240 CHAPTER 2 Graphs and Functions The graph of this function has just one intercept—the origin 10, 02. We need to find an additional point to graph the function by choosing a different value for x (or y). 4x - 5y = 0 4152 - 5y = 0 We choose x = 5. 20 - 5y = 0 Multiply. 4 = y Add 5y. Divide by 5. This leads to the ordered pair 15, 42. Complete the graph using the two points 10, 02 and 15, 42, with a third point as a check. The domain and range are both 1-∞, ∞2. See Figure 36. S Now Try Exercise 23. EXAMPLE 4 Graphing Ax +By =C 1C =02 Graph 4x - 5y = 0. Give the domain and range. SOLUTION Find the intercepts. 4x - 5y = 0 4102 - 5y = 0 Let x = 0. y = 0 The y-intercept is 10, 02. 4x - 5y = 0 4x - 5102 = 0 Let y = 0. x = 0 The x-intercept is 10, 02. x y (0, 0) (5, 4) 4x – 5y = 0 4 5 (3, ) Check point 12 5 Figure 36 −10 −10 10 10 Figure 37 To use a graphing calculator to graph a linear function, as in Figure 37, we must first solve the defining equation for y. 4x - 5y = 0 Equation from Example 4 -5y = -4x Subtract 4x. y = 4 5 x Divide by -5. 7 x y 0 (x2, y2) (x2, y1) ∆y = y2 – y1 ∆x = x2 – x1 (x1, y1) Slope = (∆x ≠ 0) = rise run run rise ∆y ∆x = y2 – y1 x2 – x1 Figure 38 Slope Slope is a numerical measure of the steepness and orientation of a straight line. (Geometrically, this may be interpreted as the ratio of rise to run.) The slope of a highway (sometimes called the grade) is often given as a percent. For example, a 10% Aor 10 100 = 1 10B slope means the highway rises 1 unit for every 10 horizontal units. To find the slope of a line, start with two distinct points 1x1, y12 and 1x2, y22 on the line, as shown in Figure 38, where x1 ≠x2. As we move along the line from 1x1, y12 to 1x2, y22, the horizontal difference x =x2 −x1 is the change in x, denoted by x (read “delta x”), where Δ is the Greek capital letter delta. The vertical difference, the change in y, can be written y =y2 −y1. The slope of a nonvertical line is defined as the quotient (ratio) of the change in y and the change in x.
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