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Determining Intercepts

Determining Intercepts

Author: Sophia Tutorial
Description:

Determine the x-intercept and y-intercept of an linear equation.

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what's covered
  1. Determining Intercepts on a Graph
  2. Determining Intercepts from a Table
  3. Determining Intercepts from an Equation

1. Determining Intercepts on a Graph

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:

terms to know
x-intercept
where a line or curve intersects the x-axis, at y = 0
y-intercept
where a line or curve intersects the y-axis, at x = 0

Below is the graph of a line, with both intercepts shown:

File:5683-intercept1.png

Notice that the x-intercept always has a y-coordinate of 0, and the y-intercept always has an x-coordinate of 0.


2. Determining Intercepts from a Table

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:

x y
-2 18
0 12
2 6
4 0
6 -6

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).


3. Determining Intercepts from an Equation

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:


y equals short dash 2 plus 8
Our Equation
b equals 8
When we have the equation y=mx+b, the variable b is the y-coordinate of the y-intercept
y minus i n t e r c e p t space equals space left parenthesis 0 comma space 8 right parenthesis
For the y-intercept, the x-coordinate is always 0
0 equals short dash 2 x plus 8
To find the x-intercept, plug 0 in for y as the y-coordinate is always 0
short dash 8 equals short dash 2 x
Subtract -8 from both sides
4 equals x
Divide by -2. This is the x-coordinate of the x-intercept
x minus i n t e r c e p t space equals space left parenthesis 4 comma space 0 right parenthesis
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).

summary
When determining intercepts on a graph, the x-intercept is where a line or a curve intersects the x-axis. It is also where y equals 0 and is written as (x, 0) as a coordinate pair. The y-intercept is where a line or curve intersects the y-axis. It is also where x equals 0 and is written as (0, y) as a coordinate pair. When determining intercepts from a table, we look for instances where x equals 0 or y equals 0. When determining intercepts from an equation, the variable b represents the y-coordinate of the y-intercept. To find the x-intercept, plug in 0 for y and solve for x.
Terms to Know
x-intercept

the location on a graph where a line or curve intersects the x-axis: (x, 0)

y-intercept

the location on a graph where a line or curve intersects the y-axis: (0, y)