11.3 Graphing with intercepts

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By the end of this section, you will be able to:
• Identify the intercepts on a graph
• Find the intercepts from an equation of a line
• Graph a line using the intercepts
• Choose the most convenient method to graph a line

Before you get started, take this readiness quiz.

1. Solve: $3x+4y=-12$ for $x$ when $y=0.$
If you missed this problem, review Solve Equations using the Division and Multiplication Properties of Equality .
2. Is the point $\left(0,-5\right)$ on the $x\text{-axis}$ or $y\text{-axis?}$
If you missed this problem, review Use the Rectangular Coordinate System .
3. Which ordered pairs are solutions to the equation $2x-y=6?$
$\phantom{\rule{0.2em}{0ex}}\left(6,0\right)\phantom{\rule{0.2em}{0ex}}$ $\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\left(0,-6\right)\phantom{\rule{0.2em}{0ex}}$ $\phantom{\rule{0.2em}{0ex}}\left(4,-2\right).$
If you missed this problem, review Use the Rectangular Coordinate System .

Identify the intercepts on a graph

Every linear equation    has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.

At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an intercept of the line .

Intercepts of a line

Each of the points at which a line crosses the $x\text{-axis}$ and the $y\text{-axis}$ is called an intercept of the line.

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

First, notice where each of these lines crosses the x - axis:

Figure: The line crosses the x-axis at: Ordered pair of this point
42 3 (3,0)
43 4 (4,0)
44 5 (5,0)
45 0 (0,0)

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 the line. The x- intercept occurs when y is zero.

Now, let's look at the points where these lines cross the y-axis.

Figure: The line crosses the y-axis at: Ordered pair for this point
42 6 (0,6)
43 -3 (0,-3)
44 -5 (0,-5)
45 0 (0,0)

x- Intercept and y- Intercept of a line

The $x\text{-intercept}$ is the point, $\left(a,0\right),$ where the graph crosses the $x\text{-axis}.$ The $x\text{-intercept}$ occurs when $\text{y}$ is zero.

The $y\text{-intercept}$ is the point, $\left(0,b\right),$ where the graph crosses the $y\text{-axis}.$

The $y\text{-intercept}$ occurs when $\text{x}$ is zero.

• The x-intercept occurs when y is zero.
• The y-intercept occurs when x is zero.

Find the $x\text{- and}\phantom{\rule{0.2em}{0ex}}y\text{-intercepts}$ of each line:

$\phantom{\rule{0.2em}{0ex}}x+2y=4$

$\phantom{\rule{0.2em}{0ex}}3x-y=6$

$\phantom{\rule{0.2em}{0ex}}x+y=-5$

Solution

 ⓐ The graph crosses the x -axis at the point (4, 0). The x -intercept is (4, 0). The graph crosses the y -axis at the point (0, 2). The x -intercept is (0, 2).
 ⓑ The graph crosses the x -axis at the point (2, 0). The x -intercept is (2, 0) The graph crosses the y -axis at the point (0, −6). The y -intercept is (0, −6).
 ⓒ The graph crosses the x -axis at the point (−5, 0). The x -intercept is (−5, 0). The graph crosses the y -axis at the point (0, −5). The y -intercept is (0, −5).

Find the $x\text{-}$ and $y\text{-intercepts}$ of the graph: $x-y=2.$

x -intercept (2,0): y -intercept (0,−2)

Find the $x\text{-}$ and $y\text{-intercepts}$ of the graph: $2x+3y=6.$

x -intercept (3,0); y -intercept (0,2)

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