Slope of a Line
The slope is the constant rate of change of a linear equation. It can be thought of as the ratio of the vertical change to the horizontal change between two points on the graph of a line. If the two points are (x1, y1) and (x2, y2), then the vertical change is y2 - y1 and the horizontal change is x2 - x1. Hence we can use the formula
to determine the slope of a line if we know two points on that line. See Figure 1 below.
Explore the following interactive example to see how the slope of a line is computed when two points are known. What observations can you make?
In general, when the slope of a line is positive, the line will be increasing. That is, the graph will be a line that rises from left to right. When the slope is negative, the line will be decreasing and the graph will be a line that falls from left to right. If the slope of a line is 0, then it is neither increasing nor decreasing, but is constant. The graph of a constant is a horizontal line.
Use the next interactive example to see how different values of the slope, m, affect the graph of the line y = mx + b.
Horizontal and Vertical Intercepts
The points where a line crosses the vertical and horizontal axes are known as the vertical and horizontal intercepts. (These points are often referred to as the x-intercept and the y-intercept.) Given a linear functiony = mx + b,
- The vertical intercept (y-intercept) is found by evaluating the equation when the input variable, x, is 0 and is always the same as the constant b.
The horizontal intercept (x-intercept) is the value of the variable x when the y value is 0. It is found by solving the equation 0 = mx + b.
Interactive Example
Now explore how the values of the y-intercept, b, affect the graph of the line y = mx + b.
Algebraic Example
Find the vertical and horizontal intercept of the line y = -7.2x + 250.
Solution
When x = 0, y = -7.2(0) + 250 = 250, so the vertical intercept is 250. This means that the graph crosses the horizontal axis at the point (0, 250). Also notice that this is the value of b in the general form y = mx + b.
To find the horizontal intercept we can replace y with 0 and solve the linear equation 0 = -7.2x + 250. The solution is given below.
0 = -7.2x + 250
-250 = -7.2x
34.7 = x
The horizontal intercept is 34.7. This is the point (34.7, 0) on the graph of the line. A graph of the line is shown in Figure 2.
Figure 2
