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Point-Slope Form and Writing Linear Equations LESSON 5-4 Lesson Quiz 2 3 1. Graph the equation y + 1 = –( x – 3). 2. Write an equation of the line with slope – that passes through the point (0, 4). 3. Write an equation for the line that passes through (3, –5) and (–2, 1) in point-slope form and slope-intercept form. 4. Is the relationship shown by the data linear? If so, model that data with an equation. – 10 – 7 0 5 20 – 3 – 1 5 x y 2 5 yes; y + 3 = ( x – 0) 6 5 6 5 7 5 y + 5 = – ( x – 3); y = – x – y – 4 = – ( x – 0), or y = – x + 4 2 3 2 3
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5-5 Parallel and Perpendicular Lines California Content Standards 7.0 Derive linear equations by using the point-slope formula. Master 8.0 Understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Find the equation of a line perpendicular to a given line that passes through a given point. Develop Master Page 86
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What You’ll Learn…And Why <ul><li>To determine whether lines are parallel </li></ul><ul><li>To determine whether lines are perpendicular </li></ul><ul><li>To use parallel and perpendicular lines to plan a bike path. </li></ul>
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Slopes of Parallel Lines <ul><li>Nonvertical lines are parallel if </li></ul><ul><li>Any two vertical lines are parallel. </li></ul><ul><li>Example the equations </li></ul><ul><li>Have the same slope, , and different y-intercept. The graphs of the two equations are parallel. </li></ul>They have the same slope and different y-intercepts
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Slopes of Perpendicular Lines <ul><li>Two lines are perpendicular if the product of their slopes is </li></ul><ul><li>A vertical and horizontal line are also perpendicular. </li></ul><ul><li>Example The slope of </li></ul><ul><li>The slope of y = 4x + 2 is 4. Since </li></ul><ul><li>The graphs of the two equations are perpendicular. </li></ul>- 1
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Vocabulary <ul><li>Parallel lines are lines in the same plane that never intersect. Equations have the same slope, different y-intercepts. </li></ul><ul><li>Perpendicular lines are lines that intersect to form right angles. Equations have the opposite and reciprocal slopes. </li></ul><ul><li>The product of a number and its negative reciprocal is -1. </li></ul>
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Write an equation for the line that contains (–2, 3) and is parallel to y = x – 4. 5 2 5 2 Step 1 Identify the slope of the given line. y = x – 4 slope
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Write an equation for the line that contains (–2, 3) and is parallel to y = x – 4. Step 2 Write the equation of the line through (–2, 3) using slope-intercept form. (Continued) y – y 1 = m ( x – x 1 ) Use point-slope form. y – 3 = x + 5 Simplify. 5 2 y = x + 8 Add 3 to each side and simplify. 5 2 y – 3 = ( x + 2 ) Substitute (2,3) for ( x 1 , y 1 ) and for m . 5 2 5 2 5 2
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The line in the graph represents the street in front of a new house. The point is the front door. The sidewalk from the front door will be perpendicular to the street. Write an equation representing the sidewalk. Step 1 Find the slope m of the street. Step 2 Find the negative reciprocal of the slope. m = = = = – Points (0, 2) and (3, 0) are on the street. y 2 – y 1 x 2 – x 1 0 – 2 3 – 0 – 2 3 2 3 The equation for the sidewalk is y = x – 3. 3 2 The negative reciprocal of – is . So the slope of the sidewalk is . The y- intercept is –3. 2 3 3 2 3 2
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CA Standards Check <ul><li>1) Write an equation for the line that contains (2,-6) and is parallel to y=3x +9. </li></ul>Parallel lines: same slope, different y-intercepts. Point Slope y – y 1 = m (x – x 1 ) - 6 2 3 y + 6 = 3(x – 2 ) y + 6 = 3 x – 6 – 6 – 6 y = 3 x – 12
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CA Standards Check <ul><li>2a) Write an equation for the line that contains (1,8) and is perpendicular to y=3/4x + 1. </li></ul>Perpendicular lines: Opposite and reciprecal slopes. Point Slope y – y 1 = m (x – x 1 ) 8 1 24 y – 8 = (x – 1 ) y – 8 = x + y = x + 8 = 3 + 8 + .
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CA Standards Check <ul><li>2b) Using the diagram in Ex 2, rite an equation in slope-intercept form for a new sidewalk perpendicular to the street from a front door at (-1, -2). </li></ul>Point Slope y – y 1 = m (x – x 1 ) (-1) (-2) -3 y + 1 = (x + 2 ) -1 = 3 y = x - 3 y + 1 = x - - 1 - . y = x -
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Closure: Compare the equations of non-vertical parallel line, and perpendicular lines. Parallel lines have the same slope, but different y-intercepts. Perpendicular lines have slopes that are negative reciprocals of each other.
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