{"id":772,"date":"2025-07-15T16:08:23","date_gmt":"2025-07-15T16:08:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=772"},"modified":"2026-03-05T16:12:46","modified_gmt":"2026-03-05T16:12:46","slug":"linear-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions-learn-it-4\/","title":{"raw":"Linear Functions: Learn It 4","rendered":"Linear Functions: Learn It 4"},"content":{"raw":"

Graphing a Linear Function from Point-Slope Form<\/h2>\r\nAnother way to graph a linear function is by using point-slope form<\/strong>. This form is useful when you know the slope and a single point on the line.\r\n\r\n
The point-slope form of a linear function is [latex]y - y_1 = m(x - x_1)[\/latex] where [latex]m[\/latex] is the slope and [latex](x_1,y_1)[\/latex] is a point on the line.<\/section>
Graph the function [latex]y - 2 = 3(x + 1)[\/latex].\r\n\r\nMethod 1:\r\n<\/strong>\r\nWe can determine from the equation that the slope is [latex]m=3[\/latex] and a point on the line is [latex](-1,2)[\/latex].Step 1: Plot the point [latex](-1,2)[\/latex].\"AStep 2: Plot a second point using the slope. A slope of 3 means rise 3 and run 1.\"AStep 3: Draw the line through the two points\r\n\r\n\"A\r\n\r\n \r\n\r\nMethod 2:\r\n\r\nTo verify, we can rewrite the equation in slope-intercept form:\r\n\r\n[latex]\\begin{align} y - 2 &= 3(x + 1) \\\\ y - 2 &= 3x + 3 \\\\ y &= 3x + 5 \\end{align}[\/latex]\r\n\r\nThis tells us the y-intercept is [latex](0, 5)[\/latex] so now we can graph the line using the slope and y-intercept.\r\n\r\n<\/section>
Graphing from point-slope form:\r\n
    \r\n \t
  1. Identify the point and the slope<\/li>\r\n \t
  2. Graph the point<\/li>\r\n \t
  3. Use the slope to graph a second point<\/li>\r\n \t
  4. Draw the line through the two points<\/li>\r\n<\/ol>\r\n<\/section>
    [ohm_question hide_question_numbers=1]314645[\/ohm_question]<\/section>\r\n

    Graphing a Linear Function from Standard Form<\/h2>\r\n
    The standard form of a linear equation is: [latex]Ax + By = C[\/latex] where [latex]A[\/latex] must be a nonnegative integer.<\/section>\u00a0You can graph a line from standard form in two ways: by finding intercepts or by rewriting the equation in slope-intercept form.\r\n

    Method 1: Using Intercepts<\/h3>\r\n
    Graph [latex]2x + 3y = 6[\/latex].Method 1: Intercepts<\/strong>Step 1: Find the [latex]y[\/latex]-intercept by setting [latex]x=0[\/latex]:[latex]\\begin{align}2(0)+3y&=6 \\\\ 3y &= 6 \\\\ y &= 2\\end{align}[\/latex]\r\n
    <\/pre>\r\nStep 2: Find the [latex]x[\/latex]-intercept by setting [latex]y=0[\/latex]:\r\n\r\n[latex]\\begin{align} 2x + 3(0) &= 6 \\\\ 2x &= 6 \\\\ x &= 3 \\end{align}[\/latex]\r\n\r\nStep 3: Plot the points [latex](0,2)[\/latex] and [latex](0,3)[\/latex]\r\n\r\n\"A\r\n\r\nthen draw the line through them.\r\n\r\n\"A\r\n\r\nMethod 2: Slope-Intercept Form<\/strong>\r\n\r\nYou can also solve for [latex]y[\/latex] and graph as usual.\r\n\r\n[latex]\r\n\\begin{array}{l}\r\n2x + 3y = 6 \\\r\n3y = -2x + 6 \\\r\ny = -\\frac{2}{3}x + 2\r\n\\end{array}\r\n[\/latex]\r\n\r\nNow plot the y-intercept [latex](0, 2)[\/latex] and use the slope [latex]-\\frac{2}{3}[\/latex] (down 2, right 3) to plot the next points\r\n\r\n<\/section>
    Graphing from standard form using intercepts:\r\n
      \r\n \t
    1. Set [latex]x = 0[\/latex] and solve for [latex]y[\/latex] to find the [latex]y[\/latex]-intercept<\/li>\r\n \t
    2. Set [latex]y = 0[\/latex] and solve for [latex]x[\/latex] to find the [latex]x[\/latex]-intercept<\/li>\r\n \t
    3. Plot both intercepts<\/li>\r\n \t
    4. Draw the line through the two points<\/li>\r\n<\/ol>\r\n<\/section>
      [ohm_question hide_question_numbers=1]314646[\/ohm_question]<\/section>","rendered":"
      Graphing a Linear Function from Point-Slope Form<\/h2>\n
      Another way to graph a linear function is by using point-slope form<\/strong>. This form is useful when you know the slope and a single point on the line.<\/p>\n
      The point-slope form of a linear function is [latex]y - y_1 = m(x - x_1)[\/latex] where [latex]m[\/latex] is the slope and [latex](x_1,y_1)[\/latex] is a point on the line.<\/section>\n
      Graph the function [latex]y - 2 = 3(x + 1)[\/latex].
      \n
      \nMethod 1:
      \n<\/strong>
      \nWe can determine from the equation that the slope is [latex]m=3[\/latex] and a point on the line is [latex](-1,2)[\/latex].Step 1: Plot the point [latex](-1,2)[\/latex].\"AStep 2: Plot a second point using the slope. A slope of 3 means rise 3 and run 1.\"AStep 3: Draw the line through the two points<\/p>\n
      \"A<\/p>\n
       <\/p>\n
      Method 2:<\/p>\n
      To verify, we can rewrite the equation in slope-intercept form:<\/p>\n
      [latex]\\begin{align} y - 2 &= 3(x + 1) \\\\ y - 2 &= 3x + 3 \\\\ y &= 3x + 5 \\end{align}[\/latex]<\/p>\n
      This tells us the y-intercept is [latex](0, 5)[\/latex] so now we can graph the line using the slope and y-intercept.<\/p>\n<\/section>\n
      Graphing from point-slope form:<\/p>\n
        \n
      1. Identify the point and the slope<\/li>\n
      2. Graph the point<\/li>\n
      3. Use the slope to graph a second point<\/li>\n
      4. Draw the line through the two points<\/li>\n<\/ol>\n<\/section>\n