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KS3 Mathematics

KS3 Mathematics. A5 Functions and graphs. A5.5 Graphs of functions. Coordinate pairs. When we write a coordinate, for example,. (3, 5). ( 3 , 5). ( 3 , 5 ). x -coordinate. y -coordinate. the first number is called the x -coordinate and the second number is called the y -coordinate.

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KS3 Mathematics

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  1. KS3 Mathematics A5 Functions and graphs

  2. A5.5 Graphs of functions

  3. Coordinate pairs When we write a coordinate, for example, (3, 5) (3, 5) (3, 5) x-coordinate y-coordinate the first number is called the x-coordinateand the second number is called the y-coordinate. the first number is called the x-coordinateand the second number is called the y-coordinate. Together, the x-coordinate and the y-coordinate are called a coordinate pair.

  4. Graphs parallel to the y-axis y x What do these coordinate pairs have in common? (2, 3), (2, 1), (2, –2), (2, 4), (2, 0) and (2, –3)? The x-coordinate in each pair is equal to 2. Look what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the y-axis. Name five other points that will lie on this line. This line is called x = 2. x = 2

  5. Graphs parallel to the y-axis y x All graphs of the form x = c, where c is any number, will be parallel to the y-axis and will cut the x-axis at the point (c, 0). x = –10 x = –3 x = 4 x = 9

  6. Graphs parallel to the x-axis y x What do these coordinate pairs have in common? (0, 1), (4, 1), (–2, 1), (2, 1), (1, 1) and (–3, 1)? The y-coordinate in each pair is equal to 1. Look what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the x-axis. y = 1 Name five other points that will lie on this line. This line is called y = 1.

  7. Graphs parallel to the x-axis y x All graphs of the form y = c, where c is any number, will be parallel to the x-axis and will cut the y-axis at the point (0, c). y = 5 y = 3 y = –2 y = –5

  8. Drawing graphs of functions The x-coordinate and the y-coordinate in a coordinate pair can be linked by a function. What do these coordinate pairs have in common? (1, 3), (4, 6), (–2, 0), (0, 2), (–1, 1) and (3.5, 5.5)? In each pair, the y-coordinate is 2 more than the x-coordinate. These coordinates are linked by the function: y = x + 2 We can draw a graph of the function y = x + 2 by plotting points that obey this function.

  9. Drawing graphs of functions x –3 –2 –1 0 1 2 3 y = x +3 Given a function, we can find coordinate points that obey the function by constructing a table of values. Suppose we want to plot points that obey the function y = x + 3 We can use a table as follows: 0 1 2 3 4 5 6 (–3, 0) (–2, 1) (–1, 2) (0, 3) (1, 4) (2, 5) (3, 6)

  10. Drawing graphs of functions x –3 –2 –1 0 1 2 3 y = x – 2 For example, to draw a graph of y = x– 2: y = x - 2 1) Complete a table of values: –5 –4 –3 –2 –1 0 1 2) Plot the points on a coordinate grid. 3) Draw a line through the points. 4) Label the line. 5) Check that other points on the line fit the rule.

  11. Drawing graphs of functions

  12. The equation of a straight line y = mx + c The general equation of a straight line can be written as: The value of m tells us the gradient of the line. The value of c tells us where the line crosses the y-axis. This is called the y-interceptand it has the coordinate (0, c). For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).

  13. Linear graphswith positive gradients

  14. Investigating straight-line graphs

  15. The gradient and the y-intercept 1 x 2 2 Complete this table: 3 (0, 4) (0, –5) (0, 2) –3 (0, 0) y = x y = –2x– 7 (0, –7)

  16. Rearranging equations into the form y = mx + c The equation of a straight line is 2y+x = 4. Find the gradient and the y-intercept of the line. 1 2 –x + 4 y = 2 y = –x + 2 Sometimes the equation of a straight line graph is not given in the form y = mx + c. We can rearrange the equation by transforming both sides in the same way 2y + x = 4 2y = –x + 4

  17. Rearranging equations into the form y = mx + c 1 1 y = –x + 2 2 2 – Sometimes the equation of a straight line graph is not given in the form y = mx + c. The equation of a straight line is 2y+x = 4. Find the gradient and the y-intercept of the line. Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept. So the gradient of the line is 2. and the y-intercept is

  18. What is the equation? 10 A G H 5 B F D C -5 0 5 10 E What is the equation of the line passing through the points Look at this diagram: a) A and E x = 2 b) A and F y = x + 6 y = x – 2 c) B and E d) C and D y = 2 e) E and G y = 2 –x f) A and C? y = 10 –x

  19. Substituting values into equations A line with the equation y = mx + 5 passes through the point (3, 11). What is the value of m? To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5. This gives us: 11 = 3m + 5 Subtracting 5: 6 = 3m Dividing by 3: 2 = m m = 2 The equation of the line is therefore y = 2x + 5.

  20. Pairs

  21. Matching statements

  22. Exploring gradients

  23. Gradients of straight-line graphs m = change in y change in x y2 – y1 m = x2 – x1 The gradient of a line is a measure of how steep a line is. The gradient of a straight line y = mx + c is given by For any two points on a straight line, (x1, y1) and (x2, y2)

  24. A8 Linear and real-life graphs Contents • A A8.2 Gradients and intercepts • A A8.3 Parallel and perpendicular lines • A A8.1 Linear graphs A8.4 Interpreting real-life graphs • A A8.5 Distance-time graphs • A A8.6 Speed-time graphs • A

  25. Coordinate pairs When we write a coordinate, for example, (3, 5) (3, 5) (3, 5) x-coordinate y-coordinate the first number is called the x-coordinateand the second number is called the y-coordinate. the first number is called the x-coordinateand the second number is the y-coordinate. Together, the x-coordinate and the y-coordinate are called a coordinate pair.

  26. Graphs parallel to the y-axis y x What do these coordinate pairs have in common? (2, 3), (2, 1), (2, –2), (2, 4), (2, 0) and (2, –3)? The x-coordinate in each pair is equal to 2. Look what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the y-axis. Name five other points that will lie on this line. This line is called x = 2. x = 2

  27. Graphs parallel to the y-axis y x All graphs of the form x = c, where c is any number, will be parallel to the y-axis and will cut the x-axis at the point (c, 0). x = –10 x = –3 x = 4 x = 9

  28. Graphs parallel to the x-axis y x What do these coordinate pairs have in common? (0, 1), (4, 1), (–2, 1), (2, 1), (1, 1) and (–3, 1)? The y-coordinate in each pair is equal to 1. Look at what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the x-axis. y = 1 Name five other points that will lie on this line. This line is called y = 1.

  29. Graphs parallel to the x-axis y x All graphs of the form y = c, where c is any number, will be parallel to the x-axis and will cut the y-axis at the point (0, c). y = 5 y = 3 y = –2 y = –5

  30. Plotting graphs of linear functions The x-coordinate and the y-coordinate in a coordinate pair can be linked by a function. What do these coordinate pairs have in common? (1, –1), (4, 2), (–2, –4), (0, –2), (–1, –3) and (3.5, 1.5)? In each pair, the y-coordinate is 2 less than the x-coordinate. These coordinates are linked by the function: y = x – 2 We can draw a graph of the function y = x – 2 by plotting points that obey this function.

  31. Plotting graphs of linear functions x –3 –2 –1 0 1 2 3 y = 2x + 5 Given a function, we can find coordinate points that obey the function by constructing a table of values. Suppose we want to plot points that obey the function y = 2x + 5 We can use a table as follows: –1 1 3 5 7 9 11 (–3, –1) (–2, 1) (–1, 3) (0, 5) (1, 7) (2, 9) (3, 11)

  32. Plotting graphs of linear functions x –3 –2 –1 0 1 2 3 y = 2x + 5 –1 1 3 5 7 9 11 For example, y to draw a graph of y = 2x + 5: 1) Complete a table of values: y = 2x + 5 2) Plot the points on a coordinate grid. 3) Draw a line through the points. x 4) Label the line. 5) Check that other points on the line fit the rule.

  33. Plotting graphs of linear functions

  34. A8 Linear and real-life graphs Contents A8.1 Linear graphs • A • A A8.3 Parallel and perpendicular lines • A A8.2 Gradients and intercepts A8.4 Interpreting real-life graphs • A A8.5 Distance-time graphs • A A8.6 Speed-time graphs • A

  35. Gradients of straight-line graphs y y an upwards slope a horizontal line a downwards slope y x x x The gradient of a line is a measure of how steep the line is. The gradient of a line can be positive, negative or zero if, moving from left to right, we have Positive gradient Zero gradient Negative gradient If a line is vertical, its gradient cannot be specified.

  36. Calculating gradients

  37. Finding the gradient from two given points the gradient = (x2, y2) (x1, y1) change in y change in x x y2 – y1 the gradient = x2 – x1 If we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the gradient of the line as follows, y y2 – y1 Draw a right-angled triangle between the two points on the line as follows, x2 – x1

  38. Investigating linear graphs

  39. The general equation of a straight line The general equation of a straight line can be written as: y = mx + c The value of m tells us the gradient of the line. The value of c tells us where the line crosses the y-axis. This is called the y-interceptand it has the coordinate (0, c). For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).

  40. The gradient and the y-intercept 1 x 2 2 Complete this table: 3 (0, 4) (0, –5) (0, 2) –3 (0, 0) y = x y = –2x– 7 (0, –7)

  41. Rearranging equations into the form y = mx + c The equation of a straight line is 2y+x = 4. Find the gradient and the y-intercept of the line. 2y = –x + 4 1 subtract x from both sides: 2 divide both sides by 2: –x + 4 y = 2 y = –x + 2 Sometimes the equation of a straight line graph is not given in the form y = mx + c. Rearrange the equation by performing the same operations on both sides, 2y + x = 4

  42. Rearranging equations into the form y = mx + c 1 1 y = –x + 2 2 2 – Sometimes the equation of a straight line graph is not given in the form y = mx + c. The equation of a straight line is 2y+x = 4. Find the gradient and the y-intercept of the line. Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept. So the gradient of the line is (0, 2). and the y-intercept is

  43. Substituting values into equations A line with the equation y = mx + 5 passes through the point (3, 11). What is the value of m? To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5. This gives us, 11 = 3m + 5 6 = 3m subtract 5 from both sides: 2 = m divide both sides by 3: m = 2 The equation of the line is therefore y = 2x + 5.

  44. What is the equation of the line?

  45. Match the equations to the graphs

  46. A8 Linear and real-life graphs Contents A8.1 Linear graphs • A A8.2 Gradients and intercepts • A • A A8.3 Parallel and perpendicular lines A8.4 Interpreting real-life graphs • A A8.5 Distance-time graphs • A A8.6 Speed-time graphs • A

  47. Investigating parallel lines

  48. Parallel lines 2y = –6x + 1 subtract 6x from both sides: divide both sides by 2: –6x + 1 y = 2 If two lines have the same gradient they are parallel. Show that the lines 2y + 6x = 1 and y = –3x + 4 are parallel. We can show this by rearranging the first equation so that it is in the formy = mx + c. 2y + 6x = 1 y = –3x + ½ The gradient m is –3 for both lines, so they are parallel.

  49. Matching parallel lines

  50. Investigating perpendicular lines

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