June 18th

June 18th

Graph of a Line

We can graph the linear equation defined by y = x + 1 by finding several ordered pairs. For example, if x = 2 then y = 2 + 1 = 3, giving the ordered pair (2, 3). Also, (0, 1), (4, 5), (-2, -1), (-5, -4), (-3, -2), among many others, are ordered pairs that satisfy the equation.

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To graph y = x + 1 we begin by locating the ordered pairs obtained above, as shown in Figure 6(a). All the points of this graph appear to lie on a straight line, as in Figure 6(b). This straight line is the graph of y = x + 1.

It can be shown that every equation of the form ax + by = c has a straight line as its graph. Although just two points are needed to determine a line, it is a good idea to plot a third point as a check. It is often convenient to use the x- and y-intercepts as the two points, as in the following example.

Example

Graph of a Line

Graph 3x + 2y = 12.

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Solution

To find the y -intercept, let x = 0.

Similarly, find the x-intercept by letting y = 0 which gives x = 4. Verify that when x = 2 the result is y = 3. These three points are plotted in Figure 7(a). A line is drawn through them in Figure 7(b).

Not every line has two distinct intercepts; the graph in the next example does not cross the x-axis, and so it has no x-intercept.

Example

Graph of a Horizontal Line

Graph y = -3.

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Solution

The equation y = -3 or equivalently y = 0x -3, always gives the same y -value, - 3, for any value of x . Therefore, no value of x will make y = 0, so the graph has no x -intercept. The graph of such an equation is a horizontal line parallel to the x -axis. In this case the y -intercept is - 3, as shown in Figure 8.

In general, the graph of y = k, where k is a real number, is the horizontal line having y-intercept k.

The graph in Example 13 had only one intercept. Another type of linear equation with coinciding intercepts is graphed in Example 14.

Example

Graph of a Line Through the Origin

Graph y = -3x.

Solution

Begin by looking for the x -intercept. If y = 0 then

We have the ordered pair (0, 0). Starting with x = 0 gives exactly the same ordered pair, (0, 0). Two points are needed to determine a straight line, and the intercepts have led to only one point. To get a second point, choose some other value of x (or y ). For example, if x = 2 then

y = -3x = -3(2) = -6, (let x = 2)

giving the ordered pair (2, -6). These two ordered pairs, (0, 0) and (2, -6), were used to get the graph shown in Figure 9.