13 Graphs Positive Coordinates. Worked Example 1. Solution

Transcription

1 Graphs MEP Pupil Tet. Positive Coordinates Coordinates are pairs of numbers that uniquel describe a position on a rectangular grid. The first number refers to the horizontal (-ais) and the second the vertical (-ais). The coordinates (, ) describe a point that is units across and units up on a grid from the origin (, ). (, ) Worked Eample Plot the points with coordinates (, 8), (, ) and (, ) Solution 9 For (, 8) move across and 8 up. 8 (, 8) For (, ) move across and up. For (, ) move across and up. (, ) 8 (, )

2 . MEP Pupil Tet Worked Eample Write down the coordinates of each point in the diagram below. B A C D 8 9 Solution A is across and up, so the coordinates are (, ). B has no movement across and is straight up, so the coordinates are (, ). C is across and up, so the coordinates are (, ). D is 8 across and no movement up, so the coordinates are (8, ). Eercises. Write down the coordinates of each point on the diagram below. A B E F C D

3 MEP Pupil Tet. The map of an island has been drawn on a grid. 8 Rock Point Café Camp Site Old Ben's Cottage Landing Stage Old Tower Sand Beach 8 9 Write down the coordinates of each place marked on the map.. On a grid, join the points with the following coordinates and write down the name of the shape ou draw. (, ) (8, ) (8, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (d) (, ) (, ) (, ) (, ) (, ) (e) (, ) (, ) (, ) (, ) (, ) (, ). Jenn writes her initials on a grid. Write down the coordinates of the corners of each letter. Write our initials in the same wa and write down the coordinates of our initials.

4 . MEP Pupil Tet. The pattern below is made up of circles. Write down the coordinates of the centre of each circle.. B A On the co-ordinate grid above plot the following points P (,), Q (,) R (,) Write down the co-ordinates of the points (i) A, (ii) B. (LON). Coordinates The coordinates of a point are written as a pair of numbers, (, ), which describe where the point is on a set of aes. The -ais is alwas horizontal (i.e. across the page) and the -ais alwas vertical (i.e. up the page). The -coordinate is alwas given first and the -coordinate second.

5 MEP Pupil Tet Worked Eample On a grid, plot the point A which has coordinates (, ), the point B with coordinates (, ) and the point C with coordinates (, ). Solution For A, begin at (, ), where the two aes cross. Move in the direction. Move in the direction. Points B and C are plotted in a similar wa. For B, move in the direction and in the direction. For C, move in the direction and in the direction. A (, ) B C Worked Eample Write down the coordinates of each place on the map of the island. Shop Telephone bo 8 Café Lifeboat station Jett 8 Church Campsite Lighthouse Solution Lighthouse (, ) Jett (, ) All coordinates positive Church (, ) Camp Site (, ) Negative - coordinates Shop (, ) Telephone Bo (, ) Negative - coordinates

6 . MEP Pupil Tet Café (, ) Lifeboat Station (, ) All coordinates negative Eercises. Write down the coordinates of each point marked on the grid below. G I H A C B J D K E F L. The map shows some Australian towns and cities. Broome Cairns 8 Alice Springs Brisbane Canberra Perth Alban Write down the coordinates of Canberra, Brisbane and Perth. A plane flies from the place with coordinates (, ) and lands at the place with coordinates (, ). From where does the plane take off and where does it land? A ship has coordinates (, ) at the start of a voage and coordinates (, ) at the end. Where does it start and where does it finish?

7 MEP Pupil Tet. The map shows some of the tors (rock outcrops) on Dartmoor in the south west of England. Sourton Tor 9 8 Shelstone Tor Higher Tor Row Tor West Mill Tor Yes Tor Oke Tor 8 9 Black Tor Kitt Tor Dinger Tor 8 9 Steeperton Tor (d) (e) Write down the coordinates of the following tors. West Mill Tor Steeperton Tor Shelstone Tor Black Tor Dinger Tor The highest tor marked on this map is Yes Tor. Write down the coordinates of this tor. A bo and his dog walk from Oke Tor to Kitt Tor. Write down the coordinates of the point where the start and the point where the finish. Sourton Tor is the tor that is the farthest west on this map. What are the coordinates of this tor? Higher Tor is the tor that is farthest north. What are the coordinates of this tor?. Draw a set of aes with -values from to and -values from to 9. Join together the points with coordinates (, ), (, 9) and (, ). What shape do ou get? On the same diagram, join together the points (, ), (, ) and (, ).

8 . MEP Pupil Tet. Draw a set of aes with -values from to and -values from to. Join each set of points below in the order listed. (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ). (, ), (, ), (, ), (, ). (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ). (d) (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ).. Draw a set of aes with -values from to and -values from to. Plot the following points and join them in the order listed. (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ).. Three corners of a square have coordinates (, ), (, ) and (, ). Draw a set of aes with -values from to and -values from to. Plot the three points and draw the square. Write down the coordinates of the centre of the square. 8. Two corners of a rectangle have coordinates (, ) and (, ). The centre of the rectangle has coordinates (, ). Plot the three points given and draw the rectangle. Write down the coordinates of the other two corners of the rectangle. 9. A dodecagon is a twelve-sided, plane shape. Draw a set of aes that have -values from to and -values from to 8. Plot the points listed below and join them to draw half a dodecagon. (, ), (, ), (, ), (, ), (, ), (, ), (, 8), (, 8) (d) Draw the other half of the dodecagon. It is smmetric about the line joining (, ) to (, 8). Write down the coordinates of the si new corners of the dodecagon that ou have drawn.. A set of aes are arranged so that the -ais runs from west to east and the -ais from south to north. N A ship is at the point A which has coordinates (, ). A How far south can the ship travel before its -coordinate becomes negative? W E How far west can the ship travel before its -coordinate becomes negative? If the ship travels SW, how far does it travel before both coordinates become negative? S 8

9 MEP Pupil Tet. Plotting Straight Lines B calculating values of coordinates ou can find points and draw a graph for an relationship, such as =. Worked Eample Cop and complete the following pairs of coordinates using the relationship =. (,? ), (,? ), (,? ) Plot the points on a set of aes. Draw a straight line through these points. Solution For the first point =, so So the first point is (, ). For the second point =, so So the second point is (, ). For the third point = So the third point is (, ). = = = = = = The points are plotted on these aes. (, ) (, ) (, ) The three points lie on a straight line as shown on these aes. 9

10 . MEP Pupil Tet Worked Eample Draw the graph of Solution =. The first step is to find the coordinates of three points on the line. Choose an three -values for the coordinates. Three possible values are given below. (,? ), (,? ), (,? ) Now calculate the -values using =. For the first point =, so = = The coordinates of this point are (, ). For the second point =, so = = The coordinates of the second point are (, ). For the third point =, so The third point is (, ). = ( ) = These points are now plotted on the aes below. Then a straight line can be drawn through them. (, ) (, ) = (, ) Note In fact, just two points uniquel define a straight line but it is safer to use three as a check.

11 MEP Pupil Tet Eercises. Cop and complete the coordinates below using the relationship = +. (,? ), (,? ), (,? ) Draw a set of aes with -values from to and -values from to. Plot the points with the coordinates found in. (d) Check that the points lie on a straight line and draw a straight line through the points. (e) Write down the coordinates of the point where the line crosses the -ais.. Use the relationship = + to complete the coordinates below. (,? ), (,? ), (,? ) Draw a set of -aes with values from to and -values from to. Then plot the points with the coordinates obtained in. Draw a straight line through the points in. (d) Does the line in go through the point (, )? Check if these coordinates satisf the relationship = +.. Use the relationship = to complete the coordinates below. (d) (,? ), (,? ), (,? ) Plot these points on a set of aes and draw a straight line through the points. Write down the coordinates of two other points on the line. Check that the satisf the relationship =. Does the point (, ) lie on the line? Check if these coordinates satisf the relationship =.. Use a different set of aes with -values from to to draw the graph of each relationship. You will need to decide what -values to use for each graph. = + = = (d) = + (e) =

12 . MEP Pupil Tet. Draw the lines with the equations listed below. You should decide what size aes ou need to use to plot the points. = + 8 = + = (d) = +. The relationship d = t gives the distance walked b a person. The distance in kilometres is d and t is the time in hours for which the person has been walking. Complete a cop of the table below. t d (d) (e) (f) (g) Complete and plot the coordinates below. (,? ), (,? ), (,? ) Draw a straight line through the points. Which ais has the time (t) values? Which ais has the distance (d) values? Use the graph to find the time taken to walk km. Use the graph to find the distance walked in hours.. A teacher uses the relationship p = m to convert the marks obtained on a test to percentages. Here m is the mark and p is the percentage. Complete a cop of the table below. m p (d) Complete the coordinates below using the contents of our table. (,? ), (,? ), (,? ) Using m on the horizontal ais and p on the vertical ais, plot the points with the coordinates ou obtained in. Cop the following table and then use the graph to fill in the missing entries.

13 MEP Pupil Tet Mark Percentage John Stuart Jenn 8 Karen 8 Mike 8. The relationship F = +. 8C can be used to convert temperatures in degrees Celsius, C, to temperatures in degrees Fahrenheit, F. Complete a cop of the table below. Temperature in Celsius Temperature in Fahrenheit Use the information in the table to draw a graph of F = +. 8C. (d) The recommended temperature for a greenhouse is 8 F. Use our graph to convert this to Celsius. The temperature at a holida resort is C. Use the graph to convert this to Fahrenheit. 9. Complete the coordinates below for the relationship =. (,? ), (,? ), (,? ) Draw a set of aes with and values from to and plot the points with coordinates obtained in. Write down the coordinates of the points where the line crosses the -ais and the -ais.. Write down the coordinates of three points on the line =. Plot the points ou obtained on a set of aes and draw a straight line through them. Find the area of the triangle that is formed b the aes and the line.. Draw the lines = and =. State the coordinates of the points where the lines cross. Find the area of the triangle formed b the two lines and the -ais. (d) Repeat to for = + and =. (e) Repeat to for = and =.

14 . MEP Pupil Tet. Plot the points (, ), (, ), (, ) and (, ) on a cop of the grid above. Draw a straight line through the four points. Draw the line = + on a cop of the grid above. (MEG). Cop and complete the following table for the rule. "To find, double and add." Plot the values from the table onto a cop of the following coordinate grid. Join our points with a straight line.

15 MEP Pupil Tet Write, in smbols, the rule "To find, double and add." (d) Use our rule from part to calculate the value of when = 9. (LON). Two points A and B are shown on the grid below. Write down their coordinates. A B Cop and complete the table of values below for = Draw the graph of = on our cop of the graph above. (d) On the grid, draw the straight line AB. Write down the coordinates of the point where the graph of = cuts the line AB. (MEG)

16 . MEP Pupil Tet. This machine multiplies all numbers b and then subtracts. IN OUT Complete a cop of the table below for this machine b putting values into the two empt boes. IN? OUT? OUT P IN (i) Using the values from the table, plot four points on a cop of the grid above. (ii) Draw the line joining the four points that ou have plotted. (iii) Write down the coordinates of the point labelled P on the grid. (MEG). IN Multipl b Subtract OUT When ou put in to this number machine, out comes. Fill in the missing numbers in a cop of the table below. IN OUT

17 MEP Pupil Tet Plot points on a cop of the following grid to represent our answers in part. The point (, ) has been plotted for ou. 8 OUTPUT 8 (, ) 8 9 INPUT What do ou notice about the points that ou have plotted? (MEG). Plotting Curves Some relationships produce curves rather than straight lines when plotted. Worked Eample Complete the table below using the relationship =. Write a list of coordinates using the data in the table. Plot the points and draw a smooth curve through them. Solution For each value of the value can be calculated using =. For eample: If = then = = 9 =

18 . MEP Pupil Tet If = then = = = ( ) If = then = = Calculating all the values gives the table. = The coordinates of the points to plot are (, ), (, ), (, ), (, ), (, ), (, ) and (, ). These points are plotted on the graph below and have been joined b a smooth curve. (, ) (, ) (, ) (, ) (, ) (, ) (, ) Worked Eample Draw the graph of = for values of from to. Solution The first step is to draw up and complete a table of values using the relationship =, as below. 8

19 MEP Pupil Tet ( ) ( ) For eample: If = then = = + = If = then = Each pair of values can be written as coordinates, = 8 8 = (, ), (, ), (, ), (, ), (, ), (, ), (, ) These can then be plotted and a smooth curve drawn through the points as shown in the following graph

20 . MEP Pupil Tet Eercises. Complete a cop of the table below for =. Write a list of coordinates. Plot these points and draw a smooth curve through the points.. Complete a cop of the table below using the relationship =. 8 Plot the graph of = using the data in the table.. Complete a cop of the table and draw the graph of =.. Complete a cop of the table and draw the graph of =. What is the value of at the lowest point of the curve? Use our answer to to calculate the corresponding value of.. Complete a cop of the table using the relationship =. Using the information in the table, sketch the graph of =. (d) Complete the following pairs of coordinates. (.,? ), (.,? ) Check that our graph passes through the points with the coordinates calculated in.. Complete a cop of the table and then draw the graph of. =.

21 MEP Pupil Tet. A manufacturer of postcards decides to eperiment with cards of different shapes. The cards should be rectangles with an area of cm. The height of the cards is cm and the width is cm. Eplain wh =. Complete a cop of the table. (d) (e) Draw a graph of =. Use our graph to find the width of a postcard that has a height of cm. If the height of a postcard must be no greater than cm, what is the least width it can have? 8. Some water tanks have square bases and a height of m. Eplain wh the volume, V, of a tank is. m Complete a cop of the table and draw the graph of V = V. (d) If the base of a tank is a square sheet of metal. m b. m, find from the graph the volume of the tank. What should be the size of the base in order to give a volume of (i) m (ii) m? 9. The height, h, in metres, of the distance travelled b a ball hit straight up into the air is given b h = 8t t where t is the time in seconds. Complete a cop of this table and draw a graph of h against t. t h Use our graph to estimate when the ball hits the ground. What is the maimum height reached b the ball?

22 . MEP Pupil Tet. Complete the table below using the relationship =. No value Draw the graph of = using the points calculated in. Find the coordinates of etra points on the curve between = and =. (d) Describe what happens to the curve as the values of get closer and closer to. (e) Investigate what happens between and.. Use the equation = + to complete the table of values. Use the equation = to complete the table of values. Draw the graphs of on a cop of the grid below. = + and = 9 8 (LON)

23 MEP Pupil Tet. Use the formula = to complete a cop of the table Use our table of values to draw the graph of =. Use our graph to find the values of when =. (SEG). The annual cost of the heat lost through a wall depends on the length of the wall. When the wall is a square of length m the annual cost,, is given b the equation = Calculate the cost,, when is 8 m. The table shows the cost,, for different values of m. Length, (m) Cost, ( ) 8 8 Use the table of values to draw the graph of = on a cop of the grid on the following page.

24 . MEP Pupil Tet Cost ( ) Length (m) 8 The annual cost of the heat lost through a square wall is. Use our graph to estimate the length of the wall (SEG). Angela is investigating the area of squares. cm cm cm She makes a table. Length of side (cm) Area (cm ) 9

25 MEP Pupil Tet Draw a graph of area against length of side. A Area (cm ) Length of side (cm) The side of a square is cm. The area of a square is A cm. Write down the formula which ma be used to calculate the area from the length of the side. A square has an area of cm. Angela wants to use the graph to find the length of the side of this square. She draws a line on the graph to help her do this. (i) Draw this line. (ii) Write down the length of the side of the square whose area is cm. (SEG). Gradient The gradient of a line describes how steep it is. The diagram below shows two lines, one with a positive gradient and the other with a negative gradient. POSITIVE GRADIENT NEGATIVE GRADIENT

26 . MEP Pupil Tet The gradient of a line between two points, A and B, is calculated using B (, ) Vertical CHANGE gradient of AB = vertical change horizontal change A (, ) - coordinate of B - coordinate of A = ( ) ( ) ( - coordinate of B) ( - coordinate of A) Horizontal CHANGE = Worked Eample Find the gradient of the line shown in the diagram. Solution Draw a triangle under the line below to show the horizontal and vertical distances. Here the vertical distance is and the horizontal distance is. Gradient = = = vertical change horizontal change or Horizontal CHANGE = Vertical CHANGE =

27 MEP Pupil Tet Worked Eample Find the gradient of the line joining the point A with coordinates (, ) and the point B with coordinates (, ). Solution Gradient - coordinate of B - coordinate of A = ( ) ( ) ( - coordinate of B) ( - coordinate of A) = = = Worked Eample Find the gradient of the line that joins the points with coordinates (, ) and (, ). Solution The diagram shows the line. It will have a negative gradient because of the wa it slopes. Gradient So the gradient is. Eercises - coordinate of B - coordinate of A = ( ) ( ) ( - coordinate of B) ( - coordinate of A) = = = ( ). Find the gradient of the line shown on this graph. (, ) A 9 8 (, ) B

28 . MEP Pupil Tet. Find the gradient of each line in the diagram below. 9 8 A C B 8 9 D E F. Which of the lines in the diagram below have a positive gradient? D B C E I F A L H 8 J K 8 G Which lines have a negative gradient? Find the gradient of each line.. The diagram shows a side view of a ramp in a multistore car part. Find the gradient of the ramp. m m 8

29 MEP Pupil Tet. The diagram shows the cross-section of a roof. Find the gradient of each part of the roof. m m m m m m. Find the gradient of the line that joins the points with the coordinates: (, ) and (9, ), (, ) and (, ), (, ) and (, ), (d) (, ) and (, ), (e) (, ) and (, 8), (f) (, ) and (, ).. Find the gradient of the line joining the points with the coordinates: (, ) and (, ), (, ) and (, ), (, ) and (, ), (d) (, ) and (, 8), (e) (, ) and (, ), (f) (8, ) and (, ). 8. A quadrilateral is formed b joining the points A, B, C and D. The coordinates of each point are: A (, ) B (, ) C (, ) D (, ) Find the gradient of each side of the quadrilateral. 9

30 . MEP Pupil Tet 9. For the line = +, complete the coordinates (,? ), (,? ) and (,? ) Plot the points in and draw a line through them. What is the gradient of the line ou have drawn? (d) Repeat to for = + and =. (e) What do ou notice?. Calculate the coordinates for three points on the line = +. Plot these points and draw a straight line through them. Find the gradient of the line that ou have drawn. (d) Repeat to for the lines = and = +. (e) What is the connection between the equation of a line and its gradient? (f) What do ou think the gradient of the line = + will be?. For each pair of coordinates below, find the gradient of the straight line that joins them. (, ) and (a, a) (a, a) and ( a, a) (a, a) and (, a) (d) ( a, a) and (a, a). Applications of Graphs In this section some applications of graphs are considered, particularl conversion graphs and graphs to describe motion. The graph below can be used for converting pounds sterling (British pounds) into and from Spanish pesetas. Pounds sterling 8 Pesetas

31 MEP Pupil Tet A distance-time graph of a car is shown below. The gradient of this graph gives the speed of the car. The gradient is steepest from A to B, so this is when the care has the greatest speed. The gradient BC is zero, so the car is not moving. Distance (m) D B C A Time (s) The area under a speed-time graph gives the distance travelled. Finding the shaded area on the graph below would give the distance travelled. Speed (m/s) Time (s) D Worked Eample A temperature of C is equivalent to 8 F and a temperature of C is equivalent to a temperature of F. Use this information to draw a conversion graph. Use the graph to convert: C to Fahrenheit, 8 F to Celsius. Solution Taking the horizontal ais as temperature in C and the vertical ais as temperature in F gives two pairs of coordinates, (, 8) and (, ). These are plotted on a graph and a straight line drawn through the points.

32 . MEP Pupil Tet F (, ) 8 8 (, 8) 8 9 C Start at C, then move up to the line and across to the vertical ais, to give a temperature of about 8 F. Start at 8 F, then move across to the line and down to the horizontal ais, to give a temperature of about 8 C. Worked Eample The graph shows the distance travelled b a girl on a bike. Distance (m) B C D E A Time (s) Find the speed she is travelling on each stage of the journe.

33 MEP Pupil Tet Solution For AB the gradient = =. So the speed is. m/s. Note The units are m/s (metres per second), as m are the units for distance and s the units for time. For BC the gradient = = So the speed is m/s. For CD the gradient is zero and so the speed is zero For DE the gradient is = =. So the speed is. m/s. Worked Eample The graph shows how the speed of a bird varies as it flies between two trees. How far apart are the two trees? I Speed (m/s) Time (s) Solution The distance is given b the area under the graph. In order to find this area it has been split into three sections, A, B and C.

34 . MEP Pupil Tet I Speed (m/s) 8 B A C 8 9 Time (s) Area of A = = 8 Area of B = = Area of C = = Total Area = = So the trees are m apart. Note that the units are m because the units of speed are m/s and the units of time are s. Eercises. Use the approimation that kg is about the same as lbs to draw a graph for converting between pounds and kilograms. Use the graph to convert the following: lbs to kilograms, 8 lbs to kilograms, kg to pounds, (d) kg to pounds.. Use the approimation that gallons is about the same as litres to draw a conversion graph. Use the graph to convert: gallons to litres, litres to gallons.

35 MEP Pupil Tet. The graph shows how the distance travelled b a bus increased. H F G Distance (m) B C D E A Time (s) How man times did the bus stop? Find the speed of the bus on each section of the journe. On which part of the journe did the bus travel fastest?. The distance-time graph shows the distance travelled b a car on a journe to the shops. F 8 D E Distance (m) B C A 8 Time (s) The car stopped at two sets of traffic lights. How long did the car spend waiting at the traffic lights? On which part of the journe did the car travel fastest? Find its speed on this part. On which part of the journe did the car travel at its lowest speed? What was this speed?

36 . MEP Pupil Tet. The graph below shows how the speed of an athlete varies during a race. 8 Speed (m/s) 8 Time (s) What was the distance of the race?. The graph below shows how the speed of a lorr varies as it sets off from a set of traffic lights. 8 Speed (m/s) 8 8 Time (s) Find the distance travelled b the lorr after 8 seconds, seconds, seconds.. The graph shows how the distance travelled b a snail increases. D Distance (m) B C A Time (hours)

37 MEP Pupil Tet Find the speed of the snail on each section in m/hour. 8. Hannah runs at ms for seconds and then her speed decreases to zero at a stead rate over the net seconds. Find the distance that Hannah runs. 9. Ian runs at a constant speed for seconds. He has then travelled m. He then walks at a constant speed for 8 seconds until he is 8 m from his starting point. Find the speed at which he runs and the speed at which he walks. If he had covered the complete distance in the same time, with a constant speed, what would that speed have been?. The graph shows how the distance travelled b Wend and Jodie changes during a race from one end of the school field to the other end, and back. 8 Wend Jodie Distance from start (m) Time (s) Describe what happens during the race.. Find the area under each graph below and state the distance that it represents. Speed (mph) Time (mins) Speed (mph) Time (hours)

38 . MEP Pupil Tet (d) Speed (mm/s) Time (mins) Speed (m/s) Time (mins). For each distance-time graph, find the speed in the units used on the graph and in m/s. Distance (km) Distance (mm) Time (hours) Time (s) (d) Distance (m) Distance (m) Time (hours) Time (mins). Jennifer walks from Corfe Castle to Wareham Forest and then returns to Corfe Castle. The following travel graph shows her journe. At what time did Jennifer leave Corfe Castle? How far from Wareham Forest did Jennifer make her first stop? Jennifer had lunch at Wareham Forest. For how man minutes did she stop for lunch? (d) At what average speed did Jennifer walk back from Wareham Forest to Corfe Castle? 8

39 MEP Pupil Tet Distance from Corfe Castle (km) 9 8 Time (hours) (SEG). The graph represents a swimming race between Robert and James. Distance from start (m) Robert James 8 Time (seconds) At what time did James overtake Robert for the second time? What was the maimum distance between the swimmers during the race? Who was swimming faster at seconds? How can ou tell? (SEG) 9

40 . MEP Pupil Tet. The graph illustrates the journe of a car. Speed in m/s 8 9 Time in seconds Estimate the area under the graph taking into account the scales of the graph. State the units of the quantit represented b the area under the graph. Another car did the same journe in the same time at constant speed. On a cop of the grid above, draw the graph which illustrates the second car's journe. (MEG). Scatter Plots and Lines of Best Fit When there might be a connection between two different quantities, a scatter plot can be used. If there does appear to be a connection, a line of best fit can be drawn. The following diagrams show different scatter plots. Positive correlation No correlation Negative correlation If there is a relationship between the two quantities, there is said to be a correlation between the two quantities. This ma be positive or negative, as shown in the eamples above.

41 MEP Pupil Tet Worked Eample A salesman records, for each working da, how much petrol his car uses and how far he travels. The table shows his figures for das. Da Petrol used (litres) Distance travelled (miles) Plot a scatter graph and describe an connection that is present. Draw a line of best fit. Eplain wh it is sensible for the line to go through (, ) (d) Estimate how much petrol would be used on journe of miles. Solution Petrol used (litres) Each point has been plotted on the graph below. This is an eample of positive correlation. Distance (miles) (d) A line of best fit has been drawn. There are roughl the same number of points above and below the line. This is sensible because a car will not use an petrol if it is not used. The dashed lines on the graph predict that. litres are needed for a journe of miles.

42 . MEP Pupil Tet Worked Eample The table shows how the Olmpic record for the marathon has decreased since 9. Draw a scatter graph to illustrate this. Draw a line of best fit. Estimate what the Olmpic record will be in the ear. Solution Record (mins) Each point has been plotted on the scatter graph below. This is an eample of negative correlation. Year Record (minutes) Distance (miles) A line of best fit has been added to the graph. The first point has been ignored as it would have moved the line a long wa from the other points. Using the line ou can predict that in the ear the record will drop to just under minutes. Note The line of best fit becomes unreliable as time progresses; for eample, the straight line will predict a zero-minute mile in about the ear! Eercises. The table gives the scores obtained b students on three different tests. Maths Test Science Test French Test

43 MEP Pupil Tet (d) Draw a scatter graph for maths against science. Draw a scatter graph for maths against French. Which set of points lie closer to a straight line? Would it be reasonable to draw a line of best fit in both cases?. A firm records how long it takes a driver to make deliveries at warehouses at different distances from the factor. Distance (miles) Time taken (hours) (d) Draw a scatter graph of time taken against distance, and describe an correlation between the two quantities. Draw a line of best fit. A deliver takes hours. Use our line to estimate how far the driver has travelled. How long would ou epect a deliver to take if the driver has to travel miles?. The table shows the fling time and costs for holidas in some popular resorts. Destination Fling time (hours) Cost of holida ( ) Algarve. 9 Benidorm. 9 Gambia. Majorca. 8 Morocco. Mombasa 8. Tenerife. 8 Torremolinos. Tunisia. 9 Draw a scatter graph of cost against time. Draw a line of best fit. Estimate the cost of a holida with a fling time of hours. (d) Estimate the fling time for a holida that costs.. Ten children were weighed and then had their height measured. The results are in the table. Weight (kg) Height (cm)

44 . MEP Pupil Tet (d) Draw a scatter graph of height against weight. Draw a line of best fit and comment on how well it can be applied to the data. Estimate the height of a bo who weighs kg. Estimate the weight of a girl who is cm tall.. A group of children were tested on their tables. The time in seconds taken to do a test on the times table and a test on the times table were recorded. Time for times table Time for times table Draw a scatter graph and a line of best fit. Ben missed the test for the times table but took seconds for the times table test. Estimate how long he would have taken for the times table test. Emma completed her times table test in seconds. She missed the test for the times table. How long do ou estimate that she would have taken for this test?. The table below shows how the Olmpic record for the men's and women's m freestle has decreased. The times are given to the nearest second. Year Men's record (s) Women's record (s) On the same set of aes, plot scatter graphs for both the men's and the women's records. What tpe of correlation do ou see in the plot?

45 MEP Pupil Tet Draw a line of best fit for each sets of records against time. Use our lines to estimate what the records will be in the ear. (d) Do the graphs suggest that the women's record will be less than the men's record? Is this a realistic prediction?. Describe two quantities that ou would epect to have a positive correlation, no correlation, a negative correlation. 8. The table gives ou the marks scored b pupils in a French test and in a German test. French German 9 8 On a cop of the grid below, draw a scatter graph of the marks scored in the French and German tests. German French Describe the correlation between the marks scored in the two tests. (LON) 9. The table gives information about the age and value of a number of cars of the same tpe. Age (ears) Value ( ) Use this information to draw a scatter graph on a cop of the following grid.

46 . MEP Pupil Tet 8 Value ( ) Age (ears) What does the graph tell ou about the value of theses cars as the get older? (SEG). Ten people entered a craft competition. Their displas of work were awarded marks b two different judges. Competitor First judge Second judge A 9 B C D E 9 F G H 8 I J The table shows the marks that the two judges gave to each of the competitors. (i) On a cop of the following grid, draw a scatter diagram to show this information.

47 MEP Pupil Tet 9 8 Marks from second judge 8 9 Marks from first judge (ii) Draw a line of best fit. A late entr was given marks b the first judge. Use our scatter diagram to estimate the mark that might have been given b the second judge. (Show how ou found our answer.) (NEAB). The height and arm length for each of eight pupils are shown in the table. Height (cm) Arm length (cm) On a cop of the grid below, plot a scatter graph for these data. 8 8 Arm length (cm) Height (cm)

48 . MEP Pupil Tet (i) Peter gives his height as cm. Use the scatter graph to estimate Peter's arm length. (ii) Eplain wh our answer can onl be an estimate. (SEG). A group of schoolchildren took a Mathematics test and a Phsics test. The results for children were plotted on a scatter diagram. Phsics mark * * * * ** * ** * * * Mathematics mark Does the scatter diagram show the results ou would epect? Eplain our answer. (i) Add a line of best fit, b inspection, to the scatter diagram. (ii) (iii) One pupil scored marks for Mathematics but missed the Phsics test. Use the line of best fit to estimate the mark she might have score for Phsics. One pupil was awarded the prize for the best overall performance in Mathematics and Phsics. Put a ring around the cross representing that pupil on the scatter diagram. (MEG). Megan wanted to find out if there is a connection between the average temperature and the total rainfall in the month of August. She obtained weather records for the last ears and plotted a scatter graph. 9 8 Average temperature ( C) Total rainfall (mm) 8

49 MEP Pupil Tet What does the graph show about a possible link between temperature and rainfall in August? Estimate the total rainfall in August when the average temperature is C..8 The Equation of a Straight Line The equation of a straight line is usuall written in the form = m + c where m is the gradient and c is the intercept. = m + c Gradient = m c Worked Eample Find the equation of the line shown in the diagram. Solution The first step is to find the gradient of the line. Drawing the triangle shown under the line gives gradient = = = vertical change horizontal change So the value of m is. The line crosses the -ais at, so the value of c is. The equation of a straight line is = m + c In this case = +. Vertical change = Horizontal = change 9

50 .8 MEP Pupil Tet Worked Eample Find the equation of the line that passes through the points (, ) and (, ). Solution Plotting these points gives the straight line shown. Using the triangle drawn underneath the line allows the gradient to be found. gradient = The intercept is. = vertical change horizontal change = So m = and c = and the equation of the line is Eercises = = or =. Find the equation of the straight line with: gradient = and -intercept =, gradient = and -intercept =, Intercept = Horizontal = change Vertical change = gradient = and -intercept =, (d) gradient = and -intercept =, (e) gradient = and -intercept =.. Write down the gradient and -intercept of each line. = + = = + (d) = ( ) ( ) (e) = + (f) = (g) = + (h) =

51 MEP Pupil Tet. The diagram shows the straight line that passes through the points (, ) and (, ). Find the gradient of the line. Write down the -intercept. Write down the equation of the straight line.. Find the equation of each line shown in the diagram below. A B C 8 D E F 8 9. Write down the gradient of each line and the coordinates of the -intercept. = 8 = + = (d) = + (e) = 8 (f) = (g) + = 8 (h) = ( )

52 .8 MEP Pupil Tet. Find the equation of the line that passes through the points with the coordinates below. (, ) and (, ) (, ) and (8, ) (, ) and (, ) (d) (, ) and (, ). Find the equation of each line in the diagram below. A 8 B C E D F 8 8 Length (cm) 8. The graph shows the results of an eperiment. Masses were hung on a spring and the length of the spring recorded. Find the equation of the line of best fit that has been drawn. Mass (grams)

53 MEP Pupil Tet The data in the following table was collected using a different spring. Mass (grams) Length (cm) 8 Draw a graph and find the equation of the straight line that passes through these points. 9. Some students made a simple thermometer and recorded the results below. Temperature ( C) Height of mercur (mm) 9 9 Plot the data points with temperature on the horizontal ais. Draw a straight line through the points and find its equation.. The graph below can be used for converting gallons to litres. Litres 8 9 Gallons Find the equation of the line. Draw a similar graph for converting litres to pints, given that litres is approimatel pints. Use the horizontal ais for pints. Find the equation of the line drawn in.. The velocit of a ball thrown straight up into the air was recorded at half second intervals. Time (s) Velocit (ms ) Plot a graph with time on the horizontal ais. Draw a line through the points and find its equation. What was the velocit of the ball when it was thrown upwards?

54 .8 MEP Pupil Tet. The line = + c passes through the point (, ). Find the value of c.. The point (, ) lies on the line = + c. Find the value of c.. The line = m + passes through the point (, ). Find the value of m.. Television repair charges depend on the length of time taken for the repair, as shown on the graph. c Charge ( ) 8 9 t Time (minutes) The charge is made up of a fied amount plus an etra amount which depends on time. What is the charge for a repair which takes minutes? (i) Calculate the gradient of the line. (ii) What does the gradient represent? Write down the equation of the line. (d) Mr Banks' repair will cost 8 or less. Calculate the maimum amount of time which can be spent on the repair. (SEG). A set of rectangular tiles is made. The length l cm, of each tile is equal to its width, w cm, plus two centimetres so l = w +. Draw the graph of l = w + to show this rule. Label this Line A.

55 MEP Pupil Tet 9 Line B 8 Length l 8 9 Width w Line B shows the rule connecting the length and width of a different set of tiles. What is this rule? (SEG). The table shows the largest quantit of salt, w grams, which can be dissolved in a beaker of water at temperature t C. t C w grams 8 On a cop of the following grid, plot the points and draw a graph to illustrate this information. Use our graph to find (i) the lowest temperature at which g of salt will dissolve in the water. (ii) the largest amount of salt that will dissolve in the water at C. (i) The equation of the graph is of the form (ii) w = at + b. Use our graph to estimate the values of the constants a and b. Use the equation to calculate the largest amount of salt which will dissolve in the water at 9 C.

56 .8 w 8 MEP Pupil Tet quantit of salt (grams) t temperature of water ( C) 8. The time, T minutes, for cooking a piece of meat, weighing P pounds, is found using the instruction 'cook for minutes then add minutes for each pound'. Write down a formula for T in terms of P. The graph shows cooking times for pieces of meat which weigh from to pounds. T (minutes) P (pounds)

57 MEP Pupil Tet (i) (ii) Plot the point to show the cooking time for a piece which weighs pounds. A piece of meat took minutes to cook. Use the graph to estimate the weight of the meat..9 Horizontal and Vertical Lines Horizontal lines have equations like = or =. = = Vertical lines have equations like = or =. Worked Eample = = Draw the line =. Draw the line =. Write down the coordinates of the points where these lines cross. Solution For the line = the -coordinate of ever point will alwas be. So the points (, ) (, ) (, ) all lie on the line =. For the line = the -coordinate of ever point will alwas be. So the points (, ) (, ) and (, ) all lie on the line =. = (, ) (, ) (, ) = (, ) (, ) (, ) =

58 .9 MEP Pupil Tet The graph in shows that the lines cross at the point with coordinates (, ). Eercises. Write down the equation of each line in the diagram below. A B C D E F. Draw the lines =, =, = and =. Write down the coordinates of the points where the lines cross. Find the area of the rectangle formed b the four lines. Draw the lines =, =, = and =. What are the coordinates for the centre of the square ou obtain?. Draw the rectangle which has corners at the points with coordinates (, ), (, ), (, ) and (, ). Write down the equations of the lines that form the sides of the rectangle.. Draw the lines =, = and = 8. Find the area of the triangle that ou obtain. 8

59 MEP Pupil Tet. Solution of Simultaneous Equations b Graphs Solutions to pairs of simultaneous equations can be found b plotting lines and finding the coordinates of the point where the cross. Worked Eample Solve the pair of simultaneous equations Solution + = 8 and + = First it can be helpful to write the two equations in the form =.... For the first equation, + = 8 = 8 (subtracting ) For the second equation, + = = (subtracting ) = (dividing b ) Now three pairs of coordinates can be found for each line. For = 8 If =, = 8 so (, 8) lies on the line If =, = 8 If = 8, = 8 8 = so (, ) lies on the line. = so (8, ) lies on the line. For = If =, = = so (, ) lies on the line. 9

60 . MEP Pupil Tet If =, = = If =, = = so (, ) lies on the line. = = so (, ) lies on the line. These points are then used to plot the lines shown below. 8 = 8 = 8 9 The two lines cross at the point (, ), so the solution is = and = Worked Eample Tickets for a school concert cost for adults and for children. If tickets are sold and a total of is paid, find the number of each tpe of ticket sold. Solution Let and = number of adult tickets sold = number of children' s tickets sold. As tickets are sold in total, + = As is taken in pament for the tickets, + = These equations can be written in the form =....

61 MEP Pupil Tet For the first equation, + = = (subtracting ) For the second equation, + = = (subtracting ) = (dividing b ) Now the coordinates of some points can be found for each equation. For = If =, = If =, = If =, = = so (, ) lies on the line = so (, ) lies on the line. = so (, ) lies on the line. For = If =, = If =, = = If =, = = so (, ) lies on the line = 8 so (, 8) lies on the line. = = so (, ) lies on the line. The graph shows these points plotted and lines drawn through them.

62 . MEP Pupil Tet = 8 8 = 8 8 The point where the lines cross has coordinates (, 8), so the solution is =, = 8. That is, adult tickets and 8 children's tickets. Eercises. Use the graph below to solve the simultaneous equations. = 8 = 8 = = = (d) = = = (e) = (f) = = =

63 MEP Pupil Tet = 8 = 9 8 = = =. The points with coordinates (, ), (, ) and (, ) lie on the line =. Plot these on a set of aes with -values from to and -values from to. Draw a straight line through these points. The points with coordinates (, ), (, ) and (, ) lie on the line = +. Plot these points using the set of aes in, and draw a line through them. Find the solution of the simultaneous equations = = +. Complete these coordinates for = +. (,? ), (,? ) and (,? ) Use these points to draw the line = +. Complete these coordinates for = (,? ), (,? ) and (,? ) (d) Use the points to draw the line =.

64 . MEP Pupil Tet (e) What is the solution of the simultaneous equations = + =?. Draw a set of aes with -values from to and -values from to. Write down the coordinates of three points on the line = and use them to draw the line =. On the same set of aes draw the lines = and = +. (d) Write down the solution of each set of simultaneous equations. (i) = (ii) = (iii) = = = + = +. Use a graph to solve each set of simultaneous equations below. = = = = + = =. Write the equations = and + = in the form =... Draw the graphs of both equations. What is the solution of the simultaneous equations?. Solve the following sets of simultaneous equations. + = + = + = + = + = + = 8. When two numbers and are added together ou obtain, and when is subtracted from ou obtain. Write down two equations involving and. Write both equations in the form =.... Draw the graphs of both equations and find the values of and which satisf both equations. 9. A discount store sells CDs and tapes. The price of ever CD is and the price of ever tape is. Jane bus CDs and tapes which cost a total of. Write down an equation involving and using this information.

65 MEP Pupil Tet Write down an equation in the form =.... Christopher bus CDs and tapes which cost a total of. Write down a second equation using this information. (d) Write this equation in the form =.... (e) (f) (g) Draw the graphs of both equations on the same set of aes. What is the price of a CD? What is the price of a tape?. An enterprising schoolbo charges to wash a car and to wash and polish a car. One da he earns 8 and cleans 8 cars. Let be the number of cars that he onl washed, and the number of cars washed and polished. Write down two equations involving and. Write these equations in the form =.... Draw the graphs of both equations and find out how man cars were washed and polished.. = + + = The diagram shows the graphs of the equations = + and + =. Use the diagram to solve the simultaneous equations = +, + =. (LON)

66 . MEP Pupil Tet. A longlife batter and a standard batter were both tested for their length of life. The longlife batter lasted for hours. The standard life batter lasted for hours. The combined length of life of the two batteries was hours. Eplain wh + = The longlife batter lasted hours longer than the standard batter. Write down another equation connecting and. The graph of + = has been drawn below. + = 8 8 Complete a cop of the table of values for our equation in and use it to draw the graph of our equation. (d) Use our graphs to find the length of life of each tpe of batter. (SEG)

67 MEP Pupil Tet. For off-peak electricit, customers can choose to pa b Method A or Method B. Method A: Method B: 8 per quarter plus p per unit. per quarter plus p per unit. A customer uses units. The quarterl cost is. The cost,, b Method B is =. +. (i) Complete the table of values of =. +. (ii) Draw the graph of =. +. Method A Cost ( ) Number of units used For a certain number of units both methods give the same cost. Use the graphs to find (i) this number of units, (ii) this cost. Cop and complete the following statement "Method... is alwas cheaper for customers who use more than... units". (SEG)

68 . MEP Pupil Tet. 8 (i) The graph of + = is shown on the diagram above. On a cop of the diagram, draw the graph of =. (ii) Use the graphs to find the solution of the simultaneous equations + =, =. Give the value of and the value of to one decimal place. Calculate the eact solution of the simultaneous equations + =, =. (MEG). The line with equation + = is drawn on the following grid. Solve the simultaneous equations b a graphical method. =, + =, 8

69 MEP Pupil Tet (MEG). Graphs of Common Functions Linear Functions Linear functions are alwas straight lines and have equations which can be put in the form = m + c c Gradient = m Quadratic Functions Quadratic functions contain an term as well as multiples of and a constant. Some eamples are: = = + = The following graphs show eamples = = 8 = + + 9

70 . MEP Pupil Tet Note that each curve has either a maimum or a minimum point which lies on its ais of smmetr. The curve has a maimum point when the coefficient of is negative as in the second eample, or minimum if the coefficient of is positive. Also the curve can cross the -ais twice, just touch it once or never meet the-ais. Cubic Functions Cubic functions involve an term and possibl, and constant terms as well. Some eamples are: =, = + + 8, =, = + The graphs below show some eamples. = = + = + (d) (e) (f) = + = = + + The graph of a cubic function can cross the -ais once as in eamples, and (e), touch the ais once and cross it once as in eample or cross the -ais three times as in eamples (d) and (f). In eamples, (d) and (f) the curve has a local minimum and a local maimum. Note how the shape of the curve changes when a and (e). is introduced. Compare eamples

71 MEP Pupil Tet Reciprocal Functions Reciprocal functions have the form of a fraction with as the denominator. Eamples of reciprocal functions are: = =, =, = The graphs below show some eamples. = = = The curves are split into two distinct parts. The curves get closer and closer to the aes as is clear in the diagrams. The curves have two lines of smmetr, = and =. Eercises. State whether each equation below would produce the graph of a linear, quadratic, cubic or reciprocal function. = = = (d) = + + (e) = + (f) =. Each of the following graphs is produced b a linear, quadratic, cubic or reciprocal function. State which it is for each graph. (d) (e) (f)

72 . MEP Pupil Tet. One of the graphs shown below is = +. Which one? A B C D. Which of the graphs shown below are reciprocal functions? A B C D. Each equation below has been plotted. Select the correct graph for each equation. = + A = = B (d) = + C D

73 MEP Pupil Tet. Match each graph below to the appropriate equation. A = + C B = + = D = + (d). Which of the following equations are illustrated b the graphs shown? Write the equation illustrated beside the number of each graph. = = = = = + = (i) (ii) (iii) (iv) Sketch a graph of the equation = + on a cop of this graph. (SEG)