36 CHAPTER 1 Graphs Figure 56 x y 5 21 (3, 3) (3, 2) (3, 1) (3, 21) 4 21 (a) (3, 0) 8 28 25 (b) (c) 5 x 5 3 x 5 3 4 Use the Point–Slope Form of a Line; Identify Horizontal Lines Let L be a nonvertical line with slope m that contains the point ( ) x y , . 1 1 See Figure 57. For any other point ( ) x y , on L , we have ( ) = − − − = − m y y x x y y m x x or 1 1 1 1 Figure 57 x y (x, y) L (x1, y1) x – x1 y – y1 THEOREM Point–Slope Form of an Equation of a Line An equation of a nonvertical line with slope m that contains the point ( ) x y , 1 1 is ( ) − = − y y m x x 1 1 (2) Example 3 suggests the following result: THEOREM Equation of a Vertical Line A vertical line is given by an equation of the form = x a where a is the x -intercept. 3 Find the Equation of a Vertical Line Graphing a Line Graph the equation: = x 3 Solution EXAMPLE 3 To graph = x 3, we find all points ( ) x y , in the plane for which = x 3. No matter what y -coordinate is used, the corresponding x -coordinate always equals 3. Consequently, the graph of the equation = x 3 is a vertical line with x -intercept 3 and an undefined slope. See Figure 56(a). For an equation to be graphed using some graphing utilities, the equation must be expressed in the form { } = y x expression in . But = x 3 cannot be put into this form. To overcome this, some graphing utilities have special commands for drawing vertical lines. DRAW, LINE, PLOT, and VERT are among the more common ones. Consult your manual to determine the correct methodology for your graphing utility. Figure 56(b) shows the graph on a TI-84 Plus CE. Figure 56(c) shows the graph using Desmos.
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