Basic Functions and Graphs: Background You’ll Need 2

  • Calculate and interpret the slope of a linear function

Linear Functions

Linear functions have the form [latex]f(x)=ax+b[/latex], where [latex]a[/latex] and [latex]b[/latex] are constants. In Figure 1, we see examples of linear functions when [latex]a[/latex] is positive, negative, and zero. Note that if [latex]a>0[/latex], the graph of the line rises as [latex]x[/latex] increases. In other words, [latex]f(x)=ax+b[/latex] is increasing on [latex](−\infty, \infty)[/latex]. If [latex]a<0[/latex], the graph of the line falls as [latex]x[/latex] increases. In this case, [latex]f(x)=ax+b[/latex] is decreasing on [latex](−\infty, \infty)[/latex]. If [latex]a=0[/latex], the line is horizontal.

An image of a graph. The y axis runs from -2 to 5 and the x axis runs from -2 to 5. The graph is of the 3 functions. The first function is “f(x) = 3x + 1”, which is an increasing straight line with an x intercept at ((-1/3), 0) and a y intercept at (0, 1). The second function is “g(x) = 2”, which is a horizontal line with a y intercept at (0, 2) and no x intercept. The third function is “h(x) = (-1/2)x”, which is a decreasing straight line with an x intercept and y intercept both at the origin. The function f(x) is increasing at a higher rate than the function h(x) is decreasing.
Figure 1. These linear functions are increasing or decreasing on [latex](-\infty, \infty)[/latex] and one function is a horizontal line.

Slope

The graph of any linear function is a line. One of the distinguishing features of a line is its slope. The slope is the change in [latex]y[/latex] for each unit change in [latex]x[/latex]. The slope measures both the steepness and the direction of a line.

To calculate the slope of a line, we need to determine the ratio of the change in [latex]y[/latex] versus the change in [latex]x[/latex]. To do so, we choose any two points [latex](x_1,y_1)[/latex] and [latex](x_2,y_2)[/latex] on the line and calculate [latex]\dfrac{y_2-y_1}{x_2-x_1}[/latex]. In Figure 2, we see this ratio is independent of the points chosen.

An image of a graph. The y axis runs from -1 to 10 and the x axis runs from -1 to 6. The graph is of a function that is an increasing straight line. There are four points labeled on the function at (1, 1), (2, 3), (3, 5), and (5, 9). There is a dotted horizontal line from the labeled function point (1, 1) to the unlabeled point (3, 1) which is not on the function, and then dotted vertical line from the unlabeled point (3, 1), which is not on the function, to the labeled function point (3, 5). These two dotted have the label “(y2 - y1)/(x2 - x1) = (5 -1)/(3 - 1) = 2”. There is a dotted horizontal line from the labeled function point (2, 3) to the unlabeled point (5, 3) which is not on the function, and then dotted vertical line from the unlabeled point (5, 3), which is not on the function, to the labeled function point (5, 9). These two dotted have the label “(y2 - y1)/(x2 - x1) = (9 -3)/(5 - 2) = 2”.
Figure 2. or any linear function, the slope [latex](y_2-y_1)/(x_2-x_1)[/latex] is independent of the choice of points [latex](x_1,y_1)[/latex] and [latex](x_2,y_2)[/latex] on the line.

slope of a linear line

Consider line [latex]L[/latex] passing through points [latex](x_1,y_1)[/latex] and [latex](x_2,y_2)[/latex]. Let [latex]\Delta y=y_2-y_1[/latex] and [latex]\Delta x=x_2-x_1[/latex] denote the changes in [latex]y[/latex] and [latex]x[/latex], respectively. The slope of the line is

[latex]m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{\Delta y}{\Delta x}[/latex]

If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal.

Slope-Intercept Form

The linear equation [latex]f(x)=ax+b[/latex] encapsulates two crucial pieces of information about its graph: the slope and the [latex]y[/latex]-intercept. The coefficient ‘[latex]a[/latex]‘ is the slope, dictating the angle and direction of the line, while ‘[latex]b[/latex]‘ gives us the [latex]y[/latex]-intercept, the point where the line crosses the [latex]y[/latex]-axis. This equation is the essence of the slope-intercept form, commonly written as [latex]f(x)=mx+b[/latex], with ‘[latex]m[/latex]‘ signifying the slope. It succinctly represents the linear function, offering a clear view of its gradient and starting point on a graph.

slope-intercept form

Consider a line with slope [latex]m[/latex] and [latex]y[/latex]-intercept [latex](0,b)[/latex]. The equation

[latex]y=mx+b[/latex]
 

is an equation for that line in slope-intercept form.

Consider the line passing through the points [latex](11,-4)[/latex] and [latex](-4,5)[/latex], as shown in Figure 3.

An image of a graph. The x axis runs from -5 to 12 and the y axis runs from -5 to 6. The graph is of the function that is a decreasing straight line. The function has two points plotted, at (-4, 5) and (11, 4).
Figure 3. Finding the equation of a linear function with a graph that is a line between two given points.
 
  1. Find the slope of the line.
  2. Find an equation for this linear function in slope-intercept form.

Aisha leaves her house at 5:50 a.m. and goes for a [latex]9[/latex]-mile run. She returns to her house at 7:08 a.m. Answer the following questions, assuming Aisha runs at a constant pace.

  1. Describe the distance [latex]D[/latex] (in miles) Aisha runs as a linear function of her run time [latex]t[/latex] (in minutes).
  2. Sketch a graph of [latex]D[/latex].
  3. Interpret the meaning of the slope.