPlot the ordered pairs (shown below).\r\n\r\n\"Coordinate\r\n\r\nDraw a line through the points to indicate all of the points on the line.\r\n

      Answer<\/h4>\r\n\"ANotice that [latex]y=\u22123[\/latex]\u00a0graphs as a horizontal line.[\/hidden-answer]\r\n\r\n<\/div>\r\nIt is worth noting that there was nothing particularly special about the [latex]-3[\/latex] in the above example.\u00a0 What you want to take from this example is that if you encounter an equation of the form [latex]y=constant[\/latex], regardless of the constant, it will always result in a horizontal line.\r\n\r\nWe can take a similar approach with equations of the form [latex]x=constant[\/latex].\r\n
      \r\n

      Example 10<\/h3>\r\nGraph [latex]x=2[\/latex].\r\n\r\n[reveal-answer q=\"70161\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"70161\"]\r\n\r\nOne difference with this example compared to all others we have seen up until now is that we cannot plug in values for [latex]x[\/latex].\u00a0 This is because we are told [latex]x[\/latex] is\u00a0always<\/em> 2.\u00a0 However, we have always had the option of plugging in values for [latex]y[\/latex] instead, and in this case, we must go that route.\r\n\r\nAs mentioned earlier, we could view this as [latex]x=0y+2[\/latex].\u00a0 However, applying what we learned from the previous example, this means that no matter what values we plug in for [latex]y[\/latex], we will always get [latex]x=2[\/latex].\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      y<\/em> values<\/strong><\/td>\r\nx<\/i> values<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]0[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]2[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]3[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBeing careful with the order in our ordered pairs, we can now plot the points [latex](2,0)[\/latex],\u00a0[latex](2,1)[\/latex],\u00a0[latex](2,2)[\/latex], and\u00a0[latex](2,3)[\/latex] to obtain the graph below.\r\n

      \"Vertical<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nOnce again, there was nothing special about the particular constant.\u00a0 We conclude that linear equations of the form [latex]x=constant[\/latex] will always result in a vertical line.\r\n

      \r\n

      HORIZONTAL AND VERTICAL LINES<\/h3>\r\n
        \r\n \t
      • Horizontal Lines:\u00a0 Any linear equation of the form [latex]y=a[\/latex], where [latex]a[\/latex] is any real number, is a horizontal line that crosses the [latex]y[\/latex]-axis at the point [latex](0,a)[\/latex]<\/li>\r\n \t
      • Vertical Lines:\u00a0\u00a0Any linear equation of the form [latex]x=b[\/latex], where [latex]b[\/latex] is any real number, is a vertical line that crosses the [latex]x[\/latex]-axis at the point [latex](b,0)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n

        In the following video you will see more examples of graphing horizontal and vertical lines.<\/p>\r\nhttps:\/\/youtu.be\/2A2fhImjOBc\r\n

        Using function notation to express equations of lines<\/h2>\r\nBecause all non-vertical lines are functions, we often express the equation of a line using function notation.\u00a0Recall from section 3.2, function notation\u00a0<\/span>can written as [latex] f(x) = [\/latex]<\/span>. This is read\u00a0<\/span>as \u201cf of x\u201d. It is important to note that [latex] f(x)[\/latex]<\/span>\u00a0does not mean [latex] f [\/latex]\u00a0<\/span>times [latex] x [\/latex]\u00a0<\/span>\u00a0<\/span>but\u00a0<\/span>is merely a notation indicating a function using the variable [latex] x [\/latex]\u00a0. <\/span>A function is not always <\/span>indicated by [latex] f [\/latex]\u00a0<\/span>\u00a0<\/span>but can be any letter, often [latex] g [\/latex]\u00a0<\/span>\u00a0<\/span>or <\/span>[latex] h [\/latex]\u00a0<\/span>. In linear equations the dependent variable\u00a0[latex] y [\/latex] is replaced with an\u00a0 [latex] f(x) [\/latex], as seen in the example below.\u00a0<\/span>\r\n
        \r\n

        Example 11<\/h3>\r\nGraph the line\u00a0 [latex] f(x)= \\frac{1}{2}x -3 [\/latex]<\/span>\r\n\r\n[reveal-answer q=\"205687\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"205687\"]\r\n\r\nLet's make a table of values where we evaluate the function for different values for x.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        x <\/em>values<\/strong><\/td>\r\n[latex] f(x)= \\frac{1}{2}x -3 [\/latex]<\/span><\/td>\r\n[latex]f(x)[\/latex]values<\/strong><\/td>\r\n<\/tr>\r\n
        [latex]0[\/latex]<\/td>\r\n[latex] f(0)= \\frac{1}{2}(0) -3 [\/latex]<\/span><\/td>\r\n[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]1[\/latex]<\/td>\r\n[latex] f(1)= \\frac{1}{2}(1) -3 [\/latex]<\/span><\/td>\r\n[latex]-2\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]2[\/latex]<\/td>\r\n[latex] f(2)= \\frac{1}{2}(2) -3 [\/latex]<\/span><\/td>\r\n[latex]\u22122[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]3[\/latex]<\/td>\r\n[latex] f(3)= \\frac{1}{2}(3) -3 [\/latex]<\/span><\/td>\r\n[latex]\u22121\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf we plot these points and connect them, we will have the graph of our line. Notice the x-intercept is not showing on this graph, but we were still able to graph the line based on other points we found in our table of values.\r\n\r\n\"Linear\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

        Summary<\/h2>\r\nA solution to a linear equation is an ordered pair [latex](x,y)[\/latex] that, when substituted into the equation, results in a true statement.\u00a0 Any linear equation in two variables has an infinite number of solutions.\u00a0 Graphing these solution results in a line.\r\n\r\nWe can graph lines by finding several solutions, often organized in a table of values.\u00a0 By convention, we typically plug in values for [latex]x[\/latex], which can be made easier by first solving for [latex]y[\/latex], expressing the equation in the form [latex]y=mx+b[\/latex].\r\n\r\nThe intercepts of a graph are the points at which the line crosses the axes.\u00a0 To find the [latex]y[\/latex]-intercept, plug in [latex]x=0[\/latex] and to find the [latex]x[\/latex]-intercept, plug in [latex]y=0[\/latex].\u00a0 These intercepts are often quick to determine if the equation is in the standard form, [latex]Ax+By=C[\/latex].\u00a0 If this results in two distinct intercepts, we can also use these to help us graph the line.\r\n\r\nThere are two special cases to watch out for, where one of the variables is missing from the equation.\u00a0 If an equation is of the form [latex]y=constant[\/latex], it will correspond to a horizontal line.\u00a0 If an equation is of the form [latex]x=constant[\/latex], it will correspond to a vertical line.","rendered":"
        \n

        section 3.3 Learning Objectives<\/h3>\n

        3.3: Graphing Linear Functions<\/strong><\/p>\n

          \n
        • Identify if an ordered pair is a solution to a linear equation<\/li>\n
        • Graph a linear equation by plotting points<\/li>\n
        • Identify the x-intercept and y-intercept of a linear equation<\/li>\n
        • Use the intercepts to graph a linear equation<\/li>\n
        • Graph horizontal and vertical lines<\/li>\n<\/ul>\n<\/div>\n

           <\/p>\n

          Identifying ordered pairs as solutions to linear equations<\/h2>\n

          A line is a visual representation of a linear equation, and the line itself is made up of an infinite number of points (or ordered pairs). The picture below shows the line of the linear equation [latex]y=2x\u20135[\/latex]\u00a0with some of the specific points on the line.<\/p>\n

          \"Line<\/p>\n

          Every point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. You can try plugging in any of the points that are labeled, like the ordered pair, [latex](1,\u22123)[\/latex].<\/p>\n

          [latex]\\begin{array}{l}\\,\\,\\,\\,y=2x-5\\\\-3=2\\left(1\\right)-5\\\\-3=2-5\\\\-3=-3\\\\\\text{This is true.}\\end{array}[\/latex]<\/p>\n

          You can also try ANY of the other points on the line. Every point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. All this means is that determining whether an ordered pair is a solution of an equation is pretty straightforward. If the ordered pair is on the line created by the linear equation, then it is a solution to the equation. But if the ordered pair is not on the line\u2014no matter how close it may look\u2014then it is not a solution to the equation.<\/p>\n

          \n

          Identifying Solutions<\/h3>\n

          Graphing a linear equations (using techniques learned later in this and upcoming sections) produces a line.\u00a0 Every point on the line is a solution to the linear equation.\u00a0 The line continues forever in both directions and has an infinite number of solutions.<\/p>\n

          To find out whether a specific ordered pair (x,y)<\/em>\u00a0is a solution of a linear equation, you can do the following:<\/p>\n

            \n
          • Substitute the x<\/em>\u00a0and\u00a0y\u00a0<\/em>values into the equation. If the equation yields a true statement, then the ordered pair is a solution of the linear equation. If the ordered pair does not yield a true statement then it is not a solution.<\/li>\n<\/ul>\n<\/div>\n

            <\/h3>\n
            \n

            Example 1<\/h3>\n

            Determine whether [latex](\u22122,4)[\/latex] is a solution to the equation [latex]4y+5x=3[\/latex].<\/p>\n

            Show Solution<\/span><\/p>\n
            For this problem, you will use the substitution method. Substitute [latex]x=\u22122[\/latex]\u00a0and [latex]y=4[\/latex]\u00a0into the equation.<\/p>\n

            [latex]\\begin{array}{r}4y+5x=3\\\\4\\left(4\\right)+5\\left(\u22122\\right)=3\\end{array}[\/latex]<\/p>\n

            Evaluate.<\/p>\n

            [latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\n

            The statement is not true, so [latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].<\/p>\n

            Answer<\/h4>\n

            [latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n

            <\/h3>\n