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 species of bamboo. These members of the grass family are the fastest growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour.^[1] In a twenty-four hour period this bamboo plant grows about 36 inches or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.

Recall that a function is a relation that assigns to every element in the domain exactly one element in the range. A linear function is a specific type of function that can be used to model many real-world applications such as plant growth over time. In this chapter we will explore linear functions, their graphs and how to relate them to data.

Characteristics of Linear Functions

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.^[2]

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.

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.

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.

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).

Writing a Linear Function

Calculate and Interpret Slope

In the examples we have seen so far, we have had the slope provided for us. However, we often need to calculate the slope given input and output values. Given two values for the input, [latex]{x}_{1}[/latex] and [latex]{x}_{2}[/latex], and two corresponding values for the output, [latex]{y}_{1}[/latex] and [latex]{y}_{2}[/latex] which can be represented by a set of points, [latex]\left({x}_{1}\text{, }{y}_{1}\right)[/latex] and [latex]\left({x}_{2}\text{, }{y}_{2}\right)[/latex], we can calculate the slope [latex]m[/latex], as follows:

[latex]m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex]

where [latex]\Delta y[/latex] is the change in output and [latex]\Delta x[/latex] is the change in input. In function notation, [latex]{y}_{1}=f\left({x}_{1}\right)[/latex] and [latex]{y}_{2}=f\left({x}_{2}\right)[/latex] so we could write:

[latex]m=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex]

The graph below indicates how the slope of the line between the points, [latex]\left({x}_{1,}{y}_{1}\right)[/latex] and [latex]\left({x}_{2,}{y}_{2}\right)[/latex], is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.

Point-Slope Form

Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. Now we will learn another way to write a linear function called point-slope form which is given below:

[latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex]

where [latex]m[/latex] is the slope of the linear function and [latex]({x}_{1},{y}_{1})[/latex] is any point which satisfies the linear function.

The point-slope form is derived from the slope formula.

[latex]\begin{array}{llll}{m}=\frac{y-{y}_{1}}{x-{x}_{1}}\hfill & \text{assuming }{ x }\ne {x}_{1}\hfill\\{ m }\left(x-{x}_{1}\right)=\frac{y-{y}_{1}}{x-{x}_{1}}\left(x-{x}_{1}\right)\hfill & \text{Multiply both sides by }\left(x-{x}_{1}\right)\hfill\\{ m }\left(x-{x}_{1}\right)=y-{y}_{1}\hfill & \text{Simplify}\hfill \\ y-{y}_{1}={ m }\left(x-{x}_{1}\right)\hfill &\text{Rearrange}\hfill \end{array}[/latex]

Keep in mind that slope-intercept form and point-slope form can be used to describe the same linear function. We can move from one form to another using basic algebra. For example, suppose we are given the equation [latex]y - 4=-\frac{1}{2}\left(x - 6\right)[/latex] which is in point-slope form. We can convert it to slope-intercept form as shown below.

[latex]\begin{array}{llll}y - 4=-\frac{1}{2}\left(x - 6\right)\hfill & \hfill \\ y - 4=-\frac{1}{2}x+3\hfill & \text{Distribute the }-\frac{1}{2}.\hfill \\ \text{}y=-\frac{1}{2}x+7\hfill & \text{Add 4 to each side}.\hfill \end{array}[/latex]

Therefore, the same line can be described in slope-intercept form as [latex]y=-\frac{1}{2}x+7[/latex].

Point-slope form is particularly useful if we know one point and the slope of a line. For example, suppose we are told that a line has a slope of 2 and passes through the point [latex]\left(4,1\right)[/latex]. We know that [latex]m=2[/latex] and that [latex]{x}_{1}=4[/latex] and [latex]{y}_{1}=1[/latex]. We can substitute these values into point-slope form.

[latex]\begin{array}{l}y-{y}_{1}=m\left(x-{x}_{1}\right)\\ y - 1=2\left(x - 4\right)\end{array}[/latex]

If we wanted to rewrite the equation in slope-intercept form, we apply algebraic techniques.

[latex]\begin{array}{llll} y - 1=2\left(x - 4\right)\hfill & \hfill \\ y - 1=2x - 8\hfill & \text{Distribute the }2.\hfill \\ \text{}y=2x - 7\hfill & \text{Add 1 to each side}.\hfill \end{array}[/latex]

Both equations [latex]y - 1=2\left(x - 4\right)[/latex] and [latex]y=2x - 7[/latex] describe the same line. The graph of the line can be seen below.

Writing the Equation of a Line Using Two Points

Point-slope form of an equation is also useful if we know any two points through which a line passes. Suppose, for example, we know that a line passes through the points [latex]\left(0,\text{ }1\right)[/latex] and [latex]\left(3,\text{ }2\right)[/latex]. We can use the coordinates of the two points to find the slope.

[latex]\begin{array}{l}{m}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\ \text{}{m}=\frac{2 - 1}{3 - 0}\hfill \\ \text{}{m}=\frac{1}{3}\hfill \end{array}[/latex]

Now we can use the slope we found and the coordinates of one of the points to find the equation for the line. Let’s use (0, 1) for our point.

[latex]\begin{array}{l}y-{y}_{1}=m\left(x-{x}_{1}\right)\\ y - 1=\frac{1}{3}\left(x - 0\right)\end{array}[/latex]

As before, we can use algebra to rewrite the equation in slope-intercept form.

[latex]\begin{array}{lll}y - 1=\frac{1}{3}\left(x - 0\right)\hfill & \hfill \\ y - 1=\frac{1}{3}x\hfill & \text{Distribute the }\frac{1}{3}.\hfill \\ \text{}y=\frac{1}{3}x+1\hfill & \text{Add 1 to each side}.\hfill \end{array}[/latex]

Both equations describe the line graphed below.

Key Equations

slope-intercept form of a line	[latex]y=mx+b[/latex]
slope	[latex]m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex]
point-slope form of a line	[latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex]

Key Concepts

• The ordered pairs given by a linear function represent points on a line.
• Linear functions can be represented in words, function notation, tabular form and graphical form.
• The rate of change of a linear function is also known as the slope.
• An equation in slope-intercept form of a line includes the slope and the initial value of the function.
• The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point where the line crosses the y-axis.
• An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
• A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
• A constant linear function results in a graph that is a horizontal line.
• Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.
• The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line.
• The slope and initial value can be determined given a graph or any two points on the line.
• One form of a linear function is slope-intercept form.
• Point-slope form is useful for finding the equation of a linear function when given the slope of a line and one point.
• Point-slope form is also convenient for finding the equation of a linear function when given two points through which a line passes.
• The equation for a linear function can be written in slope-intercept form if the slope m and initial value b are known.
• A linear function can be used to solve real-world problems.
• A linear function can be written from tabular form.

Glossary

decreasing linear function

a function with a negative slope: If [latex]m<0, \text{then }f\left(x\right)=mx+b[/latex] is decreasing.

increasing linear function

a function with a positive slope: If [latex]m>0, \text{then }f\left(x\right)=mx+b[/latex] is increasing.

linear function

a function with a constant rate of change that is a polynomial of degree 1 whosegraph is a straight line

point-slope form

the equation of a linear function of the form [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex]

slope

the ratio of the change in output values to the change in input values; a measure of the steepness of a line

slope-intercept form

the equation of a linear function of the form [latex]f\left(x\right)=mx+b[/latex]

y-intercept

the value of a function when the input value is zero; also known as initial value

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