{"id":1739,"date":"2024-06-05T22:12:16","date_gmt":"2024-06-05T22:12:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1739"},"modified":"2025-08-09T13:04:46","modified_gmt":"2025-08-09T13:04:46","slug":"graphs-of-linear-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-of-linear-functions-learn-it-4\/","title":{"raw":"Graphs of Linear Functions: Learn It 4","rendered":"Graphs of Linear Functions: Learn It 4"},"content":{"raw":"

Graphing Linear Functions Cont.<\/h2>\r\n

Graphing a Linear Function Using Transformations<\/h3>\r\nAnother option for graphing is to use transformations<\/strong> on the identity function [latex]f\\left(x\\right)=x[\/latex]. A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.\r\n

Vertical Stretch or Compression<\/h4>\r\nIn the equation [latex]f\\left(x\\right)=mx[\/latex], the [latex]m[\/latex]\u00a0is acting as the vertical stretch<\/strong> or compression<\/strong> of the identity function. When [latex]m[\/latex] is negative, there is also a vertical reflection of the graph. Notice that multiplying the equation [latex]f\\left(x\\right)=x[\/latex] by [latex]m [\/latex] stretches the graph of [latex]f [\/latex] by a factor of [latex]m[\/latex] units if [latex]m > 1[\/latex] and compresses the graph of [latex]f [\/latex] by a factor of [latex]m[\/latex] units if [latex]0 {\\lt} m {\\lt} 1[\/latex]. This means the larger the absolute value of [latex]m[\/latex], the steeper the slope.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]\"Graph Vertical stretches and compressions and reflections on the function [latex]f\\left(x\\right)=x[\/latex].[\/caption]\r\n

Vertical Shift<\/h4>\r\nIn [latex]f\\left(x\\right)=mx+b[\/latex], the [latex]b[\/latex]\u00a0acts as the vertical shift<\/strong>, moving the graph up and down without affecting the slope of the line. Notice that adding a value of [latex]b[\/latex]\u00a0to the equation of [latex]f\\left(x\\right)=x[\/latex] shifts the graph of\u00a0[latex]f [\/latex] a total of [latex]b[\/latex]\u00a0units up if [latex]b[\/latex]\u00a0is positive and\u00a0[latex]|b|[\/latex] units down if [latex]b[\/latex]\u00a0is negative.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]\"graph This graph illustrates vertical shifts of the function [latex]f\\left(x\\right)=x[\/latex].[\/caption]Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.\r\n\r\n
How To: Given the equation of a linear function, use transformations to graph the linear function in the form [latex]f\\left(x\\right)=mx+b[\/latex].<\/strong>\r\n
    \r\n \t
  1. Graph [latex]f\\left(x\\right)=x[\/latex].<\/li>\r\n \t
  2. Vertically stretch or compress the graph by a factor m<\/em>.<\/li>\r\n \t
  3. Shift the graph up or down b<\/em>\u00a0units.<\/li>\r\n<\/ol>\r\n<\/section>
    Graph [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex] using transformations.[reveal-answer q=\"750947\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"750947\"]The equation for the function shows that [latex]m=\\frac{1}{2}[\/latex] so the identity function is vertically compressed by [latex]\\frac{1}{2}[\/latex]. The equation for the function also shows that [latex]b=-3[\/latex], so the identity function is vertically shifted down [latex]3[\/latex] units.First, graph the identity function, and show the vertical compression.[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"graph The function [latex]y=x[\/latex] compressed by a factor of [latex]\\frac{1}{2}[\/latex].[\/caption]Then, show the vertical shift.[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The function [latex]y=\\frac{1}{2}x[\/latex] shifted down 3 units.[\/caption][\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]114584[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]294296[\/ohm_question]<\/section>","rendered":"

    Graphing Linear Functions Cont.<\/h2>\n

    Graphing a Linear Function Using Transformations<\/h3>\n

    Another option for graphing is to use transformations<\/strong> on the identity function [latex]f\\left(x\\right)=x[\/latex]. A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.<\/p>\n

    Vertical Stretch or Compression<\/h4>\n

    In the equation [latex]f\\left(x\\right)=mx[\/latex], the [latex]m[\/latex]\u00a0is acting as the vertical stretch<\/strong> or compression<\/strong> of the identity function. When [latex]m[\/latex] is negative, there is also a vertical reflection of the graph. Notice that multiplying the equation [latex]f\\left(x\\right)=x[\/latex] by [latex]m[\/latex] stretches the graph of [latex]f[\/latex] by a factor of [latex]m[\/latex] units if [latex]m > 1[\/latex] and compresses the graph of [latex]f[\/latex] by a factor of [latex]m[\/latex] units if [latex]0 {\\lt} m {\\lt} 1[\/latex]. This means the larger the absolute value of [latex]m[\/latex], the steeper the slope.<\/p>\n

    \"Graph
    Vertical stretches and compressions and reflections on the function [latex]f\\left(x\\right)=x[\/latex].<\/figcaption><\/figure>\n

    Vertical Shift<\/h4>\n

    In [latex]f\\left(x\\right)=mx+b[\/latex], the [latex]b[\/latex]\u00a0acts as the vertical shift<\/strong>, moving the graph up and down without affecting the slope of the line. Notice that adding a value of [latex]b[\/latex]\u00a0to the equation of [latex]f\\left(x\\right)=x[\/latex] shifts the graph of\u00a0[latex]f[\/latex] a total of [latex]b[\/latex]\u00a0units up if [latex]b[\/latex]\u00a0is positive and\u00a0[latex]|b|[\/latex] units down if [latex]b[\/latex]\u00a0is negative.<\/p>\n

    \"graph
    This graph illustrates vertical shifts of the function [latex]f\\left(x\\right)=x[\/latex].<\/figcaption><\/figure>\n

    Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.<\/p>\n

    How To: Given the equation of a linear function, use transformations to graph the linear function in the form [latex]f\\left(x\\right)=mx+b[\/latex].<\/strong><\/p>\n
      \n
    1. Graph [latex]f\\left(x\\right)=x[\/latex].<\/li>\n
    2. Vertically stretch or compress the graph by a factor m<\/em>.<\/li>\n
    3. Shift the graph up or down b<\/em>\u00a0units.<\/li>\n<\/ol>\n<\/section>\n
      Graph [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex] using transformations.<\/p>\n