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Section 4: Point-Slope Form of a Line

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 3.4 The Point-Slope Form of a Line 293 Version: Fall 2007 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( y = mx + b ). The slope-intercept form of a line is applicable when you’re given the slope and y -intercept of the line. However, there will be times when the y -intercept is unknown. Suppose for example, that you are asked to find the equation of a line that passes through a particular point P ( x 0 , y 0 ) with slope = m . This situation is pictured in Figure 1 . x y P ( x 0 ,y 0 ) Q ( x,y ) Figure 1. A line through ( x 0 , y 0 ) with slope m . Let the point Q ( x, y ) be an arbitrary point on the line. We can determine the equation of the line by using the slope formula with points P and Q . Hence, Slope = y x = y y 0 x x 0 . Because the slope equals m , we can set Slope = m in this last result to obtain m = y y 0 x x 0 . If we multiply both sides of this last equation by x x 0 , we get m ( x x 0 ) = y y 0 , or exchanging sides of this last equation, y y 0 = m ( x x 0 ) . This last result is the equation of the line. Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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294 Chapter 3 Linear Functions Version: Fall 2007 The Point-Slope Form of a Line . If line L passes through the point ( x 0 , y 0 ) and has slope m , then the equation of the line is y y 0 = m ( x x 0 ) . (1) This form of the equation of a line is called the point-slope form . To use the point-slope form of a line, follow these steps. Procedure for Using the Point-Slope Form of a Line . When given the slope of a line and a point on the line, use the point-slope form as follows: 1. Substitute the given slope for m in the formula y y 0 = m ( x x 0 ) . 2. Substitute the coordinates of the given point for x 0 and y 0 in the formula y y 0 = m ( x x 0 ) . For example, if the line has slope 2 and passes through the point (3 , 4) , then substitute m = 2 , x 0 = 3 , and y 0 = 4 in the formula y y 0 = m ( x x 0 ) to obtain y 4 = 2( x 3) . l⚏ Example 2. Draw the line that passes through the point P ( 3 , 2) and has slope m = 1 / 2 . Use the point-slope form to determine the equation of the line. First, plot the point P ( 3 , 2) , as shown in Figure 2 (a). Starting from the point P ( 3 , 2) , move 2 units to the right and 1 unit up to the point Q ( 1 , 1) . The line through the points P and Q in Figure 2 (a) now has slope m = 1 / 2 . x y P ( - 3 , - 2) Q ( - 1 , - 1) Δ x =2 Δ y =1 x y R (0 , - 0 . 5) (a) The line through P ( 3 , 2) with slope m = 1 / 2 . (b) Checking the y -intercept. Figure 2.
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Section 3.4 The Point-Slope Form of a Line 295 Version: Fall 2007 To determine the equation of the line in Figure 2 (a), we will use the point-slope form of the line y y 0 = m ( x x 0 ) . (3) The slope of the line is m = 1 / 2 and the given point is P ( 3 , 2) , so ( x 0 , y 0 ) = ( 3 , 2) . In equation (3) , set m = 1 / 2 , x 0 = 3 , and y 0 = 2 , obtaining y ( 2) = 1 2 ( x ( 3)) , or equivalently, y + 2 = 1 2 ( x + 3) . (4) This is the equation of the line in Figure 2 (a).
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