Characteristics of Linear Functions
Learning Objectives
- Represent a linear function in words, tabular form, with function notation, and with a graph
- Determine whether a linear function is increasing, decreasing, or constant
![Front view of a subway train, the maglev train.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18223047/CNX_Precalc_Figure_02_01_0112.jpg)
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 constant speed.
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 the slope-intercept form of a line, where is the input value, is the rate of change, and is the initial value of the dependent variable.In the example of the train, we might use the notation in which the total distance is a function of the time . The rate, , is 83 meters per second. The initial value 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.
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.![Table with the first row, labeled t, containing the seconds from 0 to 3, and with the second row, labeled D(t), containing the meters 250 to 499. The first row goes up by 1 second, and the second row goes up by 83 meters.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18223049/CNX_Precalc_Figure_02_01_0152.jpg)
Q & A
Can the input in the previous example be any real number? No. The input represents time, so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. The input consists of non-negative real numbers.Representing a Linear Function in Graphical Form
Another way to represent linear functions is visually, using a graph. We can use the function relationship from above, , to draw a graph, represented in the graph 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 constant, determines the slant, or slope of the line. The point at which the input value is zero is the vertical intercept, or y-intercept, of the line. We can see from the graph that the y-intercept in the train example we just saw is and represents the distance of the train from the station when it began moving at a constant speed.![A graph of an increasing function with points at (-2, -4) and (0, 2).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18223051/CNX_Precalc_Figure_02_01_0122.jpg)
A General Note: Linear Function
A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a linewhere is the initial or starting value of the function (when input, ), and is the constant rate of change, or slope of the function. The y-intercept is at .
Example: Using a Linear Function to Find the Pressure on a Diver
The pressure, , in pounds per square inch (PSI) on the diver in Figure 3 depends upon her depth below the water surface, , in feet. This relationship may be modeled by the equation, . Restate this function in words.![Scuba diver.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18223054/CNX_Precalc_Figure_02_01_0032.jpg)
Answer: 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.Try it now
Use Desmos to graph the function: .![](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2017/02/13193222/calculator.png)
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. 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).![Three graphs depicting an increasing function, a decreasing function, and a constant function.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18223056/CNX_Precalc_Figure_02_01_004abc2.jpg)
A General Note: Increasing and Decreasing Functions
The slope determines if the function is an increasing linear function, a decreasing linear function, or a constant function.- .
- .
- .
Example: Deciding whether a Function Is Increasing, Decreasing, or Constant
Some recent studies suggest that a teenager sends an average of 60 texts per day.[footnote]http://www.cbsnews.com/8301-501465_162-57400228-501465/teens-are-sending-60-texts-a-day-study-says/[/footnote] For 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.- The total number of texts a teen 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.
- A teen has a limit of 500 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.
- A teen has an unlimited number of texts in his or her data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month.
Answer: Analyze each function.
- The function can be represented as where is the number of days. The slope, 60, is positive so the function is increasing. This makes sense because the total number of texts increases with each day.
- The function can be represented as where 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 days.
- The cost function can be represented as because the number of days does not affect the total cost. The slope is 0 so the function is constant.