STRAND: GRAPHS Unit 1 Straight Lines

Transcription

1 CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text STRAND: GRAPHS Unit Straight Lines TEXT Contents Section. Gradient. Gradients of Perpendicular Lines. Applications of Graphs. The Equation of a Straight Line

2 CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Straight Lines. 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 B (x, ) Vertical CHANGE The gradient of a line between two points, A and B, is calculated using A (x, ) Horizontal CHANGE x gradient of AB = vertical change horizontal change - coordinate of B - coordinate of A = ( ) ( ) ( x- coordinate of B) ( x- coordinate of A) = x x

3 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Worked Example Find the gradient of the line shown in the diagram. x 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 x Horizontal CHANGE = Vertical CHANGE = Worked Example 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 = ( ) ( ) ( x- coordinate of B) ( x- coordinate of A) = = =

4 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Worked Example 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 - coordinate of B - coordinate of A = ( ) ( ) ( x- coordinate of B) ( x- coordinate of A) = = = ( ) So the gradient is. (, ) A (, ) B x Parallel lines have the same gradient. Worked Example The coordinates of the points A, B, C and D are listed below. A (, ) B (, ) C (, ) D (, ) Solution Show that the line segments AB and CD are parallel. Are the line segments AC and BD parallel? Gradient of AB = = = Gradient of CD = ( ) ( ) ( ) = = The line segments have the same gradients and so must be parallel. ( ) ( ) Gradient of AC = = 9 = ( ) Gradient of BD = = 9 = The line segments have the same gradient and so are parallel. The results in Worked Example mean that the quadrilateral ABDC is, in fact, a parallelogram.

5 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Exercises. Find the gradient of the line shown on the graph opposite. 9 x. Find the gradient of each line in the diagram below. 9 A C B 9 D E F x. Find the gradient of each line in the diagram below. 9 C A E F D B 9 x

6 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. Which of the lines in the diagram below have a positive gradient? D B C E I F A L H M J K N G x 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. The diagram shows the cross-section of a roof. Find the gradient of each part of the roof. m m m m m m

7 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. Find the gradient of the line that joins the points with the coordinates: (, ) and (9, ), (, ) and (, ), (, ) and (, ), (d) (, ) and (, ), (e) (, ) and (, ), (f) (, ) and (, ).. 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. Draw the line segment joining the points (, ) and (, ). Draw the line segment joining the points (, ) and (, ). (d) Are these lines parallel? Calculate the gradient of each line.. Calculate the gradient of the line segment joining the points (, ) and (, ). Calculate the gradient of the line segment joining the points (, ) and (, ). Are the two line segments parallel?. In each case below the coordinates of the points A, B, C and D are listed. Determine whether or not AB is parallel to CD. A (, ) B (, ) C (, ) D (, ) A (, ) B (, ) C (, ) D (, ) A (, ) B (, ) C (9, ) D (, ) (d) A (, ) B (, ) C (, ) D (, ). The vertices A B, C and D of a quadrilateral have coordinates A (, ) B (, 9) C (, ) D (, ) Calculate the gradient of each side of the quadrilateral. Calculate the length of each side of the quadrilateral. What is the name of this tpe of quadrilateral?. A line segment joins the points A (, ) and B (, ). Calculate the gradient of AB. The line segment CD is twice as long as AB and parallel to it. If point C has coordinates (, ), determine the coordinates of point D.

8 CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. Calculate the coordinates for three points on the line = x +. 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 = x and = x +. (e) What is the connection between the equation of a line and its gradient? (f) What do ou think the gradient of the line = x + will be?. Gradients of Perpendicular Lines In this section we explore the relationship between the gradients of perpendicular lines and line segments. Worked Example Plot the points A (, ) and B (, ), join them to form the line AB and then calculate the gradient of AB. On the same set of axes, plot the points P (, ) and Q (, ), join them to form the line PQ and then calculate the gradient of PQ. Measure the angle between the lines AB and PQ. What do ou notice about the two gradients? Solution The points are shown in the diagram. Gradient of AB = 9 = = 9 A B 9 x

9 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text The points P and Q can now be added to the diagram as shown below. 9 A B P Q Gradient of PQ = = 9 x The line PQ has been extended on the diagram, so that the angle between the two lines can be measured. The angle is 9, a right angle. Note In this case, the gradient of AB = ; the gradient of PQ = and the gradients multipl to give =. The product of the gradients of two perpendicular lines will alwas be, unless one of the lines is horizontal and the other is vertical. In the example above, gradient of AB = Note gradient of PQ = = Gradient of PQ =. gradient AB If the gradient of a line is m, and m, then the gradient of a perpendicular line will be m.

10 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Worked Example Show that the line segment joining the points A (, ) and B (, ) is perpendicular to the line segment joining the points P (, ) and Q (, ). Solution Gradient of AB = Gradient of PQ = = = Gradient of AB Gradient of PQ = = =. So the line segments AB and PQ are perpendicular. Exercises. On a set of axes, draw the lines AB and PQ where the coordinates of these points are A (, ) B (, ) (d) P (, 9) Q (, ) Are the lines perpendicular? Calculate the gradient of AB. Calculate the gradient of PQ. (e) Check that the product of these gradients is.. In each case, decide whether the lines AB and PQ are parallel, perpendicular or neither. A (, ) B (, ) P (, ) Q (, ) A (, ) B (, 9) P (, ) Q (, ) A (, ) B (, ) P (, ) Q (, ) (d) A (, ) B (, ) P (, ) Q (, ). The points P (, ), Q (, ), R (, ) and S (, ) are the vertices of a quadrilateral. Calculate the gradient of each side of the quadrilateral. Is the quadrilateral a parallelogram? Is the quadrilateral a rectangle?. A triangle has vertices A (, ), B (, ) and C (, ). Show that the triangle is a right-angled triangle. 9

11 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. Show that the triangle with vertices A (, ), B (, ) and C (, ) is not a rightangled triangle.. The coordinates of the point A, B, C and D are listed below. A (, ) B (, ) C (, ) D (, ) Show that ABCD is a square.. The points A (, ), B (, ), C (, ) and D (, ) are the vertices of a quadrilateral. Show that this is not a rectangle. Show that this is a parallelogram.. The lines AB and PQ are perpendicular. The coordinates of the points are A (, ) B (, ) P (, ) and Q (, q) Determine the value of q.. Applications of Graphs In this section some applications of graphs are considered, particularl conversion graphs and graphs to describe motion. The graph on the right can be used for converting US Dollars into and from Jamaican Dollars. (As currenc exchange rates are continuall changing, these rates might not be correct now.) US Dollars Jamaican Dollars Distance (m) D A distance-time graph of a car is shown opposite. The gradient of this graph gives the velocit (speed) of the car. The gradient is steepest from A to B, so this is when the car has the greatest speed. The gradient BC is zero, so the car is not moving. B C A Time (s)

12 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text The area under a velocit-time graph gives the distance travelled. Speed Finding the shaded area on the (m/s) graph shown opposite would give the distance travelled. The gradient of this graph gives the acceleration of the car. There is constant acceleration from to seconds, then zero acceleration from to seconds (when the car has constant speed), constant deceleration from to seconds, etc. Time (s) Worked Example A temperature of C is equivalent to 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: Solution C to Fahrenheit, F to Celsius. Taking the horizontal axis as temperature in C and the vertical axis as temperature in F gives two pairs of coordinates, (, ) and (, ). These are plotted on a graph and a straight line drawn through the points. D F (, ) (, ) 9 C

13 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Start at C, then move up to the line and across to the vertical axis, to give a temperature of about F. Start at F, then move across to the line and down to the horizontal axis, to give a temperature of about C. Worked Example 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. 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.

14 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Worked Example The graph shows how the velocit of a bird varies as it flies between two trees. How far apart are the two trees? I Velocit Speed (m/s) 9 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. I Velocit Speed (m/s) B A C 9 Time (s) Area of A = = Area of B = = Area of C = = Total Area = + + =

15 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text So the trees are m apart. Note that the units are m (metres) because the units of velocit are m/s and the units of time are s (seconds). Worked Example B C Velocit (m/s) A H D G time (sec) E F The graph shows the velocit of a to train for a time period of seconds. State (i) (ii) (iii) the maximum velocit attained b the train the total number of seconds, during which the train travelled at the maximum speed. the time period during which the velocit is negative. Calculate the acceleration of the train. (i) during the first seconds (ii) for the time period of x seconds where x 9. Calculate the distance travelled (i) between and seconds (ii) over the entire journe of seconds.

16 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Solution (i) m/s (from B to C) (ii) (iii) seconds (as B is reached at seconds and C at seconds) Velocit is negative from C (time seconds) to E (time 9 seconds) i.e. from 9 = seconds (i) Acceleration is the gradient of AB = = m/s (ii) For x 9, there is acceleration = ) = m/s (or deceleration of m/s ) (i) Distance travelled from to seconds + = = + + = m ( ) + + (ii) Distance travelled from A to D = + = m B smmetr, distance travelled from D to G = m Distance travelled for G to H = = m So total distance travelled = + + = m Worked Example V ms - B C A D t secs The velocit-time graph above, not drawn to scale, shows that a train stops at two stations, A and D. The train accelerates uniforml from A to B, maintains a constant speed from B to C and decelerates uniforml from C to D.

17 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Using the information on the graph, (i) calculate, in ms, the train's acceleration (ii) show that the train took secs from C to D if it decelerated at ms. (iii) If the time taken from A to D is seconds, calculate the distance in metres between the two stations. An express train stops at station P and station Q. On leaving P, the train accelerates uniforml at the rate of ms for seconds. Then it decelerates uniforml for 9 seconds and comes to a stop at Q. (i) (ii) Using unit to represent units on BOTH axes, sketch the velocit-time graph to represent the information above. Using our graph, determine the maximum velocit the rate at which the train decelerates. Solution (i) Acceleration = = ms (ii) If it takes t seconds from C to D, then = t = seconds t (iii) Time from B to C = ( + ) = seconds Total distance = area under graph = + + m = + + m = m = m (i) See diagram on following page. (ii) Maximum velocit = ms Deceleration = 9 = ms

18 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text (i) Velocit (ms -) t (seconds) Exercises. Use the approximation 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, lbs to kilograms, kg to pounds, (d) kg to pounds.. Use the approximation that gallons is about the same as litres to draw a conversion graph. Use the graph to convert: gallons to litres, litres to gallons.. The graph shows how the distance travelled b a taxi increased. H F G Distance (m) B C D E A Time (s)

19 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text How man times did the taxi stop? Find the velocit of the taxi on each section of the journe. On which part of the journe did the taxi travel fastest?. The distance-time graph shows the distance travelled b a car on a journe to the shops. F D E Distance (m) B C A 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 velocit on this part. On which part of the journe did the car travel at its lowest velocit? What was this velocit?. The graph below shows how the speed of an athlete varies during a race. Velocit Speed (m/s) Time (s) What was the distance of the race?

20 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. The graph below shows how the velocit of a truck varies as it sets off from a set of traffic lights. Velocit Speed (m/s) Time (s) Find the distance travelled b the truck after seconds, seconds, seconds.. Kevin runs at a constant speed for seconds. He has then travelled m. He then walks at a constant speed for seconds until he is 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 Zoe and Janice changes during a race from one end of the school field to the other end, and back. Wend Zoe Jodie Janice Distance from start (m) Time (s) Describe what happens during the race. 9

21 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text 9. Find the area under each graph below and state the distance that it represents. Velocit Speed (mph) Time (mins) Velocit Speed (mph) Time (hours) (d) Velocit Speed (mm/s) Time (mins) Velocit Speed (m/s) Time (mins). For each distance-time graph, find the velocit 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)

22 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. The graph represents a swimming race between Vincent and Damion. Distance from start (m) Vincent Damion Time (seconds) At what time did Damion overtake Vincent for the second time? What was the maximum distance between the swimmers during the race? Who was swimming faster at seconds? How can ou tell?. The speed-time graph below shows the journe of a car from : am to : am. Speed (km/h) : am 9: am : am : am Time Using the graph, determine the time at which the speed of the car was km/h the distance the car travelled for the entire journe the average speed of the car for the entire journe.

23 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. The graph below represents the -hour journe of an athlete. Distance (km) Time (h) What was the average speed during the first hours? What did the athlete do between and hours after the start of the journe? What was the average speed on the return journe?. The Equation of a Straight Line The equation of a straight line is usuall written in the form = mx + c where m is the gradient and c is the intercept. = mx + c Gradient = m c x Worked Example Find the equation of the line shown in the diagram. x

24 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text 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 Vertical change = Horizontal = change x So the value of m is. The line intersects the -axis at, so the value of c is. The equation of a straight line is = mx + c In this case = x + Worked Example 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. vertical change gradient = horizontal change The intercept is. = = So m = and c = and the equation of the line is Worked Example Draw the line x =. Draw the line =. = x + or = x Intercept = x Write down the coordinates of the point of intersection of these lines. Horizontal = change Vertical change =

25 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Solution For the line x = the x-coordinate of ever point will alwas be. So the points (, ) (, ) (, ) all lie on the line x =. For the line = the -coordinate of ever point will alwas be. So the points (, ) (, ) and (, ) all lie on the line =. The graph in shows that the lines intersect at the point with coordinates (, ). x = (, ) (, ) (, ) x x = (, ) (, ) (, ) = x Worked Example The point with coordinates (, 9) lies on the line with equation = x +. Determine the equation of the perpendicular line that also passes through this point. Solution The line = x + has gradient. The perpendicular line will have gradient and so its equation will be of the form = x + c As the line passes through (, 9), we can use x = and = 9 to determine the value of c. 9 = + c 9 = + c c = The equation of the perpendicular line is therefore = x + The graph shows both lines. 9 9 = x + = x + x

26 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Worked Example A (, 9) G B (, ) O x ( ) In the diagram above, not drawn to scale, AB is the straight line joining A 9, and B (, ). (d) (e) Calculate the gradient of the line, AB. Determine the equation of the line, AB. Write the coordinates of G, the point of intersection of AB and the -axis. Write the equation of the line through O, the origin, that is perpendicular to AB. Write the equation of the line through O that is parallel to AB. Solution Gradient of AB = 9 = ( ) = Equation of line AB = mx + c = x + c As it passes through (, ), = + c c = i.e. = x + x =, so = + =. G is point (, ). (d) Gradient = = m = So = x (as it passes through the origin) (e) The line parallel to AB has gradient, so the line through O has equation = x

27 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text Exercises. Find the equation of the straight line with: gradient = and -intercept =, gradient = and -intercept =, gradient = and -intercept =, (d) gradient = and -intercept =, (e) gradient = and -intercept =.. Write down the gradient and -intercept of each line. = x + = x = x + (d) = x ( ) ( ) (e) = x + (f) = x (g) = x + (h) = x. The diagram shows the straight line that passes through the points (, ) and (, ). x Find the gradient of the line. Write down the -intercept. Write down the equation of the straight line.

28 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. Find the equation of each line shown in the diagram below. A B C D E F 9 x. Write down the gradient of each line and the coordinates of the -intercept. = x = x + = x (d) = x + (e) = x (f) = x (g) x + = (h) = x ( ). Find the equation of the line that passes through the points with the coordinates below. (, ) and (, ) (, ) and (, ) (, ) and (, ) (d) (, ) and (, ). The graph opposite can be used for converting gallons to litres. Find the equation of the line. Draw a similar graph for converting litres to pints, Litres given that litres is approximatel pints. Use the horizontal axis for pints. Find the equation of the line drawn in. 9 Gallons x

29 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. Find the equation of each line in the diagram below. A B C E D 9 9 F x 9. 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 axis. Draw a line through the points and find its equation. What was the velocit of the ball when it was thrown upwards?. The line = x + c passes through the point (, ). Find the value of c. The point (, ) lies on the line = x + c. Find the value of c. The line = mx + passes through the point (, ). Find the value of m.

30 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. Television repair charges depend on the length of time taken for the repair, as shown on the graph. c Charge ($) Charge ( ) Charge ( ) 9 t Time (minutes) The charge is made up of a fixed amount plus an extra 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 or less. Calculate the maximum amount of time which can be spent on the repair.. 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 On a cop of the following grid, plot the points and draw a graph to illustrate this information. 9

31 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text w quantit of salt (grams) t temperature of water ( C) 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. ( ). A straight line passes through the point P, and has a gradient of. Write down the equation of this line in the form = mx + c. Show that this line is parallel to the line x =.

32 . CMM Subject Support Strand: GRAPHS Unit Straight Lines: Text. The cost of hiring a taxi consists of a basic charge plus a charge per km travelled. The graph below shows the total cost in pounds () for the number of km travelled (x). Cost (km) Cost ($) 9 Distance (km) x What is the cost of hiring a taxi to travel a distance of (i) km (ii) km? What distance in km was travelled when the cost was? (d) What is the amount of the basic charge? Calculate the gradient of the line. (e) Write down the equation of the line in the form = mx + c. (f) Calculate the cost of hiring a taxi to travel a distance of km.. A line has equation = x. Another line is parallel to this and passes through the point with coordinates (, ). Determine the equation of this line.