PATTERNS AND ALGEBRA. He opened mathematics to many discoveries and exciting applications.

Transcription

1 PATTERNS AND ALGEBRA The famous French philosopher and mathematician René Descartes (596 65) made a great contribution to mathematics in 67 when he published a book linking algebra and geometr for the fi rst time. Descartes showed that man relationships and formulas could be represented b simple graphs. 6 He opened mathematics to man discoveries and eciting applications

2 In this chapter ou will: Wordbank graph points on the number plane from a table coefficient the number in front of the variable 6789 of values, using an appropriate scale in an equation (for eample, in = + 7, the build a geometric pattern, identif a number coeffi cient of is ) pattern, complete a table of values, describe constant term the number that is not the pattern in words and 5678 algebraic smbols, and multiplied b a variable in an equation (for represent the relationship on a graph eample, in = + 7, the constant term is 7) etend the line joining a set of points to show that coordinates an ordered pair of numbers that there 5678 is an infi nite number of ordered 95 pairs that describes the location of a point on the number satisf a given liner relationship plane, for eample (, 5) form a table of values for a relationship (for increasing 89 graph a graph that slopes upwards eample = ) b substituting a set of from left to right appropriate values and graphing the number pairs linear equation a formula whose graph is a on a number plane straight line derive a rule for a set of points that has been satisfies follows a rule; used to describe a value graphed on a number plane b forming a table (for eample = 7) or ordered pair [for eample of values (, 5)] that can be substituted into an equation to graph more than one line on the same set of aes make that equation true and compare the graphs to determine similarities slope the steepness of a line and differences, for eample parallel, or passing -intercept the value where a line crosses the through the same point -ais of the number plane graph two intersecting lines on the same set of aes and use the scale to read their point of intersection

3 6- Brainstarters 6 - Tables of values 6- Tables of values Start up Cop and complete the following tables for the given rules: a m = p + b = c v = 6 w p w 6 m v d = -- e d = 5e + f p = m -8-7 e 5 m d p 6-6 Find the formula relating and in each of these tables: a b c Finding the rule d e - f Find the formula for each of these tables of values: a m b p c n 8 q d h e m - f z F -5 5 a - b u v Skillsheet 6- The number plane 6- The number plane 6- Number plane review 6-5 Coordinates code puzzle Write the coordinates of each point, A to L, shown on the number plane on the right. 5 Which of the points on the number plane in Question are: a in the st quadrant? b in the nd quadrant? c in the rd quadrant? d in the th quadrant? e on the -ais? f on the -ais? E 6 C D F B G A I - J K - H L -6 7 NEW CENTURY MATHS 8 STAGE ISBN:

4 6- Graphing patterns on the number plane Eample 6-7 Patterns and rules The rule = describes this pattern: Complete this table and graph the points on a number plane. Number of shapes, 5 Number of toothpicks, Solution Plot the points (, ), (, 6), (, 9), (, ) and (5, 5). You can see that the points lie on a straight line. Note: The top row of the table shows the -coordinates, and the bottom row of the table shows the -coordinates Eercise 6- For each matchstick pattern and rule below, cop and complete the table and then graph the points on a number plane. a = E Appendi Number plane grid 5 b = 6 5 ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 75

5 c = + 5 d = For each pattern below: i cop and complete the table below. Number of shapes, 5 Perimeter, ii graph the relationship on a number plane. a Squares b Pentagons TLF L 96 Bridge builder: Comple pentagons For each of the graphs drawn in Questions and, do the points fall in a straight line? 6- Graphing linear equations A rule that is represented b a straight line when graphed on a number plane is called a linear equation. ( Linear means of a line.) Some eamples are: = + = 5 = = 6 Graphing = on the number plane This is a table of values for = : NEW CENTURY MATHS 8 STAGE ISBN:

6 Graphing the table of values gives the points shown on the number plane on the right. Note that these points lie in a straight line (, -) - - (, -) (-, -) How man more points can we graph? What happens if we choose more values of, larger values such as =, or, and negative values such as -, or -5, and in-between values such as --, or --? When the values in the table are added to the graph, the look like this: (-5, -8) (-, -6) (, ) (, - ) (, - ) (The first three points are now shown in red.) Ever ordered pair of values that follows the rule = lies on the same line. Ever point on the line follows the rule =. (, 7) There is an infinite number of points that satisf the rule. ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 77

7 So we can rule a straight line through the points to graph the linear equation, and place arrows at the end to show that the line etends infinitel in both directions = Graphing linear equations Appendi Number plane grid 6- Using a graphics calculator Eample Graph = + on a number plane. Solution Step : Create and complete a table of values. Choose values close to zero for eas calculation and graphing Step : Step : Graph the table of values on a number plane. Rule a straight line through the points, place arrows at each end, and label the line with its equation. = NEW CENTURY MATHS 8 STAGE ISBN:

8 Eample Graph =. Solution = Eample Does the point (5, 8) lie on the line =? Solution We could graph the line of = to see if (5, 8) lies on it, but a much simpler wa is to check whether (5, 8) follows or satisfies the rule =. In (5, 8), = 5 and = 8. Substituting = 5 into the rule = gives: = 5 = 7 8 (5, 8) does not lie on the line =. [In fact, (5, 7) does.] Eercise 6- For each linear equation: i cop and complete the table of values ii graph the equation on a number plane. a = + b = c = E 6-8 A page of number planes Graphing linear equations Appendi Number plane grid ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 79

9 d = e = + f = 6 E E Graph each of these linear equations: a = + b = c = d = e = + f = 5 Graph each of these linear equations: a = - + b = c = d = - e = f = a Look at the graphs ou drew in Question. Wh do ou think the are called increasing graphs? b Look at the graphs ou drew in Question. Wh do ou think the are called decreasing graphs? 5 B looking at each of the following linear equations, predict whether its graph will be increasing or decreasing: a = 5 b = + 9 c = d = e = f = - 6 On which of the following lines does the point (, ) lie? Select A, B, C or D. A = - + B = + C = D = + 7 Test whether each of the following points lies on the given line b: i testing whether its coordinates satisf the linear equations ii eamining the graphs of the lines ou drew in Question and. a (, 5) = + b (, ) = c (5, ) = d (, 5) = 6 e (-, -5) = + f (, ) = 8 NEW CENTURY MATHS 8 STAGE ISBN:

10 Mental skills 6 Maths without calculators Finding the formula for a table of values Eamine this eample. 6-6 Finding the rule The values in the top row of the table are consecutive, so we can use the number pattern of the bottom row to find the multiplier. The numbers in the bottom row go up b each time, so the formula has a multiplier of. That is, the formula begins with = To find the number that needs to be added or subtracted at the end, test values of with the multiplier. For = : = So:? = 5 ( 7 = 5) So the formula is = 7. Find the formula for each of these tables of values. a b c d e f g h a b m 5 6 n 8 6 k 5 p r z d h t 5 6 w ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 8

11 6-6- Finding the equation of a line Finding linear equations 6- Linear equations matching activit Eample 5 Find the equation of the line shown Solution Choose some points from the graph to create a table of values with the values increasing in steps of Since the values go up b, the multiplier is. So the equation is in the form = +. When =, = + =. So the equation is: = +. Eercise 6- Use the graph on the right to give the coordinates of three points on each of these lines. = = a = b = 9 c + = 7 6 d = = = NEW CENTURY MATHS 8 STAGE ISBN:

12 For each graph below: i cop and complete the given table of values ii find the equation of the line. a b E c d e f ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 8

13 g h Spreadsheet 6- Graphing linear equations A spreadsheet can be used as a table to create formulas and to graph straight lines. Use the link provided to go to a spreadsheet activit involving linear equations. Working mathematicall Reasoning Non-linear graphs For each rule given below: i cop and complete the given table ii plot the points on a number plane iii join the points with a smooth curve. a = b = c = What is the difference between the formulas in Eercises 6- and 6- (which have straight line graphs) and the formulas above (which do not)? 8 NEW CENTURY MATHS 8 STAGE ISBN:

14 Just for the record Latitude and longitude Position on the Earth s surface is indicated b a coordinate sstem (latitude and longitude). Latitude measures how far north or south places are in relation to the Equator. Longitude measures how far east or west places are from the Greenwich meridian (an imaginar line from the North Pole to the South Pole passing through Greenwich in London, England). Ever town, cit or countr on Earth can be located b its latitude and longitude coordinates. For eample, the position coordinates of Canberra are 5 S 9 E. When ou look up a place in an atlas, the inde tells ou the map number, followed b the latitude and longitude. Find out the latitude and longitude of our town or cit. 6- Comparing linear equations In a linear equation, the number in front of the is called the coefficient of. For eample, in = 5 the coefficient of is, while in = + the coefficient of is. ( = ) The number without the, is called the constant term. For eample, in = + the constant term is, in = the constant term is -, and in = 6 the constant term is. The are constant terms because the do not contain a variable ( constant means not changing ). We will now eamine the similarities and differences = + between various tpes of lines and linear equations. In particular, we will look at: -intercept the slope or steepness of the line the -intercept of a line (the value where it crosses the -ais). In the graph of = + on the right, the -intercept is Linear equations matching activit ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 85

15 Eample 6 The lines = and = + have been graphed on the same aes on the right. a What is the coefficient of in =? b What is the coefficient of in = +? c What do ou notice about the slope of both lines? d What is the constant term in =? e What is the constant term in = +? f What do ou notice about the -intercepts of both lines? = + = Solution a The coefficient of in = is. b The coefficient of in = + is also. c The two lines have the same slope, the are parallel. d The constant term in = is. e The constant term in = + is. f The -intercepts of both lines are different. The -intercept of = is while the -intercept of = + is. (The -intercept of each line is the same as the constant term in its equation.) Eample 7 The lines = +, = - + and = + have been graphed on the same aes on the right. a How are the lines similar? b How are the linear equations similar? c How are the lines different? d How are the linear equations different? = = + = Solution a All three lines have the same -intercept (). b All three linear equations have the same constant term (). c The three lines have different slopes (the are not parallel). d The three equations have different coefficients of. 86 NEW CENTURY MATHS 8 STAGE ISBN:

16 Eercise 6- Graphics calculators ma be used for this eercise. On the same set of aes, graph the following equations: = = + = a What is the same about all three lines? b What is the same about all three linear equations? c What is different about the lines? d What is different about the linear equations? Repeat Question for the following equations: = - + = - = - + Repeat Question for the following equations: = = = + On the same set of aes, graph the following equations: = = = -- a What is the same about all three lines? b What is the same about all three linear equations? c What is different about the lines? d What is different about the linear equations? 5 Repeat Question for the following equations: = + = + = -- + E A page of number planes Appendi Number plane grid E 7 6 Repeat Question for the following equations: = - = - = -- 7 The graphs in Question 5 are all increasing while the graphs in Question 6 are all decreasing. What feature of their equations indicates this? 8 Cop and complete each of the following: a The equations of lines with the same slope have the same. b The graphs of lines with the same slope are. c The equations of lines with the same -intercept have the same. d The graphs of lines with the same -intercept cross the -ais. e The equations of lines that are increasing have a coefficient of. f The equations of lines that are decreasing have a coefficient of. ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 87

17 Using technolog Comparing graphs of linear equations We are going to use a spreadsheet to compare the graphs of =, = and = for the values = - to =. Step : Set up a spreadsheet as shown below. Step : In cell B enter =A (This means that =.) Then click cell B. Use Fill Down to cop this formula from cell B down to B6. Step : In column C we need a spreadsheet formula to represent =. Since the values are in column A, in cell C we enter =*A. Click cell C and Fill Down from cell C to C6. Step : In column D, we need a spreadsheet formula to represent =. Write a formula in cell D that again refers to cell A. Click cell D and Fill Down from cell D to D6. Step 5: Highlight all cells. Use Chart Wizard and XY (Scatter) and click to graph all three graphs on the same aes. Choose the option to save As new sheet. Compare the three graphs. What do ou notice? In our workbook, describe the pattern and important features. Set up a separate spreadsheet for each set of three linear equations shown below. Use appropriate formulas to complete each column and use Chart Wizard (as in Step 5 above) to graph each set of three graphs. Compare the three graphs in each set and, in our workbook, describe the pattern and important features of each set. a b c Investigate other sets of straight lines that ou have studied in this chapter. Use a spreadsheet to graph them (as shown above) and, in our workbook, describe an patterns and important features that ou see. 88 NEW CENTURY MATHS 8 STAGE ISBN:

18 6-5 Intersecting lines Eample 8 6- Intersection of lines a On the same set of aes, graph = + and =. b Write the coordinates of the point of intersection of the graphs and show that the point satisfies both equations. Solution 6 5 a Complete a table of values for each equation, then graph both lines. = = = + b The point of intersection is (, ). Substitute = into each equation and see if it is true that =. = + = = + = = = 6 = So the point (, ) satisfies both equations. = Eercise 6-5 Which of the following is the point of intersection of = + and = - + 5? Select A, B, C or D. A (, ) B (, ) C (, ) D (, 5) For each pair of equations given below: i graph both equations on the same set of aes ii give the coordinates of the point of intersection of the graphs and show that the point satisfies both equations. a = and = - + b = + and = c + = and = d = + and = e = - and = + f = -- and = - + The following three equations form a triangle when graphed on the number plane. Graph them and write the coordinates of the vertices of the triangle. = + = -- = ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS A page of number planes Appendi Number plane grid E 8

19 Graph these equations on the same set of aes: = + = = = - a What shape is formed b these lines? b Write the coordinates of each verte of this shape. Using technolog Intersecting lines We will graph the two straight lines = + 5 and =, to find their point of intersection. Step : Set up a spreadsheet as shown on the right. Step : In cell B, enter the formula =A+5. Click on cell B and Fill Down to cell B. Step : In cell C, enter the formula =-A. Click on cell C and Fill Down to cell C. Step : Highlight all three columns and use the Chart Wizard to graph as XY (Scatter) and click on. Step 5: From the graph, read the point of intersection of the two straight lines. Find the point of intersection for each pair of straight lines below, b setting up a separate spreadsheet for each pair of lines and using the Chart Wizard (as above) to graph each pair of lines on the same set of aes. a b c 6-8 A page of number planes Appendi Number plane grid Power plus Graph each of these lines on a number plane. What is special about them? a = b = - c = Graph each of these equations and state whether the graph is a straight line or a curve. a = b + = 5 c = + d = e + = - f = 9 NEW CENTURY MATHS 8 STAGE ISBN:

20 Chapter 6 review Language of maths arrows ais/aes coefficient constant term coordinates decreasing evaluate increasing infinite intersection linear linear equation non-linear number plane parallel rule satisf slope substitute table of values variable -ais -ais -intercept 6- Linear equations crossword What does the word linear mean? Which part of the number plane is the -ais? What is the name given to the place where a line crosses the -ais? What is the constant term in the equation = - +? 5 What is another word for the steepness of a line? 6 What does it mean if the coordinates of a point satisf the equation of a line? Topic overview Write in our own words what ou have learnt about linear equations and their graphs. What parts of this topic did ou have difficult with? Discuss them with a friend. Where might ou use the skills acquired in this chapter? Cop and complete this summar of the main points in this chapter. The - _ = 5 and the line is de_ - _ These lines are _ and the have the same_ 6 The constant term of = is _ = + 5 Point of _, (, ) s_ both equations = + and = _ The line = + 5 slopes up from left to right so it is _ _ - -6 The graph of = + _ ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 9

21 Topic test Chapter 6 Eercise 6- Chapter revision Cop and complete the table for this pattern and graph the points on the number plane. = + 5 Eercise 6- For each linear equation given below: i cop and complete the table of values ii graph the equation on a number plane iii name a point on the line that is not in the table. a = b = - c = - d = -- Eercise 6- Eercise 6- Eercise 6- For each graph in Question, state whether the line is increasing or decreasing. Does each of the following points lie on the line = +? a (, ) b (-, ) c (-, -) d (, ) 5 Identif the multiplier and complete the formula for each table: a = b = + c = Eercise 6-6 Find the equation of each of the following lines. a b NEW CENTURY MATHS 8 STAGE ISBN:

22 7 On the same set of aes, graph the following equations: = = + = a What is the same about the three lines? b What is the same about the three linear equations? 8 On the same set of aes, graph the following equations: = - = = a What is the same about all three lines? b What is the same about the three linear equations? 9 For each pair of equations given below: i graph both equations on the same set of aes ii give the coordinates of the point of intersection of the graphs and show that the point satisfies both equations. a = + and = - b = and = Eercise 6- Eercise 6- Eercise 6-5 ISBN: CHAPTER 6 GRAPHING LINEAR EQUATIONS 9