a8-linear-and-real-life-graphs

1. KS4 Mathematics A8 Linear and real-life graphs

2. 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

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 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 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.

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. 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.

9. 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)

10. 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.

11. Plotting graphs of linear functions

12. 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

13. 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.

14. Calculating gradients

15. 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

16. Investigating linear graphs

17. 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).

18. 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)

19. 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

20. 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

21. 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.

22. What is the equation of the line?

23. Match the equations to the graphs

24. 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

25. Investigating parallel lines

26. 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.

27. Matching parallel lines

28. Investigating perpendicular lines

29. Perpendicular lines In general, if the gradient of a line is m, then the gradient of –1 the line perpendicular to it is . m 1 1 1 4 4 4 . The equation of the line with gradient and y-intercept –5 is, y = x– 5 If the gradients of two lines have a product of –1 then they are perpendicular. Write down the equation of the line that is perpendicular to y = –4x + 3 and passes through the point (0, –5). The gradient of the line y = –4x + 3 is –4. The gradient of the line perpendicular to it is therefore

30. Matching perpendicular lines

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

32. Real-life graphs American dollars British pounds When we use graphs to illustrate real-life situations, instead of plotting y-values against x-values, we plot one physical quantity against another physical quantity. The resulting graph shows the rate that one quantity changes with another. For example, This graph shows the exchange rate from British pounds to American dollars. It is a straight line graph through the origin and so the equation of the line would be of the form y = mx. The value of m would be equal to the number of dollars in each pound. What would the value of m represent?

33. Real-life graphs investment value time This graph show the value of an investment as it gains interest cumulatively over time. The graph increases by increasing amounts. Each time interest is added it is calculated on an ever greater amount. This makes a small difference at first but as time goes on it makes a much greater difference. This is an example of an exponential increase.

34. Real-life graphs mass time This graph show the mass of a newborn baby over the first month from birth. The baby’s mass decreases slightly during the first week. Its mass then increases in decreasing amounts over the rest of the month.

35. Rates of change

36. Filling flasks

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

38. Formulae relating distance, time and speed DISTANCE distance time = SPEED TIME speed distance speed = time It is important to remember how distance, time and speed are related. Using a formula triangle can help, distance = speed × time

39. Distance-time graphs 20 distance (miles) 15 10 5 0 0 15 30 45 60 75 90 105 120 time (mins) In a distance-time graph the horizontal axis shows time and the vertical axis shows distance. For example, John takes his car to visit a friend. There are three parts to the journey: John drives at constant speed for 30 minutes until he reaches his friend’s house 20 miles away. He stays at his friend’s house for 45 minutes. He then drives home at a constant speed and arrives home 45 minutes later.

40. Finding speed from distance-time graphs change in distance gradient = change in time distance change in distance change in time time How do we calculate speed? Speed is calculated by dividing distance by time. In a distance-time graph this is given by the gradient of the graph. = speed The steeper the line, the faster the object is moving. A zero gradient means that the object is not moving.

41. Interpreting distance-time graphs

42. Distance-time graphs change in speed acceleration = time When a distance-time graph is linear, the objects involved are moving at a constant speed. Most real-life objects do not always move at constant speed, however. It is more likely that they will speed up and slow down during the journey. Increase in speed over time is called acceleration. It is measured in metres per second per second or m/s2. When speed decreases over time is often is called deceleration.

43. Distance-time graphs This distance-time graph shows an object accelerating from rest before continuing at a constant speed. distance distance time time This distance-time graph shows an object decelerating from constant speed before coming to rest. Distance-time graphs that show acceleration or deceleration are curved. For example,

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

45. Speed-time graphs 20 speed (m/s) 15 10 5 0 0 5 10 15 20 25 30 35 40 time (s) Travel graphs can also be used to show change in speed over time. For example, this graph shows a car accelerating steadily from rest to a speed of 20 m/s. It then continues at a constant speed for 15 seconds. The brakes are then applied and it decelerates steadily to a stop. The car is moving in the same direction throughout.

46. Finding acceleration from speed-time graphs speed change in speed change in speed gradient = change in time change in time time Acceleration is calculated by dividing speed by time. In a speed-time graph this is given by the gradient of the graph. = acceleration The steeper the line, the greater the acceleration. A zero gradient means that the object is moving at a constant speed. A negative gradient means that the object is decelerating.

47. Finding distance from speed-time graphs 20 15 speed (m/s) 10 5 0 0 15 30 45 60 75 90 105 120 time (s) The following speed-time graph shows a car driving at a constant speed of 30 m/s for 2 minutes. What is the area under the graph? The area under the graph is rectangular and so we can find its area by multiplying its length by its height. Area under graph = 20 × 120 = 240 What does this amount correspond to?

48. The area under a speed-time graph For example, to find the distance travelled for the journey shown in this graph we find the area under it. 15 20 speed (m/s) 15 10 5 0 0 5 10 15 20 25 30 35 40 time (s) This area under a speed-time graph corresponds to the distance travelled. The shape under the graph is a trapezium so, Area = ½(15 + 40) × 20 = ½ × 55 × 20 = 550 So, distance travelled = 550 m

49. Interpreting speed-time graphs