Slope of a Line 11-2 Warm Up Problem of the Day Lesson Presentation Pre-Algebra

Pre-Algebra 11-2 Slope of a Line Warm Up Evaluate each equation for x = –1, 0, and 1. 1. y = 3x 2. y = x – 7 3. y = 2x + 5 4. y = 6x – 2 –3, 0, 3 –8, –7, –6 3, 5, 7 –8, –2, 4

Learn to find the slope of a line and use slope to understand and draw graphs.

You looked at slope on the coordinate plane in Lesson 5-5 (p. 244). Remember!

vertical change horizontal change Linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio: vertical change horizontal change change in y change in x = This ratio is often referred to as , or “rise over run,” where rise indicates the number of units moved up or down and run indicates the number of units moved to the left or right. Slope can be positive, negative, zero, or undefined. A line with positive slope goes up from left to right. A line with negative slope goes down from left to right. rise run

If you know any two points on a line, or two solutions of a linear equation, you can find the slope of the line without graphing. The slope of a line through the points (x1, y1) and (x2, y2) is as follows: y2 – y1 x2 – x1

Additional Example 1: Finding Slope, Given Two Points Find the slope of the line that passes through (–2, –3) and (4, 6). Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6). = y2 – y1 x2 – x1 6 – (–3) 4 – (–2) Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1. 9 6 = 3 2 = The slope of the line that passes through (–2, –3) and (4, 6) is . 3 2

Try This: Example 1 Find the slope of the line that passes through (–4, –6) and (2, 3). Let (x1, y1) be (–4, –6) and (x2, y2) be (2, 3). = y2 – y1 x2 – x1 3 – (–6) 2 – (–4) Substitute 3 for y2, –6 for y1, 2 for x2, and –4 for x1. 9 6 = 3 2 = The slope of the line that passes through (–4, –6) and (2, 3) is . 3 2

Additional Example 2: Finding Slope from a Graph Use the graph of the line to determine its slope.

Additional Example 2 Continued Choose two points on the line: (0, 1) and (3, –4). Guess by looking at the graph: rise run = –5 3 = – 5 Use the slope formula. Let (3, –4) be (x1, y1) and (0, 1) be (x2, y2). –5 = y2 – y1 x2 – x1 1 – (–4) 0 – 3 3 5 –3 = 5 3 = –

Additional Example 2 Continued Notice that if you switch (x1, y1) and (x2, y2), you get the same slope: Let (0, 1) be (x1, y1) and (3, –4) be (x2, y2). = y2 – y1 x2 – x1 –4 – 1 3 – 0 –5 3 = 5 3 = – 5 3 The slope of the given line is – .

Try This: Example 2 Use the graph of the line to determine its slope.

Try This: Example 2 Continued Choose two points on the line: (1, 1) and (0, –1). Guess by looking at the graph: rise run = 2 1 = 2 1 Use the slope formula. 2 Let (1, 1) be (x1, y1) and (0, –1) be (x2, y2). = y2 – y1 x2 – x1 –1 – 1 0 – 1 –2 –1 = = 2

Recall that two parallel lines have the same slope Recall that two parallel lines have the same slope. The slopes of two perpendicular lines are negative reciprocals of each other.

Additional Example 3A: Identifying Parallel and Perpendicular Lines by Slope Tell whether the lines passing through the given points are parallel or perpendicular. A. line 1: (–6, 4) and (2, –5); line 2: (–1, –4) and (8, 4) = y2 – y1 x2 – x1 –5 – 4 2 – (–6) –9 8 = 9 8 = – slope of line 1: = y2 – y1 x2 – x1 4 – (–4) 8 – (–1) 8 9 = slope of line 2: Line 1 has a slope equal to – and line 2 has a slope equal to , – and are negative reciprocals of each other, so the lines are perpendicular. 9 8

Additional Example 3B: Identifying Parallel and Perpendicular Lines by Slope B. line 1: (0, 5) and (6, –2); line 2: (–1, 3) and (5, –4) = y2 – y1 x2 – x1 –2 – 5 6 – 0 –7 6 = 7 6 = – slope of line 1: = y2 – y1 x2 – x1 –4 – 3 5 – (–1) –7 6 = 7 6 = – slope of line 2: Both lines have a slope equal to – , so the lines are parallel. 7 6

Try This: Example 3A Tell whether the lines passing through the given points are parallel or perpendicular. A. line 1: (–8, 2) and (0, –7); line 2: (–3, –6) and (6, 2) = y2 – y1 x2 – x1 –7 – 2 0 – (–8) –9 8 = 9 8 = – slope of line 1: = y2 – y1 x2 – x1 2 – (–6) 6 – (–3) 8 9 = slope of line 2: Line 1 has a slope equal to – and line 2 has a slope equal to , – and are negative reciprocals of each other, so the lines are perpendicular. 9 8

Try This: Example 3B B. line 1: (1, 1) and (2, 2); line 2: (1, –2) and (2, -1) = y2 – y1 x2 – x1 2 – 1 1 = slope of line 1: = 1 = y2 – y1 x2 – x1 –1 – (–2) 2 – (1) –1 1 = slope of line 2: = –1 Line 1 has a slope equal to 1 and line 2 has a slope equal to –1. 1 and –1 are negative reciprocals of each other, so the lines are perpendicular.

Additional Example 4: Graphing a Line Using a Point and the Slope Graph the line passing through (3, 1) with slope 2. The slope is 2, or . So for every 2 units up, you will move right 1 unit, and for every 2 units down, you will move left 1 unit. 2 1 Plot the point (3, 1). Then move 2 units up and right 1 unit and plot the point (4, 3). Use a straightedge to connect the two points.

Additional Example 4 Continued 1 2 (3, 1)

Try This: Example 4 Graph the line passing through (1, 1) with slope 2. The slope is 2, or . So for every 2 units up, you will move right 1 unit, and for every 2 units down, you will move left 1 unit. 2 1 Plot the point (1, 1). Then move 2 units up and right 1 unit and plot the point (2, 3). Use a straightedge to connect the two points.

Try This: Example 4 Continued 1 2 (1, 1)

Lesson Quiz: Part 1 Find the slope of the line passing through each pair of points. 1. (4, 3) and (–1, 1) 2. (–1, 5) and (4, 2) 3. Use the graph of the line to determine its slope. 2 5 5 3 – 3 4 –

Lesson Quiz: Part 2 Tell whether the lines passing through the given points are parallel or perpendicular. 4. line 1: (–2, 1), (2, –1); line 2: (0, 0), (–1, –2) 5. line 1: (–3, 1), (–2, 3); line 2: (2, 1), (0, –3) perpendicular parallel