{"id":1913,"date":"2025-07-30T21:23:23","date_gmt":"2025-07-30T21:23:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1913"},"modified":"2025-10-13T20:46:47","modified_gmt":"2025-10-13T20:46:47","slug":"graphs-of-the-other-trigonometric-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-other-trigonometric-functions-learn-it-2\/","title":{"raw":"Graphs of the Other Trigonometric Functions: Learn It 2","rendered":"Graphs of the Other Trigonometric Functions: Learn It 2"},"content":{"raw":"

Graphing Transformations of y<\/em> = tan x<\/em><\/h2>\r\n

Graphing One Period of a Shifted Tangent Function<\/h3>\r\nNow that we can graph a tangent function<\/strong> that is stretched or compressed, we will add a vertical and\/or horizontal (or phase) shift. In this case, we add C<\/em> and D<\/em> to the general form of the tangent function.\r\n
\r\n
[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex]<\/div>\r\n<\/div>\r\nThe graph of a transformed tangent function is different from the basic tangent function tan x in several ways:\r\n\r\n
\r\n

features of the graph of [latex]y = A\\tan\\left(Bx\u2212C\\right)+D[\/latex]<\/h3>\r\n<\/header>\r\n
    \r\n \t
  • The period is [latex]\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t
  • The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where k<\/em> is an integer.<\/li>\r\n \t
  • The range is [latex](-\\infty,\\infty)[\/latex]<\/li>\r\n \t
  • The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where k<\/em> is an odd integer.<\/li>\r\n \t
  • There is no amplitude.<\/li>\r\n \t
  • [latex]y=A\\tan(Bx)[\/latex] is an odd function because it is the quotient of odd and even functions (sine and cosine respectively).<\/li>\r\n<\/ul>\r\n<\/section>
    How To: Given the function [latex]y=A\\tan(Bx\u2212C)+D[\/latex], sketch the graph of one period.<\/strong>\r\n
      \r\n \t
    1. Express the function given in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].<\/li>\r\n \t
    2. Identify the stretching\/compressing<\/strong> factor, |A|.<\/li>\r\n \t
    3. Identify B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t
    4. Identify C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\r\n \t
    5. Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by D<\/em>.<\/li>\r\n \t
    6. Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where k<\/em> is an odd integer.<\/li>\r\n \t
    7. Plot any three reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/section>
      Graph one period of the function [latex]y=\u22122\\tan(\\pi x+\\pi)\u22121[\/latex].[reveal-answer q=\"385350\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"385350\"]Step 1.<\/strong> The function is already written in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].Step 2.<\/strong>\u00a0[latex]A=\u22122[\/latex], so the stretching factor is [latex]|A|=2[\/latex].\r\n\r\nStep 3.<\/strong>\u00a0[latex]B=\\pi[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\pi}=1[\/latex].\r\n\r\nStep 4.<\/strong>\u00a0[latex]C=\u2212\\pi[\/latex], so the phase shift is [latex]\\dfrac{C}{B}=\\dfrac{\u2212\\pi}{\\pi}=\u22121[\/latex].\r\n\r\nStep 5\u20137.<\/strong> The asymptotes are at [latex]x=\u2212\\frac{3}{2}[\/latex] and [latex]x=\u2212\\frac{1}{2}[\/latex] and the three recommended reference points are (\u22121.25, 1), (\u22121,\u22121), and (\u22120.75, \u22123).\r\n\r\n\"A\r\n

      Analysis of the Solution<\/h4>\r\nNote that this is a decreasing function because A<\/em> < 0.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      How would the graph in Example 2\u00a0look different if we made A<\/em> = 2 instead of \u22122?[reveal-answer q=\"560477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"560477\"]It would be reflected across the line [latex]y=\u22121[\/latex], becoming an increasing function.[\/hidden-answer]<\/section>
      Find a formula for the function.\"A[reveal-answer q=\"606896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"606896\"]The graph has the shape of a tangent function.\r\n\r\nStep 1.<\/strong> One cycle extends from \u20134 to 4, so the period is [latex]P=8[\/latex]. Since [latex]P=\\frac{\\pi}{|B|}[\/latex], we have [latex]B=\\frac{\\pi}{P}=\\frac{\\pi}{8}[\/latex].\r\n\r\nStep 2.<\/strong> The equation must have the [latex]\\text{form}f(x)=A\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].\r\n\r\nStep 3.<\/strong> To find the vertical stretch A<\/em>, we can use the point (2,2).\r\n

      [latex]2=A\\tan\\left(\\frac{\\pi}{8}\\times2\\right)=A\\tan\\left(\\frac{\\pi}{4}\\right)[\/latex]<\/p>\r\nBecause [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], A<\/em> = 2.\r\n\r\nThis function would have a formula [latex]f(x)=2\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

      Find a formula for the function.\"A[reveal-answer q=\"359527\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"359527\"][latex]g(x)=4\\tan(2x)[\/latex][\/hidden-answer]\r\n\r\n<\/section>
      [ohm_question hide_question_numbers=1]129731[\/ohm_question]<\/section><\/div>","rendered":"

      Graphing Transformations of y<\/em> = tan x<\/em><\/h2>\n

      Graphing One Period of a Shifted Tangent Function<\/h3>\n

      Now that we can graph a tangent function<\/strong> that is stretched or compressed, we will add a vertical and\/or horizontal (or phase) shift. In this case, we add C<\/em> and D<\/em> to the general form of the tangent function.<\/p>\n

      \n
      [latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n

      The graph of a transformed tangent function is different from the basic tangent function tan x in several ways:<\/p>\n

      \n
      \n

      features of the graph of [latex]y = A\\tan\\left(Bx\u2212C\\right)+D[\/latex]<\/h3>\n<\/header>\n
        \n
      • The period is [latex]\\frac{\\pi}{|B|}[\/latex].<\/li>\n
      • The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where k<\/em> is an integer.<\/li>\n
      • The range is [latex](-\\infty,\\infty)[\/latex]<\/li>\n
      • The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where k<\/em> is an odd integer.<\/li>\n
      • There is no amplitude.<\/li>\n
      • [latex]y=A\\tan(Bx)[\/latex] is an odd function because it is the quotient of odd and even functions (sine and cosine respectively).<\/li>\n<\/ul>\n<\/section>\n
        How To: Given the function [latex]y=A\\tan(Bx\u2212C)+D[\/latex], sketch the graph of one period.<\/strong><\/p>\n
          \n
        1. Express the function given in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].<\/li>\n
        2. Identify the stretching\/compressing<\/strong> factor, |A|.<\/li>\n
        3. Identify B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n
        4. Identify C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n
        5. Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by D<\/em>.<\/li>\n
        6. Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where k<\/em> is an odd integer.<\/li>\n
        7. Plot any three reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/section>\n
          Graph one period of the function [latex]y=\u22122\\tan(\\pi x+\\pi)\u22121[\/latex].<\/p>\n