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Math unit28 straight lines

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eLearningJa
eLearningJa

This document covers straight lines and their properties including positive and negative gradient, the relationship between perpendicular lines, using graphs to determine speed and distance, and finding equations of lines. It provides examples of calculating gradient, determining if two lines are perpendicular, finding speed from a distance-time graph, and deriving equations of lines given points or being parallel/perpendicular to another line. The content builds understanding of key concepts involving straight lines through worked examples.

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Unit 28 Straight Lines Presentation 1 Positive and Negative Gradient Presentation 2 Gradients of Perpendicular Lines Presentation 3 Application of Graphs Presentation 4 Equation of Straight Line
Unit 28 28.1 Positive and Negative Gradient
A B Example Find the gradient of the line shown opposite. Solution -4 -3 -2 -1 0 1 2 3 5 4 3 2 1 -1 -2 -3 -4 -5 y x ? ? ? ? Vertical change = 10 Horizontal change = 6 x y
A B Example Find the gradient of the line shown opposite. Solution x y -2 -1 0 1 2 3 4 5 4 3 2 1 A B (-2, 4) (4, 1) ? ? ? ? ? x y
Unit 28 28.2 Gradient of Perpendicular Lines
? If two lines are perpendicular to one another, then the product of the two gradients is equal to -1. So if is the gradient of one line , the other line has a gradient of Example Show that the line segment joining the points A(3, 2) and B(5, 7) is perpendicular to the line segment joining the points P(2, 5) and Q(7, 3). Solution ? ?? ? ? ?? ? ??
Unit 28 28.3 Application of Graphs
Distance-time graph Distance Time The gradient gives the velocity. If the gradient is zero, the object is not moving. 2000 1500 1000 500 0 50 100 150 200 250 300 350 400 Example The graph shows the distance travelled by a girl on bicycle. Find the speed she is travelling on each stage of the journey Solution ? ? ? ? ? ? ? ? ? A B C D E Time (s) Distance(m)
Solution The distance is given by the area under the graph, which can be split into 3 sections A, B and C Velocity Time velocity-time graph The gradient gives the acceleration. If the gradient is zero, the object is moving at a constant velocity. The area under this graph is the distance travelled. 8 7 6 5 4 3 2 1 2 4 6 8 10 12 14 Example The graph shows how the speed of a bird varies as it flies between two trees. How far apart are the two trees?A B C18m 36m 6m ?? ? ???? ??? Time (s) Velocity(m/s)
Unit 28 28.4 Equation of a Straight Line
The equation a of a straight line is usually written in the form Where m is the gradient and c is the intercept. Example (a) (b) Equation of line AB: As it passes through (3, 1) and 10 8 6 4 2 -3 -2 -1 0 1 2 3 4 5 6 A B G x y ? ? ? ? ? (-1, 9) (3, 1) ? ?? ??? ? ?? O
10 8 6 4 2 -3 -2 -1 0 1 2 3 4 5 6 A B G x y (-1, 9) (3, 1) (c) Coordinates of G: so (d) Equation of line through O, the origin, perpendicular to AB: Equation: i.e. (e) Equation of line through O, parallel to AB Equation: i.e. ? ? ? O ? ? ? ? ? ? ??

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Math unit28 straight lines

  • 1. Unit 28 Straight Lines Presentation 1 Positive and Negative Gradient Presentation 2 Gradients of Perpendicular Lines Presentation 3 Application of Graphs Presentation 4 Equation of Straight Line
  • 2. Unit 28 28.1 Positive and Negative Gradient
  • 3. A B Example Find the gradient of the line shown opposite. Solution -4 -3 -2 -1 0 1 2 3 5 4 3 2 1 -1 -2 -3 -4 -5 y x ? ? ? ? Vertical change = 10 Horizontal change = 6 x y
  • 4. A B Example Find the gradient of the line shown opposite. Solution x y -2 -1 0 1 2 3 4 5 4 3 2 1 A B (-2, 4) (4, 1) ? ? ? ? ? x y
  • 5. Unit 28 28.2 Gradient of Perpendicular Lines
  • 6. ? If two lines are perpendicular to one another, then the product of the two gradients is equal to -1. So if is the gradient of one line , the other line has a gradient of Example Show that the line segment joining the points A(3, 2) and B(5, 7) is perpendicular to the line segment joining the points P(2, 5) and Q(7, 3). Solution ? ?? ? ? ?? ? ??
  • 8. Distance-time graph Distance Time The gradient gives the velocity. If the gradient is zero, the object is not moving. 2000 1500 1000 500 0 50 100 150 200 250 300 350 400 Example The graph shows the distance travelled by a girl on bicycle. Find the speed she is travelling on each stage of the journey Solution ? ? ? ? ? ? ? ? ? A B C D E Time (s) Distance(m)
  • 9. Solution The distance is given by the area under the graph, which can be split into 3 sections A, B and C Velocity Time velocity-time graph The gradient gives the acceleration. If the gradient is zero, the object is moving at a constant velocity. The area under this graph is the distance travelled. 8 7 6 5 4 3 2 1 2 4 6 8 10 12 14 Example The graph shows how the speed of a bird varies as it flies between two trees. How far apart are the two trees?A B C18m 36m 6m ?? ? ???? ??? Time (s) Velocity(m/s)
  • 10. Unit 28 28.4 Equation of a Straight Line
  • 11. The equation a of a straight line is usually written in the form Where m is the gradient and c is the intercept. Example (a) (b) Equation of line AB: As it passes through (3, 1) and 10 8 6 4 2 -3 -2 -1 0 1 2 3 4 5 6 A B G x y ? ? ? ? ? (-1, 9) (3, 1) ? ?? ??? ? ?? O
  • 12. 10 8 6 4 2 -3 -2 -1 0 1 2 3 4 5 6 A B G x y (-1, 9) (3, 1) (c) Coordinates of G: so (d) Equation of line through O, the origin, perpendicular to AB: Equation: i.e. (e) Equation of line through O, parallel to AB Equation: i.e. ? ? ? O ? ? ? ? ? ? ??