243 2.4 Linear Functions Because the slope of a line is the ratio of vertical change (rise) to horizontal change (run), if we know the slope of a line and the coordinates of a point on the line, we can draw the graph of the line. EXAMPLE 7 Graphing a Line Using a Point and the Slope Graph the line passing through the point 1-1, 52 and having slope - 5 3 . SOLUTION First locate the point 1-1, 52 as shown in Figure 39. The slope of this line is5 3 , so a vertical change of -5 units (that is, down 5 units) corresponds to a horizontal change of 3 units (that is, to the right 3 units). This gives a second point, 12, 02, which can then be used to complete the graph. Because -5 3 = 5 -3 , another point could be obtained by starting at 1-1, 52 and moving up 5 units and to the left 3 units. We would reach a different second point, 1-4, 102, but the graph would be the same line. See Figure 39. S Now Try Exercise 59. (–4, 10) –2 6 x y (2, 0) (–1, 5) Down 5 Left 3 Up 5 Right 3 Figure 39 Figure 40 shows lines with various slopes. Positive slope x y 0 x y Negative slope 0 x y Slope 0 Slopes of lines 0 y x Undefined slope 0 Figure 40 Notice the following important concepts. • A line with a positive slope rises from left to right. The corresponding linear function is increasing on its entire domain. • A line with a negative slope falls from left to right. The corresponding linear function is decreasing on its entire domain. • A line with slope 0 neither rises nor falls. The corresponding linear function is constant on its entire domain. • The slope of a vertical line is undefined. Average Rate of Change We know that the slope of a line is the ratio of the vertical change in y to the horizontal change in x. Thus, slope gives the average rate of change in y per unit of change in x, where the value of y depends on the value of x. If ƒ is a linear function defined on the interval 3a, b4, then we have the following. Average rate of change on 3 a, b4 = ƒ1b2 −ƒ1a2 b −a This is simply another way to write the slope formula, using function notation.
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