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**Unformatted text preview: **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 ,y ) with slope = m . This situation is pictured in Figure 1 . x y P ( x ,y ) Q ( x,y ) Figure 1. A line through ( x ,y ) 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 x − x . Because the slope equals m , we can set Slope = m in this last result to obtain m = y − y x − x . If we multiply both sides of this last equation by x − x , we get m ( x − x ) = y − y , or exchanging sides of this last equation, y − y = m ( x − x ) . This last result is the equation of the line. Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 294 Chapter 3 Linear Functions Version: Fall 2007 The Point-Slope Form of a Line . If line L passes through the point ( x ,y ) and has slope m , then the equation of the line is y − y = m ( x − x ) . (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 = m ( x − x ). 2. Substitute the coordinates of the given point for x and y in the formula y − y = m ( x − x ). For example, if the line has slope − 2 and passes through the point (3 , 4), then substitute m = − 2, x = 3, and y = 4 in the formula y − y = m ( x − x ) to obtain y − 4 = − 2( x − 3) . ⚏ 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 ,- . 5) (a) The line through P ( − 3 , − 2) with slope m = 1 / 2. (b) Checking the y-intercept. Figure 2. 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 = m ( x − x ) . (3) The slope of the line is m = 1 / 2 and the given point is P ( − 3 , − 2), so ( x ,y ) = ( − 3 , − 2). In equation (3) , set m = 1 / 2, x = − 3, and y = −...

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