It can be shown, using theorems for similar triangles, that the slope slope is independent of the choice of points on the line. As the figure shows, a line with a positive slope goes up from left to right, but a line with a Positive slope negative slope goes down from left to right. (An attempt to use theĭefinition of slope here would produce a zero denominator.)ĭrawing a graph through the points in Example 4(c) would produce a line that is horizontal, which suggests the following generalization.įigure 3.12 shows lines of various slopes. As mentioned above, the slope of a vertical line is not defined.

Notice that the domain of this linear relation is (-inf,inf) but the range is x)=-11/6Ī sketch would show that the line through (2, 7) and (2, -4) is vertical. The only way a straight line can have no x-intercept is for it to be parallel to the x-axis, as shown in Figure 3.8. This means that the graph has no x-intercept. Since y always equals -3, the value of y can never be 0. Click on 'Solve Similar' button to see more examples.Įxample 2 GRAPHING HORIZONTAL AND VERTICAL LINES Let’s see how our math solver generates graphs of this and similar problems. A third point could be found as a check if desired. Plotting (0,3) and (2,0) gives the graph in Figure 3.7. Similarly, a y-intercept is found by setting x=0 in the equation and solving for y.Įxample 1 GRAPHING A LINEAR RELATION USING INTERCEPTS Since y = 0 on the x-axis, an x-intercept is found by setting y equal to 0 in the equation and solving for x. A y-intercept is a y-value at which a graph crosses the y-axis. An x-intercept is an x-value at which a graph crosses the x-axis. Two points that are especially useful for sketching the graph of a line are found with the intercepts. The graph of a linear relation can be found by plotting at least two points. In the equation Ax + By = C, any number can be used for x or y, so both the domain and range of a linear relation in which neither Anor B is 0are the set of real numbers (-inf,inf), This is called the standard form of a linear relation. Note Linear relations are often written in the form Ax + By = C , where A, B, and C are real, and A and B are not both 0. LINEAR RELATION A linear relation in two variables is a relation that can be written in the form Examples of linear relations are y=2x+3 , y=x and 3x + 2y = 6 Every linear relation has a graph that is a straight line, and so we need only find two points on the graph in order to sketch it. In this section we examine one of the simplest types of relations, the linear relation.

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