What are intercepts and how do you find them? Intercepts
are where a graph
crosses either the x-axis
or the y-axis. Not all functions will have intercepts, but when you are working with the graph
of a linear function, it will have both an x-intercept
and a y-intercept. An x-intercept
is where the graph
crosses the x-axis
and a y-intercept
is where it crosses the y-axis.
Let’s think about a line
that crosses the x-axis. If it is crossing the x-axis, then the y-value of that point
will be zero. Similarly, when dealing with the y-intercept, this is where the x-value is zero. We will use this information in finding x-and y-intercepts.
The general rules are: Let's Practice:
- To find an x-intercept, let the value of y in the equation be equal to zero.
Your x-intercept will be written as a point .
- To find a y-intercept, let the value of x in the equation be equal to zero.
Your y-intercept will be written as a point .
- Find the x- and y-intercepts for .
The y-intercept is found by letting x = 0.
We write the y-intercept as ,
The x-intercept is found by letting y = 0.
We write the x-intercept as .
You may have noticed something interesting about the y-intercept
in the previous example. The y-intercept
was the number by itself in the equation
of the line. The equation
in Example 1 is called the slope-intercept form
of a line
and in general is written as
. In this equation, the slope
of the line
and the y-intercept
. You can learn more about finding the slope
of a line
by clicking here
. This slope-intercept form of a line
makes is easy to find the y-intercept. More Practice:
- Find the x-and y-intercepts for
Since this equation is in slope-intercept form, we can find the y-intercept by looking at the equation. The y-intercept is 4 and is written as .
To find the x-intercept, let y = 0 in the equation.
The x-intercept is written as .
- Find the x-and y-intercepts for .
This equation is not in slope-intercept form, so we go back to our strategy of substituting x = 0 to find the y-intercept.
The y-intercept is . To find the x-intercept, let y = 0.
The x-intercept is .
There are two special types of lines that need to be considered when talking about intercepts. These are horizontal lines and vertical lines.
- A horizontal line is in the form y = k; that is,
no matter what the x-value is, the y-value is always a constant value.
- A vertical line is in the form x = h; that is,
no matter what the y-value is, the x-value is always a constant value.
- Consider the graph of y = 3.
The horizontal line shown in this graph will never cross the x-axis. A horizontal line (other than y = 0) will not have an x-intercept. The line y = 0 is another special case since y = 0 is the equation of the x-axis.
The y-intercept will always be the number in the equation. So in this case, .
- Consider the graph of x = -4.
The vertical line shown in this graph will cross the x-axis at the number given in the equation. For this equation, the x-intercept is .
Notice this line will never cross the y-axis. A vertical line (other than x = 0) will not have a y-intercept. The line x = 0 is another special case since x = 0 is the equation of the y-axis.
Now that you have these tools to find the intercepts of a line, what does this information do for you? What good are intercepts other than just knowing points on a graph?
The x-and y-intercepts play a key role in graphing linear functions. To learn more about graphing lines, click here