{"id":1693,"date":"2024-06-03T22:23:09","date_gmt":"2024-06-03T22:23:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1693"},"modified":"2025-08-13T23:20:10","modified_gmt":"2025-08-13T23:20:10","slug":"introduction-to-linear-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-linear-functions-learn-it-3\/","title":{"raw":"Introduction to Linear Functions: Learn It 3","rendered":"Introduction to Linear Functions: Learn It 3"},"content":{"raw":"

Determine Whether a Linear Function is Increasing, Decreasing, or Constant<\/h2>\r\nA linear function may be increasing, decreasing, or constant. For an increasing function<\/strong>, as with the train example, the output values increase as the input values increase. <\/em><\/strong>The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in (a)<\/strong>. For a decreasing function<\/strong>, the slope is negative.\u00a0The output values decrease as the input values increase<\/em><\/strong>.<\/em> A line with a negative slope slants downward from left to right as in (b)<\/strong>. If the function is constant, the output values are the same for all input values, so the slope is zero. A line with a slope of zero is horizontal as in (c)<\/strong>.\r\n\r\n
\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Three Three graphs indicating increasing, decreasing, and constant functions[\/caption]\r\n\r\n<\/center>
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increasing and decreasing linear functions<\/h3>\r\nThe slope determines if a linear function function is an increasing, <\/strong>decreasing<\/strong> or constant<\/strong>.\r\n\r\n \r\n
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  • [latex]f\\left(x\\right)=mx+b\\text{ is an increasing function if }m>0[\/latex]<\/li>\r\n \t
  • [latex]f\\left(x\\right)=mx+b\\text{ is an decreasing function if }m<0[\/latex]<\/li>\r\n \t
  • [latex]f\\left(x\\right)=mx+b\\text{ is a constant function if }m=0[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>
    Some recent studies suggest that a teenager sends an average of [latex]60[\/latex] text messages per day.[footnote]\"http:\/\/www.cbsnews.com\/news\/teens-are-sending-60-texts-a-day-study-says\/\".[\/footnote]\u00a0For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then determine whether the graph of the function is increasing, decreasing, or constant.\r\n
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    1. The total number of texts a teenager sends is considered a function of time in days. The input is the number of days and output is the total number of texts sent.<\/li>\r\n \t
    2. A teenager has a limit of [latex]500[\/latex] texts per month in his or her data plan. The input is the number of days and output is the total number of texts remaining for the month.<\/li>\r\n \t
    3. A teenager has an unlimited number of texts in his or her data plan for a cost of [latex]$50[\/latex] per month. The input is the number of days and output is the total cost of texting each month.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"574628\"]Show Solution[\/reveal-answer] [hidden-answer a=\"574628\"] Analyze each function.\r\n
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      1. The function can be represented as [latex]f\\left(x\\right)=60x[\/latex] where [latex]x[\/latex] is the number of days. The slope, [latex]60[\/latex], is positive so the function is increasing. This makes sense because the total number of texts increases with each day.<\/li>\r\n \t
      2. The function can be represented as [latex]f\\left(x\\right)=500 - 60x[\/latex] where [latex]x[\/latex] is the number of days. In this case, the slope is negative so the function is decreasing. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after [latex]x[\/latex] days.<\/li>\r\n \t
      3. The cost function can be represented as [latex]f\\left(x\\right)=50[\/latex] because the number of days does not affect the total cost. The slope is [latex]0[\/latex] so the function is constant.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
        [ohm2_question hide_question_numbers=1]13520[\/ohm2_question]<\/section>
        Use an online graphing calculator to graph the function: [latex]f(x)=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].If you are using Desmos, you can add sliders to represent various aspects of your equation. Below is a short tutorial on how to add sliders to your graphs in Desmos. Other online graphing calculators may or may not have this feature.\r\n