In the context of graphing, the constant rate of change is called the slope. For a linear function, the graph is a line which has a constant slope. Earlier we saw that the pattern in table 1 can be represented by the function [latex]f(x)=3x+1[/latex], where 3 was the rate of change and 1 was the initial value. Since the rate of change was 3, the slope of the line that represents this function, [latex]y=3x+1[/latex], is also 3. That is, the slope is [latex]\dfrac{3}{1}=\dfrac{rise}{run}[/latex]. The table value [latex]y = 1[/latex] when [latex]x = 0[/latex] is represented by the point (0, 1) on the coordinate plane. This point is the y-intercept of the line that represents the function and shows the initial value of the function. Therefore, the function of rate-initial value form

 [latex]f(x) = 3x + 1[/latex]

is also called the Slope-Intercept Form because the rate of change 3 is the slope, and the initial value 1 is the [latex]y[/latex]-coordinate of the [latex]y[/latex]-intercept.

The graph of [latex]f(x)=3x+1[/latex] is shown in figure 1. To illustrate, starting at the [latex]y[/latex]-intercept (0, 1), the line rises 3 (moving up 3 units) as it runs 1 (moving right 1 unit).

Figure 1. Graph of [latex]f(x)=3x+1[/latex]

Slope-Intercept Form on the Coordinate Plane

We can write the function represented by a line on the coordinate plane as long as the slope and [latex]y[/latex]-intercept can be identified on the graph.

There are two lines graphed in figure 2. The black line has a [latex]y[/latex]-intercept at the point (0, 3) and a slope of 3 [latex]\left ( m=\frac{\Delta y}{\Delta x}=\frac{3-(-9)}{0-(-4)}=\frac{12}{4}=3 \right )[/latex]. Therefore, the function of the black line is [latex]f(x) = 3x + 3[/latex]. The black line has a [latex]y[/latex]-intercept at (0, –5), and its slope is [latex]-\frac{3}{10}[/latex] [latex]\left ( m=\frac{\Delta y}{\Delta x}=\frac{-2-(-5)}{-10-(0)}=\frac{3}{-10}=-\frac{3}{10} \right )[/latex]. Therefore, the function of the black line is [latex]g(x) = -\frac{3}{10}x -5[/latex].

Figure 2. Two lines on the coordinate plane.

Be careful not to confuse the [latex]x[/latex]-intercept and the [latex]y[/latex]-intercept. The “intercept” in the term “Slope-intercept form” means the [latex]y[/latex]-intercept. The [latex]x[/latex]-intercept is the intersection point between a graph (e.g., a line) and the [latex]x[/latex]-axis. Because the intersection point is on the [latex]x[/latex]-axis, its coordinates will be [latex](a, 0)[/latex]. For the blue line in figure 2, the [latex]x[/latex]-intercept is the point (-1, 0).

The Domain and Range of a Line

We have seen that the graph of a linear function is a line on the coordinate plane. For any line that is neither horizontal nor vertical (see figure 3), it extends to the left toward an [latex]x[/latex]-value of negative infinity and right towards an [latex]x[/latex]-value of positive infinity. Therefore, its domain, which is the set of all [latex]x[/latex]-values, is [latex]\{\;x\;|\;x\in(-\infty, \infty)\;\}[/latex]. It also extends down towards a [latex]y[/latex]-value of negative infinity and up towards a [latex]y[/latex]-value of positive infinity. Therefore, its range, which is the set of all [latex]y[/latex]-values, is [latex]\{\;f(x)\;|\;f(x)\in(-\infty, \infty)\;\}[/latex].

Figure 3. Graph of a line.

For a horizontal line, the domain is [latex]\{ \;x\;|\;x\in (-\infty, \infty)\}[/latex] because it extends to the left and right on the coordinate plane. However, its range will be a single function-value because all of the points on the horizontal line have the same [latex]y[/latex]-coordinate. For example, the range of the horizontal line in the figure 4 is [latex]\{\;f(x)\;|\;f(x)=3\;\}[/latex].

Figure 4. A horizontal line and a vertical line.

A vertical line, like the blue line in figure 4, is NOT a function. Do you see why?

However, it is a relation and therefore has a domain and a range. Its domain will be a single [latex]x[/latex]-value because all of the points on the vertical line have the same [latex]x[/latex]-coordinate. For example, the domain of the vertical line in figure 4 is [latex]\{ \;x\;|\;x=–2\;\}[/latex]. The range is [latex]\{ \;y\;|\;y\in (-\infty, \infty)\;\}[/latex] because it extends up towards a [latex]y[/latex]-value of positive infinity and down towards a [latex]y[/latex]-value of negative infinity.