- Express a linear function using an equation, words, tables, and graphs
- Determine whether a linear function is increasing, decreasing, or constant.
- Calculate and interpret slope
- Find the x-intercept of a linear function given its equation
Linear Functions
Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly [latex]1.5[/latex] inches every hour [1]. In a twenty-four hour period, this bamboo plant grows about [latex]36[/latex] inches, or an incredible [latex]3[/latex] feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.
Just as with the growth of a bamboo plant, there are many situations that involve constant change over time. For example, consider the first commercial Maglev train in the world, the Shanghai Maglev Train. It carries passengers comfortably for a [latex]30[/latex]-kilometer trip from the airport to the subway station in only [latex]8[/latex] minutes.[2]
Suppose that a Maglev train were to travel a long distance, and the train maintains a constant speed of [latex]83[/latex] meters per second for a period of time once it is [latex]250[/latex] meters from the station. How can we analyze the train’s distance from the station as a function of time? In this section, we will investigate a type of function that is useful for this purpose and use it to investigate real-world situations such as the train’s distance from the station at a given point in time.
The function describing the train’s motion is a linear function, which is defined as a function with a constant rate of change, that is, a polynomial of degree [latex]1[/latex]. There are several ways to represent a linear function including word form, function notation, tabular form and graphical form. We will describe the train’s motion as a function using each method.
linear function
A linear function is characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[/latex].