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The graph of the line $y=b$ , where $b$ is a constant, is a horizontal line that passes through the point (0, $b$ ). Every point on this line has the y-coordinate $b$ , regardless of the x-coordinate.
Graph the lines: $x=-2$ , and $y=3$ .
The graph of the line $x=-2$ is a vertical line that has the x-coordinate –2 no matter what the y-coordinate is. Therefore, the graph is a vertical line passing through (–2, 0).
The graph of the line $y=3$ , is a horizontal line that has the y-coordinate 3 regardless of what the x-coordinate is. Therefore, the graph is a horizontal line that passes through (0, 3).
In this section, you will learn to:
In the last section, we learned to graph a line by choosing two points on the line. A graph of a line can also be determined if one point and the "steepness" of the line is known. The precise number that refers to the steepness or inclination of a line is called the slope of the line.
From previous math courses, many of you remember slope as the "rise over run," or "the vertical change over the horizontal change" and have often seen it expressed as:
We give a precise definition.
If ( ${x}_{1}$ , ${y}_{1}$ ) and ( ${x}_{2}$ , ${y}_{2}$ ) are two different points on a line, then the slope of the line is
$\text{Slope}=m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$
Find the slope of the line that passes through the points (-2, 3) and (4, -1), and graph the line.
Let $({x}_{1},{y}_{1})=(-\mathrm{2,3})$ and $({x}_{2},{y}_{2})=(\mathrm{4,}-1)$ then the slope
To give the reader a better understanding, both the vertical change, –4, and the horizontal change, 6, are shown in the above figure.
When two points are given, it does not matter which point is denoted as $({x}_{1},{y}_{1})$ and which $({x}_{2},{y}_{2})$ . The value for the slope will be the same. For example, if we choose $({x}_{1},{y}_{2})=(\mathrm{4,}-1)$ and $({x}_{2},{y}_{2})=(-\mathrm{2,3})$ , we will get the same value for the slope as we obtained earlier. The steps involved are as follows.
The student should further observe that if a line rises when going from left to right, then it has a positive slope; and if it falls going from left to right, it has a negative slope.
Find the slope of the line that passes through the points (2, 3) and (2, -1), and graph.
Let $({x}_{1},{y}_{1})=(\mathrm{2,3})$ and $({x}_{2},{y}_{2})=(\mathrm{2,}-1)$ then the slope
Graph the line that passes through the point (1, 2) and has slope $-\frac{3}{4}$ .
Slope equals $\frac{\text{rise}}{\text{run}}$ . The fact that the slope is $\frac{-3}{4}$ , means that for every rise of –3 units (fall of 3 units) there is a run of 4. So if from the given point (1, 2) we go down 3 units and go right 4 units, we reach the point (5, –1). The following graph is obtained by connecting these two points.
Alternatively, since $\frac{3}{-4}$ represents the same number, the line can be drawn by starting at the point (1,2) and choosing a rise of 3 units followed by a run of –4 units. So from the point (1, 2), we go up 3 units, and to the left 4, thus reaching the point (–3, 5) which is also on the same line. See figure below.
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