{"id":10299,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10299"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"graphing-of-linear-functions-rate-of-change-growth-and-decay","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/graphing-of-linear-functions-rate-of-change-growth-and-decay\/","title":{"rendered":"Graphing Of Linear Functions, Rate Of Change, Growth And Decay"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Source: Kapdec.com<\/p>\n<\/div>\n

Unit: Linear Functions<\/strong><\/h2>\n

Graphing of Linear Functions, Rate of Change, Growth and Decay<\/strong><\/h3>\n

Linear functions are a specific type of function that create straight lines when graphed. They are fundamental in algebra and widely used to model relationships between variables.<\/p>\n

Definition and General Form<\/strong><\/p>\n

A linear function is a function that can be written in the form: f<\/em>(x<\/em>)=mx<\/em>+b<\/em> where:<\/p>\n

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  • f<\/em>(x<\/em>) or y<\/em> is the output or dependent variable.<\/li>\n
  • x<\/em> is the input or independent variable.<\/li>\n
  • m<\/em> is the slope of the line.<\/li>\n
  • b<\/em> is the y-intercept, the point where the line crosses the y-axis.<\/li>\n<\/ul>\n

    Characteristics of Linear Functions<\/strong><\/p>\n

      \n
    1. Constant Rate of Change:<\/strong>\n
        \n
      • The slope m<\/em> represents the constant rate of change of the function.<\/li>\n
      • For every unit increase in x<\/em>, y<\/em> increases by m<\/em>.<\/li>\n<\/ul>\n<\/li>\n
      • Straight Line Graph:<\/strong>\n
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        • The graph of a linear function is always a straight line.<\/li>\n<\/ul>\n<\/li>\n
        • Intercepts:<\/strong>\n
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          • Y-Intercept: The value of y<\/em> when x<\/em>=0, given by b<\/em>.<\/li>\n
          • X-Intercept: The value of x<\/em> when y<\/em>=0, found by solving mx<\/em>+b<\/em>=0.<\/li>\n<\/ul>\n<\/li>\n
          • Domain and Range:<\/strong>\n
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            • The domain of a linear function is all real numbers, (\u2212\u221e,\u221e).<\/li>\n
            • The range of a linear function is all real numbers, (\u2212\u221e,\u221e).<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

              Slope and Y-Intercept<\/strong><\/p>\n

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              1. Slope (m):\n
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                • Describes the steepness and direction of the line.<\/li>\n
                • Calculated as:<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

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                  Source: Kapdec.com<\/p>\n<\/div>\n

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                  1. \n
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                    • Positive slope: line rises from left to right.<\/li>\n
                    • Negative slope: line falls from left to right.<\/li>\n
                    • Zero slope: horizontal line.<\/li>\n
                    • Undefined slope: vertical line.<\/li>\n<\/ul>\n<\/li>\n
                    • Y-Intercept (b):<\/strong>\n
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                      • The point where the line crosses the y-axis.<\/li>\n
                      • Found by setting x<\/em>=0 in the equation f<\/em>(x<\/em>)=mx<\/em>+b<\/em>.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                        Example<\/strong><\/p>\n

                        For the linear function f<\/em>(x<\/em>)=3x<\/em>+2:<\/p>\n

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                        • Slope (m<\/em>): 3<\/li>\n
                        • Y-Intercept (b<\/em>): 2<\/li>\n<\/ul>\n

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

                          To graph a linear function, follow these steps:<\/p>\n

                            \n
                          1. Identify the Slope and Y-Intercept:<\/strong>\n
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                            • From the equation f<\/em>(x<\/em>)=mx<\/em>+b<\/em>.<\/li>\n<\/ul>\n<\/li>\n
                            • Plot the Y-Intercept:<\/strong>\n
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                              • Locate the point (0,b<\/em>) on the graph.<\/li>\n<\/ul>\n<\/li>\n
                              • Use the Slope to Find Another Point:<\/strong>\n
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                                • From the y-intercept, use the slope \ud835\udc5am<\/em> (rise over run) to find another point.<\/li>\n
                                • Example: For m<\/em>=3, from (0,2), move up 3 units and right 1 unit to (1,5).<\/li>\n<\/ul>\n<\/li>\n
                                • Draw the Line:\n
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                                  • Connect the points with a straight line extending in both directions.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                                    Example<\/strong><\/p>\n

                                    Graph the function f<\/em>(x<\/em>)=\u22122x<\/em>+4:<\/p>\n

                                      \n
                                    • Slope m<\/em>=\u22122<\/li>\n
                                    • Y-Intercept b<\/em>=4<\/li>\n<\/ul>\n
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                                      1. Plot the y-intercept (0,4).<\/li>\n
                                      2. Use the slope to find another point: From (0,4), move down 2 units and right 1 unit to (1,2).<\/li>\n
                                      3. Draw the line through (0,4) and (1,2).<\/li>\n<\/ol>\n

                                        Linear Function Applications<\/strong><\/p>\n

                                        Linear functions are used in various real-life scenarios, including:<\/p>\n

                                          \n
                                        1. Business and Economics:\n
                                            \n
                                          • Cost Functions: Representing the total cost as a function of the number of units produced.<\/li>\n
                                          • Revenue Functions: Representing total revenue as a function of the number of units sold.<\/li>\n<\/ul>\n<\/li>\n
                                          • Physics:\n
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                                            • Motion: Representing the relationship between distance and time for objects moving at constant speed.<\/li>\n<\/ul>\n<\/li>\n
                                            • Everyday Situations:\n
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                                              • Budgeting: Modelling expenses over time.<\/li>\n
                                              • Conversion: Converting units, such as temperature or currency exchange rates.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                                                Example<\/p>\n

                                                A taxi company charges a flat fee of $3 plus $2 per mile driven. The cost C<\/em> of a trip that covers x<\/em> miles can be modelled by the linear function: C<\/em>(x<\/em>)=2x<\/em>+3<\/p>\n

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                                                • Slope (m<\/em>): $2 per mile.<\/li>\n
                                                • Y-Intercept (b<\/em>): $3 (flat fee).<\/li>\n<\/ul>\n

                                                  Summary<\/p>\n

                                                    \n
                                                  • Definition: Linear functions are written as f<\/em>(x<\/em>)=mx<\/em>+b<\/em>.<\/li>\n
                                                  • Characteristics: Constant rate of change, straight-line graph, intercepts, and an infinite domain and range.<\/li>\n
                                                  • Slope and Y-Intercept: The slope measures steepness and direction, while the y-intercept indicates where the line crosses the y-axis.<\/li>\n
                                                  • Graphing: Identify slope and y-intercept, plot points, and draw the line.<\/li>\n
                                                  • Applications: Used in business, physics, and daily life to model linear relationships.<\/li>\n<\/ul>\n

                                                    Understanding linear functions is crucial for solving problems and modelling relationships in various disciplines.<\/p>\n

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