MEP Practice Book ES Find the gradient of each line in the diagram below. 3 C

Transcription

1 Graphs MEP Practice Book ES.5 Gradient. Find the gradient of each line in the diagram below A C B Which of the following lines have positive negative (c) zero gradient in the grid below? 4 A B C D 4 4 H E G 4 F

2 .5 MEP Practice Book ES. Find the gradient of the four lines shown on the grid below. 4 A C B 4 4 E D 4 4. Find the gradient of the line that joins the points with coordinates (, ) to (4, 8) (, ) to (7, ) (c) (, 4) to (, ) (d) (, ) to (, 8) (e) (4, ) to (, ) (f) (5, ) to (, 5) 5. For the line =, complete the coordinates A (,? ), B (,? ), C (,? ) D (,? ) What is the gradient of the line AB, BC, CD? What do ou notice? 6. A quadrilateral is formed b joining the points A to B, B to C, C to D, D to A, where A, B, C, D have coordinates A (, ), B (, ), C (, ), D (, ) Which of the four lines found has the largest gradient? Which of the four lines has the smallest gradient?

3 MEP Practice Book ES.6 Application of Graphs. The diagram shows a conversion graph between Pounds ( ) and German Deutschmarks (DM). 5 Pounds ( ) Deutschmarks (DM) 4 Use the graph to write down how man Deutschmarks can be echanged for, Pounds can be echanged for 4 Deutschmarks. (LON). 6 4 Cost ( ) Units of electricit The graph shows the cost, in pounds, of electricit used b one person. The cost is made up of a fied standing charge, plus the cost of the number of units of electricit used.

4 .6 MEP Practice Book ES Use the graph to find the standing charge in pounds, the cost, in pence, of one unit of electricit. (LON). This graph can be used to convert between pounds ( ) and French Francs French Francs 4 5 Pounds ( ) Use the graph to convert (i) 6 to French Francs, (ii) 55 French Francs to pounds. 6 7 Jim is going to the USA. The bank will echange.58 dollars for. It also charges a commission of % for the echange. Jim wants to echange 5 for dollars at the bank. Work out how man dollars he would receive. (LON) 4

5 MEP Practice Book ES 4. The table below shows volumes epressed in imperial and metric measure. Imperial measure in fluid ounces Metric measure in litres Plot these entries on a suitable grid. One of the entries in the metric column is incorrect. (i) Draw the conversion graph showing the connection between imperial and metric measure and write down the incorrect metric entr. (ii) Use our graph to estimate the correct metric entr. (NEAB) 5. Hannah goes to the shop to bu a loaf of bread. The shop is 8 m from the house. She leaves home at 5 and walks to the shop at a stead speed. She takes 6 minutes to reach the shop and then 5 minutes to bu a loaf of bread. She then walks home at a stead speed arriving at 548. On a cop of the diagram draw a distance-time graph to represent her journe. Distance from home (km) 5 5 Time 6 (SEG) 5

6 .6 MEP Practice Book ES 6. The graph below shows the distance travelled b a car on a journe to work. 4 F Distance (m) 8 D E 6 4 B C A Time (s) 48 The car stopped at two sets of traffic lights. How long did the car spend waiting at each set of lights? On which part of the journe did the car travel fastest? (c) (i) How far did the car travel? (ii) How long did it take for the whole journe? (iii) What was the car's average speed for the whole journe? 7. The graph below shows the speed of a train as it sets off from a station. 4 6 Speed (m/s) Time (s) 6

7 MEP Practice Book ES Find the distance travelled b the train after 4 seconds 8 seconds (c) 6 seconds What is the formula that connects the time of travel and distance travelled for t 6? 8. The graphs of the average weight for different heights for women and men are shown. 9 8 Men 7 Women Weight (kg) Height (cm) Jim and his wife Linda are both 6 cm in height. Use the graphs to estimate the difference in their weights. 8 Arthur and his wife Pam both weigh 75 kg. Use the graphs to estimate the difference in their heights. Show all our working. (c) The actual difference in the heights of Arthur and Pam is cm. Give a possible reason wh the graphs give a different answer. (SEG) 7

8 .6 MEP Practice Book ES 9. The table shows the repaments required on loans of different amounts, for ear. Amount of loan ( ) Monthl Repament ( ) Plot these pairs of values on a cop of the grid below. Join them with a straight line. 4 Monthl repament ( ) Amount of loan ( ) I can afford to repa 8 a month. Use the graph to find out the largest amount I could borrow. (c) Use our graph to find the monthl repament on a loan of. (d) Phil borrows 5. Altogether his monthl repaments amount to more than 5. How much more? (MEG). The following distance-time graph shows the journes made b a van and a car starting from Oford, travelling to Luton, and returning to Oford. How far had the car travelled when it met the van for the second time? Calculate, in miles per hour, the average speed of the car between 95 and. (c) During which period of time was the van travelling at its greatest average speed? 8

9 MEP Practice Book ES Distance (miles) car van Time (SEG). 5 Distance in miles from Manchester 5 am am noon pm pm pm 4pm 5pm 6pm 7pm 8pm Time The graph represents part of Mrs Hinton's journe from Manchester to London. Mrs Hinton stopped for a rest at a service station. (i) Write down the time at which she stopped. (ii) For how long did she stop? For part of her journe Mrs Hinton had to slow down because of a traffic queue. For how man miles did she travel at this lower speed? Mrs Hinton spent an hour at a meeting in London. She then returned home to Manchester, travelling at a stead speed of 5 miles an hour. (c) Use this information to complete the graph of her journe. (LON) 9

10 .6 MEP Practice Book ES. The diagram shows the travel graph of a train. 8 6 Distance from station 4 (km) Time (minutes) Find the greatest speed at which the train travelled. Give our answer in km/h. Calculate the average speed for the whole journe in km/h. (NEAB).7 Scatter Plots and Lines of Best Fit. The etension of a spring for a variet of attached weights is given in the table below. Mass (grams) 4 6 Etension (cms) Plot the data on a scatter diagram. What tpe of correlation does this table show? Draw a line of best fit. What do ou think would be the etension for a mass of 5 grams?. Annie asked a four teenagers to sa how much time the spent doing homework one evening, and how much time the spent watching TV. Here is a scatter diagram to show the results.

11 MEP Practice Book ES A D Number of hours spent doing homework B C Number of hours spent watching TV Which of the four points, A, B, C or D, represents each of the statements shown below? Write down one letter for each of the names. LUCY sas I spent most of m evening doing homework. I onl watched one programme on TV. TOM sas I watched a lot of TV last night and I also did a lot of homework CHRIS sas I went out last night. I didn't do much homework or watch TV. (c) (d) Make up a statement which matches the fourth point. What does the graph tell ou about the relationship between time spent watching TV and time spent doing homework? Annie also drew scatter diagrams which showed that:

12 .7 MEP Practice Book ES Older students tend to spend more time doing homework than ounger students. There is no relationship between the time students spend watching TV and the time students spend sleeping. On a cop of the aes below, show what Annie's scatter diagram ma have looked like. 4 Hours spent doing homework 8 Age of students (ears) Hours spent sleeping 5 Hours spent watching TV 4. Each week during the summer season, a seaside resort recorded the rainfall and the number of deckchair tickets sold. Some of the results are plotted on the scatter diagram on the net page. What does the scatter diagram tell ou about the connection between the rainfall and the number of deckchair tickets sold? On a cop of the diagram draw in a line of best fit. (c) In the first week of June onl 5 deckchair tickets were sold. How much rain do ou think the resort had that week?

13 MEP Practice Book ES number of 4 deckchair tickets sold mm of rain (NEAB) 4. Ten bos of different ages were set the same General Knowledge test. The results are shown in the table below. Bo A B C D E F G H I J Age (months) Score The mean of the ages of the bos is 58 months. Calculate the mean of their scores.

14 .7 MEP Practice Book ES The results in the table have been plotted on the following scatter graph. 4 Marks scored Age in months 8 (i) (ii) (iii) Does the scatter graph show the sort of result ou would epect? Eplain our answer. On a cop of the scatter graph draw a line of best fit. Taking age into account, to which bo would ou award a prize for the best performance? 5. The table below shows the number of Compact Discs (CDs) and the number of Long Plaing Records (LPs) that were sold from 984 to Number of CDs (millions) Number of LPs (millions) 7 million CDs were sold in 99. Write the number 7 million in figures. In which ear did the sale of CDs overtake the sale of LPs? (i) Draw a scatter graph to show the sale of CDs against the sale of LPs. (ii) What does our scatter diagram tell ou about the connection between the sale of CDs and the sale of LPs? 4

15 MEP Practice Book ES 6. Tom breeds hamsters. The number of hamsters is epected to treble each ear. Tom had hamsters on..96. Date Estimated number of hamsters On a suitable grid, draw a smooth curve to represent this information. Use our grid to estimate the number of hamsters Tom would have on (c) Write down an epression to find the estimated number of hamsters n ears after..96. (LON) 7. Below are the ears and times of some world records for running the mile. Glen Cunningham sec Roger Bannister sec Michael Jaz sec John Walker 975 sec Steve Cram sec Noureddine Morcelli 99 5 sec These data are used to plot a scatter diagram. 5 4 Time (s) Year 5

16 .7 MEP Practice Book ES (i) On a cop of the scatter diagram, draw the line of best fit. (ii) Sebastian Coe ran a new world record in 979. Use the line to estimate his time. (iii) Eplain wh the line ou have drawn can onl be used to estimate times for a limited number of ears. (c) Roger Bannister's actual time of 9. seconds is known to be correct to the nearest tenth of a second. What is the shortest time that it could actuall be? In 99 Noureddine Morcelli's time for running the mile was 5 seconds. B taking 5 miles to be equal to 8 kilometres, calculate what his time for the 5 metres would have been, assuming that his average speed was the same. Give our answer to the nearest second..8 The Equation of a Straight Line. Find the equation of the straight line with gradient = and - intercept = gradient = and - intercept = (c) gradient = and - intercept = (d) gradient = and - intercept =. Write down the gradient and -intercept of each of the following lines. = = + 4 (c) = (d) = + (e) = ( + ) (f) = +. The diagram shows the straight lines passing through the points (, ) and (4, ). Find the gradient of the line, the -intercept of the line, (c) the equation of the line

17 MEP Practice Book ES 4. Write down the gradient and -intercept of each of the following lines. = 4 = + (c) = 4 (d) = 5 (e) = 4 (f) = 5. Find the equation of each line shown in the diagram below. 5 C D E 4 B A Find the equation of the line that passes through the points with coordinates: (, ) and (, 5) (, 5) and (, ) (c) (, ) and (4, ) (d) (, ) and (, ) 7

18 .8 MEP Practice Book ES 7. The scatter diagram represents the profits made b a compan over the ears 965 to Profit (millions) Year Use the diagram to calculate an estimate of the profit the compan would epect to get in the ear if this trend continues. (NEAB) 8. The line = + c passes through the point (, 8). Find the value of c. 9. The line = m + passes through the point (, 7). Find the value of m.. The charges made b a removal firm consist of a fied charge of 5, and a variable charge of 5 per mile travelled. Write down the formula for the total cost,, in terms of the distance travelled, miles. Draw a graph of this relationship for and use it to estimate the distance travelled when the total cost is.. Complete a cop of the table of values for =. = 8

19 MEP Practice Book ES On a cop of the following grid plot our values for and. Join our points with a straight line (c) Write down the coordinates of the points where our graph crosses the -ais. (LON). This table shows the diameter, d, and the circumference, c, of four circular objects. The have been measured to the nearest centimetre. Object d c p coin cm 6 cm tin of beans 6 cm 8 cm saucer 7 cm cm plate cm cm On a cop of the following grid, plot c against d for each object. Draw a straight line to show the relationship between c and d. Write down the equation of this straight line. 9

20 .8 MEP Practice Book ES c d

21 MEP Practice Book ES.9 Horizontal and Vertical Lines. Write down the equation of each line marked in the following diagram. C B A D E F. Draw the lines =, = 7, =, =. Write down the coordinates of the points where the cross and find the area of the rectangle enclosed b these lines.. Draw the line =, =, = 4, =. What are the coordinates of the centre of the rectangle formed? 4. Draw the square which has corners at the points with coordinates (, ), (, 4), (, ) and (, 4) What are the equations of the lines that form the sides of the square? 5. Draw the lines =, = 8, =. What is the area of the triangle enclosed?

22 MEP Practice Book ES. Solutions of Simultaneous Equations b Graphs C Use the graph above to solve = + = = 7 7 (c) = + 6 = + 6 = +. Solve graphicall the following sets of simultaneous equations. + = 4 + = 5 + = = (c) 5 = (d) 4 + = + = =

23 MEP Practice Book ES The diagram shows the graphs of the equations + and = 5 Use the diagram to solve the simultaneous equations + = = 5 (LON) 4. Jane bus litres of oil and 4 litres of petrol for. Richard bus litres of oil and litres of petrol for. The cost of litre of oil is. The cost of litre of petrol is. Therefore + 4 = and + = Draw the graphs of these equations. What is the cost of one litre of petrol? (SEG) 5. Fift-one students went to a pop concert. Let represent the number of men. Let represent the number of women. The number of women is related to the number of men b the two equations On the following graph, the line + = 5 = + + = 5 has been drawn Draw the line = + on a cop of the graph. Use the graph to write down the number of men and women who went to the pop concert.

24 . MEP Practice Book ES (c) At another pop concert there were fewer than 5 students but there were at least men. B drawing another line on our graph, find the region that represents the possible values of and. Label the region with the letter R. 5 4 Number of women 4 5 Number of men (SEG) 6. Beth was asked to draw the graph of = + 4. She plotted the si points shown in the following diagram. (i) From the shape of the graph, how can ou tell that one of the points is in the wrong place? (ii) On a cop of the diagram, draw the graph of = + 4. B drawing another straight line on the diagram, solve the simultaneous equations = + 4 = 4

25 MEP Practice Book ES (MEG) 7. Errol's house has a meter which measures the amount of water he uses. Errol can pa on Tariff A for the number of water units that he uses. The graph on the net page can be used to find out how much he must pa for his water on Tariff A. Use the graph to find how much he must pa when he used (i) 6 units, (ii) 4 units. Errol uses water units. This costs c. Use the information from the graph to find a formula for c in terms of. Instead of Tariff A, Errol could pa for his water on Tariff B. The table shows how much Errol would have to pa for his water on Tariff B. Number of water units used () Cost ( c) (c) Plot a graph on a cop of the grid to show this information. 5

26 . MEP Practice Book ES Errol wants to be charged the smaller amount for the water he uses. (d) Use the graph to find how man units Errol can use before Tariff A becomes dearer than Tariff B Cost in Number of units. Graphs of Common Functions. State whether each equation below would produce the graph of a function. linear, quadratic, cubic or reciprocal = = (c) = 5 (d) = (e) = (f) = + 4 6

27 MEP Practice Book ES. Each of the following graphs is the sketch of a linear, quadratic, cubic or reciprocal function. State which it is for each graph. (c) (d) (e) (f). Each equation below has been sketched. Select the most suitable graph for each equation. A: = B: = C: D: = E: = = F: = + (c) (d) (e) (f) 7

28 . MEP Practice Book ES 4. Which of the following functions is illustrated b each of the graphs below? = +, = +, =, = +, = A B (SEG) 5. The radius, r, and value, v, of gold coins were measured and recorded. r (cm) v ( ) Which of these graphs represents the information shown in the table? v v A r B r v v C r D r Which of these equations describes the information shown in the table? v = k r, v = kr, v = kr, v = k r where kis a constant. (SEG) 8

29 MEP Practice Book ES 6. A cuboid has the dimensions shown. cm 5 cm 4 cm Not to scale The volume, V, of the bo is given b ( )( ) V = 4 5 Draw a graph of V against for 4. Use our graph to find the maimum volume of the cuboid. Find the maimum total surface area of the cuboid when the volume is cm. (SEG) ( ) can be written in the form + p q, where p and q are integers. Find the value of (i) p (ii) q Hence, or otherwise, (i) sketch the graph of = (ii) write down the minimum value of (LON). Graphical Solutions of Equations. Draw a graph of = for. Use our graph to solve the equations = 6 =. The following sketch shows the graph of = + 9

30 . MEP Practice Book ES Use the graph to write down the value of the positive solution to + 8 = (LON). = 4 Complete a cop of the table of values. On a cop of the following grid draw the graph of = 4. (c) (d) B drawing a suitable straight line on the grid, solve the equation 4 = Using the method of trial and improvement, or otherwise, solve the equation 4 = correct to one decimal place.

31 MEP Practice Book ES (LON) h s The diagram shows part of the graph of = 4 +.

32 . MEP Practice Book ES Use a cop of the graph to find approimate solutions in the range < < of 4 + = B drawing suitable straight lines on our grid, find approimate solutions in the range < < of the equations (i) 4 = (ii) 5 + = (LON) 5. Ale is using "trial and improvement" to solve the equation =. First he tries = and finds the value of values of find a solution of the equation decimal place. You must show all our working. is. B tring other =, correct to one (i) Draw the graph of =, for values of from to 4. (ii) B drawing suitable line on our graph show that the equation = has two solutions. (SEG) 6. 4

33 MEP Practice Book ES The diagram shows the graph of = for values of from to 4. Use a cop of the graph to find three values of which satisf the equation =. The equation = k is satisfied b onl one value of between and 4. What can be said about the number k? (c) (i) On a cop of the diagram, draw the reflection of the graph in the -ais. (ii) Write down the equation of this reflection. (MEG) 7. Complete the table of values for the graphs of = and = = 5 6 = (i) On graph paper draw the graphs of = and = + 6 (ii) Use our graphs to solve the equation + 4 =. (LON) 8. Draw the graph of = 5 and = for Use our graph to estimate, correct to one decimal place, the solution of + 5 = (SEG)

34 . MEP Practice Book ES 9. A child's to consists of a set of different sized blocks which are in the shape of cubes. The cost, C pence, of making each block is made up of two parts: A fied cost of 4 pence; and a cost that is proportional to the cube of the length, centimetres, of the block. When = ; C = 56. Find the equation connecting and C. The cost, D pence, of decorating each block is given b the equation D = 4 + Another child's to is in the shape of a stick. The cost, D pence, of decorating a stick of length, centimetres, is given b the equation D = Use a graphical method to find the value of when the cost of decorating the block and the stick is the same. (SEG). The graph of n = 8 5. is given. t n On a cop of the grid below draw a graph so that the equation can be solved. 8 5 t. = t t 4 5 4

35 MEP Practice Book ES (i) Use our graph to find the value of t at the point of intersection of the two curves. (ii) 8 Write the equation 5. = t in the form 4 =..... t Simplif the right hand side as far as possible. (c) For the equation n = 8 5. estimate the rate of decrease of n when t =. t (d) Estimate, b drawing, the gradient of the tangent at the point when t = 4, on the graph ou have drawn. (SEG). John places a cake in his freezer. The temperature, T C, of the cake after t minutes is given b the formula T t = ( ) 8 Cop and complete the table below. t(minutes) 4 T C Draw the graph of T against t. (c) (d) John knows that the cake's temperature is 4 C when he places it in the freezer. He does not know the formula for its temperature after t minutes. He estimates that its temperature will fall b C ever minute. On our grid, draw the graph showing how John thinks the temperature will var during the first three minutes. Use our graph to find the time when the estimated temperature is the same as the true temperature of the cake. 5