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Learn to Graph Lines and Understand Slope

This lesson presentation and homework assignment will help students learn to graph lines using linear equations and understand the concept of slope. They will also learn how to use slopes to graph linear equations and recognize direct variation and inequalities.

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Learn to Graph Lines and Understand Slope

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  1. Slope of a Line 11-2 Homework & Learning Goal Lesson Presentation Pre-Algebra

  2. HOMEWORK answers Page 543 #1-11

  3. Pre-Algebra HOMEWORK Page 548 #1-5

  4. Our Learning Goal Students will be able to graph linesusing linear equations, understand the slope of a line and graph inequalities.

  5. Our Learning Goal Assignments • Learn to identify and graph linear equations. • Learn to find the slope of a line and use slope to understand and draw graphs. • Learn to use slopes and intercepts to graph linear equations. • Learn to find the equation of a line given one point and the slope. • Learn to recognize direct variation by graphing tables of data and checking for constant ratios. • Learn to graph inequalities on the coordinate plane. • Learn to recognize relationships in data and find the equation of a line of best fit.

  6. Today’s Learning Goal Assignment Learn to find the slope of a line and use slope to understand and draw graphs.

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

  8. rise run vertical changehorizontal change change in ychange 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. Linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio:

  9. y2–y1 x2–x1 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:

  10. y2 – y1 = x2 – x1 6 – (–3) 3 9 4 – (–2) 3 The slope of the line that passes through (–2, –3) and (4, 6) is . 6 2 2 = = 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). Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1.

  11. y2 – y1 = x2 – x1 3 – (–6) 3 9 2 – (–4) 3 The slope of the line that passes through (–4, –6) and (2, 3) is . 6 2 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). Substitute 3 for y2, –6 for y1, 2 for x2, and –4 for x1.

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

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

  14. y2 – y1 = x2 – x1 The slope of the given line is – . –4 – 1 5 –5 3 – 0 3 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).

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

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

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

  18. y2 – y1 y2 – y1 = = x2 – x1 x2 – x1 9 9 8 8 8 8 4 – (–4) –5 – 4 9 9 9 –9 8 2 – (–6) 8 – (–1) 8 8 9 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. = = = – 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) slope of line 1: slope of line 2:

  19. y2 – y1 y2 – y1 = = x2 – x1 x2 – x1 7 –2 – 5 –4 – 3 6 7 –7 7 –7 5 – (–1) 6 – 0 6 6 6 6 Both lines have a slope equal to – , so the lines are parallel. = = = – = – 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) slope of line 1: slope of line 2:

  20. y2 – y1 y2 – y1 = = x2 – x1 x2 – x1 9 9 8 8 8 8 2 – (–6) –7 – 2 9 9 9 –9 8 0 – (–8) 6 – (–3) 8 8 9 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. = = = – 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) slope of line 1: slope of line 2:

  21. y2 – y1 y2 – y1 = = x2 – x1 x2 – x1 –1 – (–2) 2 – (1) 2 – 1 1 –1 2 – 1 1 1 = = Try This: Example 3B B. line 1: (1, 1) and (2, 2); line 2: (1, –2) and (2, -1) = 1 slope of line 1: = –1 slope of line 2: 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.

  22. 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 Additional Example 4: Graphing a Line Using a Point and the Slope Graph the line passing through (3, 1) with slope 2. 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.

  23. Additional Example 4 Continued 1 2 (3, 1)

  24. 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 Try This: Example 4 Graph the line passing through (1, 1) with slope 2. 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.

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

  26. 2 5 3 4 – – 5 3 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.

  27. 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

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