When we talk about a line's intercepts, we most often mean the x-intercept and the y-intercept. These are locations where the line intercepts or crosses the x- or y- axis. A more formal definition is as follows:
Below is the graph of a line, with both intercepts shown:
Notice that the x-intercept always has a y-coordinate of 0, and the y-intercept always has an x-coordinate of 0.
Keeping the above note in mind, we can easily identify intercepts from a table of x and y values when we notice that one of the coordinates is zero! Take a look:
We see that the second pair (0, 12) contains a zero. In this coordinate pair, the x-coordinate is zero, which means this represents a y-intercept. The y-intercept of the line is at the point (0, 12). We also notice a zero in the coordinate pair (4, 0). In this case, the y-coordinate is zero, so this is the x-intercept. The line crosses the x-axis at the point (4, 0).
How can we use the equation of a line to find its x– and y– intercepts? A common form of a line is in slope-intercept form: y = mx+b. This form is called slope-intercept form, because b represents the y-coordinate of the y-intercept (and we know that its x-coordinate is 0). We can also use the equation in this form to find the x-intercept. Since x-intercepts have a y-coordinate of 0, we set the equation equal to zero and solve for x. This is shown in the example below:
||When we have the equation y=mx+b, the variable b is the y-coordinate of the y-intercept|
||For the y-intercept, the x-coordinate is always 0|
||To find the x-intercept, plug 0 in for y as the y-coordinate is always 0|
||Subtract -8 from both sides|
||Divide by -2. This is the x-coordinate of the x-intercept|
||For the x-intercept, the y-coordinate is always 0|
For the equation y = -2x + 8, the y-intercept is at (0,8) and the x-intercept is at (4,0).