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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 2.4 Equations of Lines and Linear Models

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Objectives Point-Slope Form Horizontal and Vertical Lines Parallel and Perpendicular Lines

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. The line with slope m passing through the point (x 1, y 1 ) is given by y = m(x – x 1 ) + y 1 or equivalently, y – y 1 = m(x – x 1 ), the point-slope form of a line. POINT-SLOPE FORM

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line? Solution Let m = 2 and (x 1, y 1 ) = (3,1) in the point-slope form. To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y. y – y 1 = m(x – x 1 ) y 1 = 2(x – 3) 3 – 1 ? 2(4 – 3) 2 = 2 The point (4, 3) lies on the line because it satisfies the point-slope form.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Use the point-slope form to find an equation of the line passing through the points (2, 3) and (2, 5). Solution Before we can apply the point-slope form, we must find the slope.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) We can use either (2, 3) or (2, 5) for (x 1, y 1 ) in the point- slope form. If we choose (2, 3), the point-slope form becomes the following. If we choose (2, 5), the point-slope form with x 1 = 2 and y 1 = 5 becomes y – y 1 = m(x – x 1 )

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Find an equation of the line passing through (4, 2) and (1, 5). Solution First find the slope of the line.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Find an equation of the line passing through (4, 2) and (1, 5). Solution Now substitute 1 for m and (4, 2) for x and y in the slope intercept form. The point (1, 5) could be used instead. The slope-intercept form is y = x + 6.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Find a line y = mx + b that models the data in the table. Solution Carefully make a scatter plot of the data and sketch a line. x12345 f(x)f(x)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Find a line y = mx + b that models the data in the table. Solution Chose two points on the line and find the equation. The equation of the line is y = 6(x – 3) + 18 or y = 6x. x12345 f(x)f(x)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. The equation of a horizontal line with y-intercept b is y = b. The equation of a vertical line with x-intercept h is x = h. EQUATIONS OF HORIZONTAL AND VERTICAL LINES

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Find equations of the vertical and horizontal lines that pass through the point (5, 2). Graph these two lines. Solution The x-coordinate of the point (5, 2) is 5. The vertical line x = 5 passes through every point on the xy-plane including the point (5, 2). Similarly, the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (5, 2).

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Find equations of the vertical and horizontal lines that pass through the point (5, 2). Graph these two lines. Solution x = 5 y = 2

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Two lines with the same slope are parallel. Two nonvertical parallel lines have the same slope. PARALLEL LINES

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2, 1). Solution The line has a slope of 3 any parallel line also has slope 3.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Two lines with nonzero slopes m 1 and m 2 are perpendicular if m 1 m 2 = 1. If two lines have slopes m 1 and m 2 such that m 1 · m 2 = 1, then they are perpendicular. A vertical and horizontal line are perpendicular. PERPENDICULAR LINES

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Find the slope-intercept form of the line perpendicular to y = x – 3 passing through the point (4, 6). Solution The line has slope m 1 = 1. The slope of the perpendicular line is m 2 = 1. The slope-intercept form of a line having slope 1 and passing through (4, 6) can be found as follows.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Find the slope-intercept form of each line shown. Verify that the two lines are perpendicular. Solution The graph of y 1 has the slope m 1 = 3/2 because the line rises 3 units for every 2 units of run. Its y-intercept is 1/2, and its slope intercept form is The graph of y 2 has the slope m 2 = 2/3 because the line falls 2 units for every 3 units of run. Its y-intercept is 1/3, and its slope intercept form is y2y2 y1y1

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Find the slope-intercept form of each line shown. Verify that the two lines are perpendicular. Solution The graph of y 2 has the slope m 2 = 2/3 because the line falls 2 units for every 3 units of run. Its y-intercept is 1/3, and its slope intercept form is y2y2 y1y1

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