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 30-kilometer trip from the airport to the subway station in only 8 minutes.^[1]

A view of the Shanghai Maglev Train. (credit: Rolf Wilhelm Pfennig)

Suppose that a Maglev train were to travel a long distance, and the train maintains a constant speed of 83 meters per second for a period of time once it is 250 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 1. 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.

Representing a Linear Function in Word Form

Let’s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.

• The train’s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at a constant speed.

The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station.

Representing a Linear Function in Function Notation

Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as slope-intercept form of a line, where [latex]x[/latex] is the input value, [latex]m[/latex] is the rate of change, and [latex]b[/latex] is the initial value of the dependent variable. Slope-intercept form is given below:

[latex]\begin{array}{lll}\text{Equation form}\hfill & y=mx+b\hfill \\ \text{Function notation}\hfill & f\left(x\right)=mx+b\hfill \end{array}[/latex]

In the example of the train, we might use the notation [latex]D\left(t\right)[/latex] in which the total distance [latex]D[/latex] is a function of the time [latex]t[/latex]. The rate, [latex]m[/latex], is 83 meters per second. The initial value, [latex]b[/latex], of the dependent variable is the original distance from the station, 250 meters. We can write a generalized equation to represent the motion of the train.

[latex]D\left(t\right)=83t+250[/latex]

Representing a Linear Function in Tabular Form

A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in the table below. From the table, we can see that the distance changes by 83 meters for every 1 second increase in time.

Tabular representation of the function D showing selected input and output values.

Representing a Linear Function in Graphical Form

Another way to represent linear functions is visually by using a graph. We can use the function relationship from above, [latex]D\left(t\right)=83t+250[/latex], to draw a graph as seen below. Notice the graph is a line. When we plot a linear function, the graph is always a line.

The rate of change, which is always constant for linear functions, determines the slant or slope of the line. The point at which the input value is zero is the y-intercept of the line. We can see from the graph that the y-intercept in the train example we just saw is [latex]\left(0,250\right)[/latex] and represents the distance of the train from the station when it began moving at a constant speed.

The graph of [latex]D\left(t\right)=83t+250[/latex]. Graphs of linear functions are lines because the rate of change is constant.

Notice the graph of the train example is restricted since the input value, time, must always be a nonnegative real number. However, this is not always the case for every linear function. Consider the graph of the line [latex]f\left(x\right)=2{x}_{}+1[/latex]. Ask yourself what input values can be plugged into this function, that is, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled and then have one added to the product.

Example: Using a Linear Function to MODEL the Pressure on a Diver

The pressure, [latex]P[/latex], in pounds per square inch (PSI) on a diver depends on depth below the water surface, [latex]d[/latex], in feet. This relationship may be modeled by the equation [latex]P\left(d\right)=0.434d+14.696[/latex]. Restate this function in words.

A diver exploring the depths of the ocean. (credit: Ilse Reijs and Jan-Noud Hutten)

Show Solution

To restate the function in words, we need to describe each part of the equation. The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths.

Analysis of the Solution

The initial value, 14.696, is the pressure in PSI on the diver at a depth of 0 feet, which is the surface of the water. The rate of change, or slope, is 0.434 PSI per foot. This tells us that the pressure on the diver increases 0.434 PSI for each foot her depth increases.

Determine Whether a Linear Function is Increasing, Decreasing, or Constant

The linear functions we used in the two previous examples increased over time, but not every linear function does this. A linear function may be increasing, decreasing, or constant. For an increasing function, as with the train example, the output values increase as the input values increase. 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). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in (b). 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).

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