4.3 Graph with intercepts

Identify the x - and y - intercepts on a graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x - axis and the y - axis. These points are called the intercepts of the line.

Let’s look at the graphs of the lines in [link] .

First, notice where each of these lines crosses the $x$ negative axis. See [link] .

Do you see a pattern?

For each row, the y - coordinate of the point where the line crosses the x - axis is zero. The point where the line crosses the x - axis has the form $\left(a,0\right)$ and is called the x - intercept of a line . The x - intercept occurs when $y$ is zero.

Now, let’s look at the points where these lines cross the y - axis. See [link] .

What is the pattern here?

In each row, the x - coordinate of the point where the line crosses the y - axis is zero. The point where the line crosses the y - axis has the form $\left(0,b\right)$ and is called the y- intercept of the line. The y - intercept occurs when $x$ is zero.

Find the x - and y - intercepts on each graph.

Solution

1. ⓐ The graph crosses the x - axis at the point $\left(4,0\right)$ . The x - intercept is $\left(4,0\right)$ .
   The graph crosses the y - axis at the point $\left(0,2\right)$ . The y - intercept is $\left(0,2\right)$ .
   
   
2. ⓑ The graph crosses the x - axis at the point $\left(2,0\right)$ . The x - intercept is $\left(2,0\right)$
   The graph crosses the y - axis at the point $\left(0,\mathrm{-6}\right)$ . The y - intercept is $\left(0,\mathrm{-6}\right)$ .
   
   
3. ⓒ The graph crosses the x - axis at the point $\left(\mathrm{-5},0\right)$ . The x - intercept is $\left(\mathrm{-5},0\right)$ .
   The graph crosses the y - axis at the point $\left(0,\mathrm{-5}\right)$ . The y - intercept is $\left(0,\mathrm{-5}\right)$ .

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Find the x - and y - intercepts on the graph.

x - intercept: $\left(2,0\right)$ ; y - intercept: $\left(0,\mathrm{-2}\right)$

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Find the x - and y - intercepts on the graph.

x - intercept: $\left(3,0\right)$ , y - intercept: $\left(0,2\right)$

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Find the x - and y - intercepts from an equation of a line

Recognizing that the x - intercept    occurs when y is zero and that the y - intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x - intercept, let $y=0$ and solve for x . To find the y - intercept , let $x=0$ and solve for y .