Mathematics Stage 5 PAS5.1.2 Coordinate geometry

Transcription

1 Mathematics Stage PAS.. Coordinate geometr Part Graphing lines

2 Acknowledgments This publication is copright New South Wales Department of Education and Training (DET), however it ma contain material from other sources which is not owned b DET. We would like to acknowledge the following people and organisations whose material has been used: Extracts from Mathematics Sllabus Years 7-0 Board of Studies 00 Unit overview pp iii-v Part p., Part p., Part p.. Photograph Velodrome, courtes of Ccling Australia. Part p. Photograph Mount Hotham Victoria, courtes of Bob Lees Part p. Photograph Casio CFX 980GB Graphic calculator, courtes of Casio Computer Co,. Part p. 7 LTD COMMONWEALTH OF AUSTRALIA Copright Regulations 99 WARNING This material has been reproduced and communicated to ou on behalf of the New South Wales Department of Education and Training (Centre for Learning Innovation) pursuant to Part VB of the Copright Act 98 (the Act). The material in this communication ma be subject to copright under the Act. An further reproduction or communication of this material b ou ma be the subject of copright protection under the Act. CLI Project Team acknowledgement: Writer: Janine Angove Illustrators: Thomas Brown, Tim Hutchinson Editor: Ric Morante Desktop Publishing: Gale Redd Version Date: November 8, 00 Revision Date: August, 00 All reasonable efforts have been made to obtain copright permissions. All claims will be settled in good faith. Published b Centre for Learning Innovation (CLI) Wentworth Rd Strathfield NSW Copright of this material is reserved to the Crown in the right of the State of New South Wales. Reproduction or transmittal in whole, or in part, other than in accordance with provisions of the Copright Act, is prohibited without the written authorit of the Centre for Learning Innovation (CLI). State of New South Wales, Department of Education and Training 00.

3 Contents Part Introduction Part... Indicators... Preliminar quiz... Vertical lines...9 Gradients of vertical lines...9 Graphing vertical lines... Horizontal lines...7 Gradients of horizontal lines...7 Graphing horizontal lines...8 Intercepts... Graphing lines...9 How man points?... Suggested answers Part... Exercises Part...9 Part Graphing lines

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5 Introduction Part Each line on the number plane can be described using an equation. In this part ou will explore these equations and how the relate to the graph. Indicators B the end of part, ou will have been given the opportunit to work towards aspects of knowledge and skills including: understanding terms such as algebra, substitute, coordinate, horizontal, vertical, oblique, equation, intercept, linear and intersect using coordinates to graph vertical and horizontal lines identifing the x-axis as the line = 0 identifing the -axis as the line x = 0 identifing the x- and -intercepts of graphs constructing tables of values for a variet of linear equations graphing a variet of linear equations on the number plane. B the end of part, ou will have been given the opportunit to work mathematicall b: describing vertical and horizontal lines and their properties explaining wh the axes have equations. Source: Adapted from outcomes of the Mathematics Years 70 sllabus < _sllabus.pdf > (accessed 0 November 00). Board of Studies NSW, 00. Part Graphing lines

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7 Preliminar quiz Before ou start this part, use this preliminar quiz to revise some skills ou will need. Activit Preliminar quiz Tr these. Plot these points on the number plane below and label each with its capital letter. 0 x A (, ) B (, ) C (0, 0) D (, 0) E (0, ) If k = 0 find the value of these expressions. a k + b k c 8 k Part Graphing lines

8 Write the coordinates of the points shown on the number plane below. B A A B D 0 x C C D Use the diagram below to answer the following questions. Remember that to name a line ou use two capital letters for points on that line. A C H E I F D B G a b c What is the name of the vertical line? What is the name of the horizontal line? Where do EF and GH intersect? Complete the following sentences. a The x-coordinate for (, 7) is. b The -coordinate for (, 0) is. PAS.. Coordinate geometr

9 Evaluate these number sentences. (Evaluate means find the answer.) a b Solve these equations b finding the number that the pronumeral stands for. a x + 7 = 9 b = 0 c 7 = 9 d x = 0 Check our response b going to the suggested answers section. Part Graphing lines 7

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11 Vertical lines Lines can be described as either: horizontal (straight across) vertical (straight up and down) oblique (at an angle, not vertical or horizontal). In this section ou will explore aspects of vertical lines. The first feature that will be discussed about vertical lines is their gradient. Gradients of vertical lines On the number plane, oblique lines are said to have a positive gradient if the go up the graph from left to right, or a negative gradient if the go down. Positive gradients Negative gradients 0 x 0 x But what about vertical lines? Work though the following task to explore this question. Part Graphing lines 9

12 For this task, ou will need a pencil and a ruler. Use the vertical line shown on the graph below to complete this activit. 0 x Form earlier work ou know that gradient = (+ or ) rise run. The first thing to consider is whether the slope is positive or negative. Does the line go up or down the graph as it moves from left to right? Since the line doesn t move left to right, the first problem that arises in finding the gradient of a vertical line is that ou cannot decide whether it is positive or negative. Ok, put this problem aside for the moment and move on. Next ou need to calculate the number part of the gradient. Use the two points marked on the vertical line above to find the rise and the run. rise = run = The rise is eas. You need to move units from one point to the other. But what is the run? Strictl speaking, the run is zero because ou do not need to move across at all to get from one point to the other. Therefore, gradient = 0 Oops, another big problem! You cannot divide b zero. Even on a calculator the answer comes up as an error. 0 PAS.. Coordinate geometr

13 Summarising what ou have found: the gradient is neither positive nor negative the run is zero so the number for the gradient cannot be calculated. So the gradient of a vertical line cannot be calculated using the usual technique. In fact, vertical lines are said to have an undefined gradient. This means that ou cannot give a number for their gradient. The next section explores other aspects of vertical lines. Graphing vertical lines The points that make up a vertical line have something in common. Find what that is b completing the following activit. For this activit ou will need a pen and ruler. Use the diagram below to list the coordinates of all the points marked on the vertical line. A B A B 0 C x C D D What have all the points got in common? Part Graphing lines

14 You should have found that all the points on a vertical line have the same x- coordinate of. In fact, an vertical line ou draw will have the same propert, that is, all the points will share the same x-coordinate. This means that to describe which vertical line ou want to draw, all ou have to do is describe the x-coordinate. Algebra is a tool in mathematics that is used to write general rules without using words. To describe the vertical line in this diagram using English ou can sa the line where all the points have an x-coordinate of. To describe this line using algebra ou can just write the line x =. 0 x This equation, x =, is said to be the equation for the line. It is reall saing that the -coordinate can be anthing as long as the x-coordinate is. When ou are not talking about number planes and equations for lines, using x = might mean something ver different. Look at the following example. Follow through the steps in this example. Do our own working in the margin if ou wish. a Graph the line x =. b Does the point (, ) lie on this line? PAS.. Coordinate geometr

15 Solution a To draw the line ou need to recognise the equation as a vertical line then use a ruler to draw it at the correct place on the graph. For this example, the line goes through on the x-axis. The graph below shows the line. 0 x b The point (, ) does not lie on the line. The x-coordinate (the first one) is not. You can also see this easil if ou find the point on the graph above. Continue to explore vertical lines b completing the following activit. Activit Vertical lines Tr these. How do ou know that (, 9), (, 0) and (, ) all lie on the same vertical line? Part Graphing lines

16 a Plot these points on the number plane and draw a line to show that the all lie on the same vertical line. (, ) (, ) 0 x (, ) (, ) (, ) b Complete the equation for this line. x = Write the equation for each of these vertical lines. a 0 x b x PAS.. Coordinate geometr

17 Graph these vertical lines on the number plane provided. Write the equation along each line on the graph. a x = b x = c (Harder) x = 0 x Check our response b going to the suggested answers section. You have seen that the points on vertical lines have one thing in common: the all have the same first coordinate. You have also seen that ou can describe each vertical line using a simple equation: x = a where a is a number. But there is one special vertical line that has not been discussed and that is the -axis. This line also has an equation to describe it, x = 0. So even the vertical axis can be described using algebra. Continue to explore our understanding of vertical lines on the number plane b completing this exercise. Go to the exercises section and complete Exercise. Vertical lines. Part Graphing lines

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19 Horizontal lines In this section ou will explore features of horizontal lines on the number plane. Gradients of horizontal lines What is the gradient of a horizontal line? Explore this question b completing the following activit. For this activit ou will need a pen. Use the graph of the horizontal lines below to complete this activit. 0 x Answer these three questions before reading further. a b c Is the gradient positive, negative or neither? What is the rise between the two points shown? What is the run between the two points shown? Part Graphing lines 7

20 Your answers to these questions should have been. a b The gradient is neither positive nor negative (because the horizontal line does not go up or down the graph) The rise is 0 (because ou do not move up or down to travel from one point to another) c The run is. Using this information, ou can determine the gradient of this horizontal line. gradient = (+ or ) rise run = (+ or ) 0 = (+ or ) 0 Since zero is neither positive nor negative, the problem of not being able to decide on the sign of the gradient is solved. Horizontal lines have a gradient of zero because the are perfectl flat. And this is true for all horizontal lines. Graphing horizontal lines You have alread seen that vertical lines can be described using equations like x = or x = 7. This means that all the points on a vertical line have the same x coordinate. In this section ou will consider if a similar equation can be written for horizontal lines. 8 PAS.. Coordinate geometr

21 For this activit ou will need a pen and ruler. Plot all these points and join them with a straight line. (, ) (, ) (, ) 0 x (0, ) (, ) You should have graphed a horizontal line. What do all the points have in common? What do ou think the equation of this line might be? Use the equation of vertical lines to help ou. Each point on a horizontal line has the same -coordinate (second number). Therefore to describe which horizontal line ou want to graph ou simpl have to sa what the -coordinate is. To describe the horizontal line shown in this graph ou can sa all the points with a -coordinate of. Or ou can use the algebraic equation =. 0 x The following example shows how to graph horizontal lines and how to write their equations. Part Graphing lines 9

22 Follow through the steps in this example. Do our own working in the margin if ou wish. Use the number plane below for these tasks. 0 x a What is the equation of the horizontal line shown on the number plane? b Graph the horizontal line =. Solution a b All the points on the line have a -coordinate of so the equation is =. (You can also see that it cuts the -axis at.) You can plot some points that have a -coordinate of like (, ), (, ) and (, ). Then ou would draw the line through them. Or, if ou know where it will go alread, ou don t need to plot points. Just graph the line. The answer is shown below. = 0 x 0 PAS.. Coordinate geometr

23 Practise graphing horizontal lines and finding their equations b completing this activit. Activit Horizontal lines Tr these. Without graphing these points, explain how ou know the will all lie on the same horizontal line. (, ), (, ), (0, ) and, Write the equation for each line shown on the graph below. A B C x D E F 7 The equation of AB is The equation of CD is The equation of EF is Part Graphing lines

24 The x-axis is a horizontal line so ou can describe it using an equation. What would its equation be? Graph and label these lines on the number plane below. MN is = and PQ is = 0 x Check our response b going to the suggested answers section. Now ou know how to describe horizontal and vertical lines in both English and using the language of algebra. You have seen the patterns on the graph and what all the points have in common. The following website provides ou with more practice plotting points and writing equations for both vertical and horizontal lines. Access an interactive site dealing with vertical and horizontal lines on the number plane b visiting the CLI webpage < Select Mathematics then Stage. and follow the links to resources for this unit Patterns and algebra, PAS.. Coordinate geometr, Part. Show our understanding of this work b completing the following exercise. Go to the exercises section and complete Exercise. Horizontal lines. PAS.. Coordinate geometr

25 Intercepts In English, the word intercept means to take or seize on the wa from one place to another, like when ou intercept a messenger who is taking a note to someone else. In mathematics, the term intercept is used to describe the place where a curve or line meets the axes. The place where it cuts the x-axis is called the x-intercept, and not surprisingl the place where it cuts the -axis is called the -intercept. -intercept -intercept x x x-intercept two x-intercepts Sometimes the intercept cannot be seen on the graph because the number plane is not drawn large enough. Sometimes curves or lines don t have both tpes of intercepts because the don t actuall meet the axes. The -intercept does exist but it is off the graph. 0 x 0 x This curve never meets the x-axis so there are no x-intercepts. The following example shows how to describe intercepts on a number plane. Part Graphing lines

26 Follow through the steps in this example. Do our own working in the margin if ou wish. What are the intercepts of the lines shown on the graph below? C B 0 x A D Solution For the line AB, the x-intercept is and the -intercept is. You could also write the answer as: The intercepts for AB are x = and =. The line CD goes through the origin (0, 0). Therefore it cuts both axes at the same point. The answer to this question would be: The x- and -intercepts are both 0. You normall talk about points on a number plane b writing their two coordinates. But with these special intercepts ou just have to use one number for each because ou alread know the are on an axis. Intercepts can also be used to describe where a line is. Look at the following example. PAS.. Coordinate geometr

27 Follow through the steps in this example. Do our own working in the margin if ou wish. a Graph the line that has an x-intercept of and a -intercept of. b Use the intercepts to find the gradient of the line. Solution a Put a dot on the x-axis at and a dot on the -axis at. Use a ruler to carefull draw a line through the two points. Your line should go to the edges of the grid. The answer is shown in the graph below. 0 x b To find the gradient ou can draw a right angled triangle between an two points on the line and use the rule: gradient = rise run If ou use the intercepts, ou can see that the axes and the line alread form a right-angled triangle. The diagram below shows ou the triangle and the lengths of the rise and the run. Part Graphing lines

28 The gradient is negative because the line slopes down. 0 x gradient = Using the intercepts to find the gradient is onl useful when the intercepts are whole numbers. Now it is our turn to identif and use intercepts. Activit Intercepts Tr these. Use the following graph to complete these sentences. C B 0 x A D a b The intercepts of AB are x = and =. For the line CD, the x-intercept is and the -intercept is. PAS.. Coordinate geometr

29 a Graph the line with an x-intercept of and -intercept of. 0 x b (Harder) Use the intercepts to calculate the gradient of our line. (Harder) What tpe of straight line will not have a -intercept? Check our response b going to the suggested answers section. Continue to practise using intercepts b completing the following exercise. Go to the exercises section and complete Exercise. Intercepts. Part Graphing lines 7

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31 Graphing lines All lines on the number plane can be described b an algebraic equation. This equation describes how to find points on the line. It does this b describing the rule that links the two coordinates (the x and numbers). For example, the equation = x sas that the -coordinate is double the x-coordinate. Some of the points that fit this pattern are (, ), (, 0) and (, ). If ou graphed these three points ou would find that ou could draw a straight line through them, as in the diagram below x In fact, ou can pick an other point on this line, and ou would find that the -coordinate is double the x-coordinate. Tr it! The easiest wa to find a point that fits a pattern is to substitute an x value into the equation and work out the value. Part Graphing lines 9

32 You can collect our x and values into a neat table like those below for the equation = x. The table can be horizontal with the x values going across, or vertical with the x values going down. x x 0 0 Both these tables show the three points from the earlier graph: (, ), (, 0) and (, ). In this section ou will complete tables of coordinates for a variet of equations, then graph the pattern. All the patterns in this section will form a straight line when graphed. Patterns or equations that form straight lines when graphed are called linear. You have alread practised drawing horizontal and vertical lines from their equations. x = 0 x vertical line = 0 x horizontal line To graph an oblique straight line from its equation ou need to work out some points that fit the pattern, plot them and then join them. But how man points do ou need to plot? 0 PAS.. Coordinate geometr

33 How man points? How man points do ou need to plot before ou know which line to draw for an equation? There are an infinite number of straight lines that ou can draw through an single point so onl plotting one point will not be enough. Once ou plot two points, there is onl one straight line that can be drawn through them both. So ou reall onl need to know two points to draw the line. However, it is best to plot three points just to make sure ou haven t made a mistake. If the three points are all in the one line, then ou are probabl right. If the are not in a straight line, then ou know that ou have to check our working. Look at this example. Follow through the steps in this example. Do our own working in the margin if ou wish. Graph the line = x. Part Graphing lines

34 Solution Draw up a table and select three x values that will be simple to put into the equation. The ones chosen here are, and. It is a good idea to spread our x numbers out so that the dots are spread across the number plane. It is easier to draw the correct line through dots that are spread out. x Work out the values using the equation = x. You can do this in our head but the working is shown here to help ou understand. When x = = = When x = = = When x = = = 0 And so the completed table looks like this: x 0 Plot the points from the table on the number plane and join them with a straight line. The points are (, ), (, ) and (,0). 0 x PAS.. Coordinate geometr

35 You can select other values for x in the example above, but ou would still graph the same line. Tr some other values ourself and plot them on the number plane above. Now it is our turn to graph a straight line in the following activit. Activit Graphing lines Tr these. Complete the table given then graph = x on the number plane below. There is space below the graph for our working. Remember that x means x. x 0 0 x Part Graphing lines

36 Check our response b going to the suggested answers section. Sometimes ou have to solve an algebraic equation to find the value. You can do this using the guess and check method, or b working backwards. The following example shows ou both methods. Follow through the steps in this example. Do our own working in the margin if ou wish. a Complete the following table for the equation x + =. x b Use the table to graph the line x + =. Solution When ou substitute the x values ou get an algebraic equation that needs to be solved. You can solve them in our head or ou might need to use other methods. Each student below explains a different wa the solved each one. When x = the equation is + =. This sas plus what gives. I just knew the answer is =. PAS.. Coordinate geometr

37 I put x = into the equation and got + =. I guessed = and checked in m head. + = That didn t work. I needed a much bigger number. So I tried = = = 8 worked. I put x = into the equation and got + =. I had no idea what to guess so I decided to work backwards. The opposite of is + so I added to both sides of the equal sign. + + = + The answer is =. The completed table is: x 8 Part Graphing lines

38 b The graph is shown below. Notice that the line extends all the wa across the grid x Now ou know how to graph all tpes of lines: vertical, horizontal and oblique. You know about equations, tables of values, gradients and intercepts. All these ideas are included in the following activit. Activit Graphing lines Tr these. a Complete the table below for the equation = x +. x 0 PAS.. Coordinate geometr

39 b Graph = x + on the number plane below. 0 x c What are the intercepts of this line? The x-intercept is. The -intercept is. d Calculate the gradient of the line = x + b drawing a right-angled triangle between two of our points. Part Graphing lines 7

40 a Graph each of these lines on the number plane below. x = = x = Write the equation along each line to identif it. Space is provided below the graph for an working needed. 0 x b What is the gradient of the line = on the graph above? 8 PAS.. Coordinate geometr

41 c (Harder) Use the intercepts for x + = to find its gradient? (Harder) Colin tried to graph the line x =. x Oops! 0 x a Explain how Colin knew he had made a mistake. Part Graphing lines 9

42 b Correct the table and then graph the correct line. x 0 0 x Check our response b going to the suggested answers section. Combine all our knowledge about graphing straight lines to complete the following activit. Go to the exercises section and complete Exercise. Graphing lines. 0 PAS.. Coordinate geometr

43 Suggested answers Part Check our responses to the preliminar quiz and activities against these suggested answers. Your answers should be similar. If our answers are ver different or if ou do not understand an answer, contact our teacher. Activit Preliminar quiz Compare our points with the ones shown below. Did ou remember to label each point with its capital letter? A C 0 x D E B For each question, ou needed to substitute (replace) the pronumeral k with the number 0 then work out the answer. You can do the working in our head but the full solution is shown to help ou understand. a k + = 0 + = Part Graphing lines

44 b k = 0 = 0 Remember to put the sign in. c 8 k = 8 0 = 8 = A fraction line means divide. Then order of operation sas do the division first. The coordinates are the numbers that show the position. The first number tells ou how far to go across and the second number tells ou how far to go up or down. The coordinates are A (, ), B (, ), C (, ) and D (, 0). a CD (vertical means straight up and down) b c EF (horizontal means straight across) I (intersect means to cross) a b 0 The working is shown to help ou to understand. a = + = 8 b + 0 = = 7 There are man methods for solving equations. You can just know the answer, ou can guess the answer and check it b putting it into the equation, or ou can work backwards b using opposite operations. The answers are given below. a x = b = c = d x = 0 PAS.. Coordinate geometr

45 Activit Vertical lines You should have written something about the x-coordinates all being the same number,. (You could also plot them on a graph.) a 0 x b x = a x = b x = 8 The equation of each line is written along it. x = x = x = 0 x Part Graphing lines

46 Activit Horizontal lines You should have written something about all the points having the same -coordinate,. AB is =. CD is =. EF is =. = 0 (all the points on the x-axis have a -coordinate of zero) M N P 0 x Q Activit Intercepts a x = and = b The x-intercept is and the -intercept is a 0 x PAS.. Coordinate geometr

47 b The diagram below shows the right-angled triangle to be used. 0 x gradient = rise run = = The gradient is negative because the line goes down the graph. A vertical line, but not the -axis = 0. Activit Graphing lines x 0 0 Your graph should look like this no matter which points ou chose to use. 0 x Part Graphing lines

48 a x 0 b 0 x c The x-intercept is. The -intercept is. d You could use an two points on the line. The gradient is. The answers to parts a are shown on the graph. x = x = 0 x = PAS.. Coordinate geometr

49 b c The gradient is 0. (All horizontal lines have a gradient of zero.) The x-intercept is so the run is. The -intercept is so the rise is also. The gradient is positive so: gradient = = a You should have written something about the dots not being in a straight line. b The last value was wrong. x 0 0 x Part Graphing lines 7

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51 Exercises Part Exercises. to. Name Teacher Exercise. Vertical lines Write the equation for each of these vertical lines. a x b 0 x c (Harder) 0 x Part Graphing lines 9

52 Do the points (, ), (, ) and (, ) all lie on the same vertical line? Explain our reasons. Graph these vertical lines on the number plane provided. a x = b x = c (Harder) x = 0 x What is another name for the line whose equation is x = 0? 0 PAS.. Coordinate geometr

53 Exercise. Horizontal lines Write the equation of each line shown on the graph below. C D E x F 7 A 8 B 9 The equation of AB is (Harder) The equation of CD is (Harder) The equation of EF is Graph and label these lines on the number plane below. JK is x = and LM is x = x Part Graphing lines

54 (Harder) If ou draw a line through the two points (, 7) and (, 0) will the line be horizontal, vertical or oblique (neither horizontal nor vertical)? Explain our reasons. PAS.. Coordinate geometr

55 Exercise. Intercepts Use the following graph to complete these sentences. D A C 0 x B a b The intercepts of AB are x = and =. For the line CD, the x-intercept is and the -intercept is. Complete the following sentence. If a straight line has an x-intercept of 0 then it must have a -intercept of. Part Graphing lines

56 a Graph the line with an x-intercept of and -intercept of. 0 x b Use the intercepts to calculate the gradient of our line. What tpe of straight line will not have an x-intercept? PAS.. Coordinate geometr

57 Exercise. Graphing lines Rachel was asked to graph the line = x. Her table of values is shown below. I wonder if these are right. x a Plot her points and explain wh ou know she has made a mistake. 0 x b Correct the error in the table. Space is given here for an working. Plot the correct points and draw the line on the number plane above. Part Graphing lines

58 a Complete the table below for the line = x. Space is provided for working if needed. x b Use the table to graph the line on the number plane below. 0 x c d Is the line vertical, horizontal or oblique? Calculate the gradient b drawing a right-angled triangle between two points on the line. e What are the x and intercepts of the line? PAS.. Coordinate geometr

59 a (Harder) Graph the lines x = and x + = on the number plane below. Space is provided below the graph for an working and an tables ou ma want to draw x b Write the coordinates of the point where these two lines intersect (cross). Part Graphing lines 7