Chapter12. Coordinate geometry

Transcription

1 Chapter1 Coordinate geometr Contents: A The Cartesian plane B Plotting points from a table of values C Linear relationships D Plotting graphs of linear equations E Horizontal and vertical lines F Points on lines G Using graphs to solve equations

2 6 COORDINATE GEOMETRY (Chapter 1) Opening problem When Tiffan makes a mobile phone call, the cost of the call depends on the time that she spends on the call. The relationship between the cost and the time of the call is shown on the graph alongside. Things to think about: a Is the relationship between cost and time linear? b B how much does the cost of the call increase for each minute spent on the call? c Can ou use this graph to determine: i the cost of a 3 minute call ii the length of a call which costs $? cost ($) time (min) 6 8 The graph in the Opening Problem is a straight line. We sa there is a linear relationship between the variables. Historical note Sir Isaac Newton is recognised as one of the great mathematicians of all time. His achievements are remarkable considering mathematics, phsics, and astronom were enjoable pastimes after his main interests of chemistr, theolog, and alchem. Despite his obvious abilities, there is a stor from Newton s childhood which indicates that even the greatest thinkers can find sill solutions for simple problems. He was asked to go out and cut a hole in the bottom of the barn door for the cats to go in and out. He decided to cut two holes: one for the cat and a smaller one for the kittens. After completing school, Newton was initiall made to work on a farm, but when his uncle discovered his enthusiasm for mathematics it was decided that he should attend Cambridge Universit. Newton s contribution to the field of coordinate geometr included the introduction of negative values for coordinates. In his Method of Fluions, Newton suggested eight new tpes of coordinate sstems, one of which we know toda as polar coordinates.

3 COORDINATE GEOMETRY (Chapter 1) 7 A THE CARTESIAN PLANE The number grid alongside is a Cartesian plane, named after René Descartes. The numbers or coordinates on it allow us to locate the eact position of a point on the plane. We start with a point of reference O called the origin. Through it we draw two fied lines which are perpendicular to each other. The are a horizontal line called the -ais, and a vertical line called the -ais. The -ais is an ordinar number line with positive numbers to the right of O and negative numbers to the left of O. Similarl, the -ais has positive numbers above O and negative numbers below O. This vertical line is the -ais This is the origin, O. This horizontal line is the -ais. PLOTTING POINTS ON THE CARTESIAN PLANE To specif the position of a point on the number plane, we use an ordered pair of numbers in the form (, ). For eample, the position of the point alongside is described b the ordered pair (3, ). We sa that the point has coordinates (3, ). To help us identif particular points, we often refer to them using a capital letter. For eample, consider the points A(1, 3), B(, 3), and C(, ). To plot the point A(1, 3): ² start at the origin O ² move right along the -ais 1 unit ² then move upwards 3 units. We sa that and 1 is the -coordinate of A 3 is the -coordinate of A. To plot the point B(, 3): ² start at the origin O ² move right along the -ais units ² then move downwards 3 units. To plot the point C(, ): ² start at the origin O ² move left along the -ais units ² then move downwards units C(-, -) DEMO The -coordinate is alwas given first. It indicates horizontal movement awa from the origin A(1, 3) (3, ) B(, -3)

4 8 COORDINATE GEOMETRY (Chapter 1) QUADRANTS The and -aes divide the Cartesian plane into four regions referred to as quadrants. These quadrants are numbered in an anti-clockwise direction as shown alongside. nd quadrant 3rd quadrant 1st quadrant th quadrant Eample 1 Self Tutor Plot the points A(3, 5), B( 1, ), C(0, 3), D( 3, ), and E(, ) on the same set of aes. B(-1, ) A(3, 5) Start at O and move horizontall, then verticall. is positive is negative is positive is negative. - - D(-3, -) - E(, -) C(0, -3) EXERCISE 1A 1 State the coordinates of the points P, Q, R, S, and T: S P - 6 R - Q - T On the same set of aes, plot the following points: a A(, 5) b B(, ) c C( 5, 0) d D( 1, ) e E(, 3) f F(0, 3) g G(5, 1) h H( 5, 1) 3 State the quadrant in which each point in lies.

5 COORDINATE GEOMETRY (Chapter 1) 9 Eample Self Tutor On a Cartesian plane, show all the points with positive -coordinate and negative -coordinate. This region is the th quadrant. This shaded region contains all points where is positive and is negative. The points on the aes are not included. On different sets of aes, show all the points with: a -coordinate equal to 1 b -coordinate equal to 3 c -coordinate equal to 0 d -coordinate equal to 0 e negative -coordinate f positive -coordinate g negative and -coordinates h negative and positive -coordinates. 5 State the quadrants in which I might find a point with: a coordinates with the same sign b coordinates with different signs. 6 Consider the set of points f(0, 0), (1, 3), (, 6), (3, 9)g. a b c Plot the points on a set of aes. Determine whether the points lie in a straight line. Determine which of the following rules fits the set of points: A = +1 B = +3 C =3 D =3 1 E =3 7 Consider the set of points f(0, 3), (1, ), (, 1), (3, 0), (, 1)g. a b c Plot the points on a set of aes. Determine whether the points lie in a straight line. Determine which of the following rules fits the set of points: A = +1 B = +3 C =3 D =3 1 E =3 8 a Plot the points ( 1, ) and (, 1) on the same set of aes, and draw the line passing through these points. b On the same set of aes as a, plot (0, ) and (, ), and draw the line passing through these points. c Determine the coordinates of the point where the two lines meet.

6 50 COORDINATE GEOMETRY (Chapter 1) B PLOTTING POINTS FROM A TABLE OF VALUES Ton plas football for his local club. The number of goals he has scored so far this season is shown below in a table of values: Game number () Goals scored () We can displa these values graphicall b plotting the and -values on a number plane. We plot the points (1, 3), (, ), (3, 0), (, 3), and (5, 1) as shown. 5 (, ) (1, 3) (, 3) (3, 0) (5, 1) 5 EXERCISE 1B 1 For each of the following tables of values, plot the points on a number plane: a c b d Which of the sets of points in 1 form a straight line? 3 A school teacher records the number of students absent from class ever da for a week: Da number () Absent students () Plot these values on a number plane. The table below shows the average minimum temperature for Canberra over the last 1 months. Month number () Average minimum temperature ( ± C) a b c Plot these values on a number plane. Do these points lie in a straight line? Comment on our answer to b. Would it be reasonable for the points to lie in a straight line?

7 COORDINATE GEOMETRY (Chapter 1) 51 5 The height of a seedling is measured Week number () each week, and the results recorded in the table shown. Height ( cm) a Plot these points on a number plane. b Do the points lie in a straight line? c If this pattern continues, predict the height of the seedling after: i 6 weeks ii 8 weeks. 6 The set of points in the table alongside lie in a straight line. a b c Use the first two points to draw the line on a number plane. Complete the table of values. Identif the point on the straight line which lies on the: i -ais ii -ais C LINEAR RELATIONSHIPS Consider the pattern: 1st nd 3rd... We can construct a table of values which connects the diagram number n to the number of dots D. To go from one diagram to the net we need to add two more dots. The equation which connects n and D in this case is D =n +1. n 1 3 D The number of dots D depends on the diagram number n. We sa that n is the independent variable and D is the dependent variable. PLOTTING A LINEAR RELATIONSHIP We can plot the relationship between D and n on a number plane. We place the independent variable n on the horizontal ais, and the dependent variable D on the vertical ais. This is an eample of a linear relationship because the points lie in a straight line. In this case the values in between the points are meaningless. For eample, we cannot have a 1 th diagram. We therefore do not connect the points with a straight line. However, there are man situations where it is sensible to connect the points with a straight line. In these cases, we can use the line to answer questions involving values between the given points. 9 D n

8 5 COORDINATE GEOMETRY (Chapter 1) Eample 3 Self Tutor Ma has 10 litres of fuel left in his car s petrol tank. When he fills it at the petrol station, petrol runs into the tank at 15 litres per minute. The petrol tank can hold 70 litres. a b c Identif the independent and dependent variables. Make a table of values for the number of litres L of petrol in the tank after time t minutes, and plot the graph of L against t. Is the relationship between L and t linear? d Is it sensible to join the points graphed with a straight line? e For ever time increase of 1 minute, what is the change in L? f g Find the number of litres of petrol in the tank after 1:5 minutes. At what time will there be 50 litres of petrol in the tank? a b The number of litres of petrol in the tank depends on the time it has been filling. ) time is the independent variable and the number of litres of petrol is the dependent variable. Each minute the tank is filling adds another 15 litres of petrol. t (min) L (litres) L (litres) t (min) The independent variable is placed on the horizontal ais. c d e f g The points lie in a straight line so the relationship is linear. Yes, as Ma could add petrol for :5 minutes, sa, or put 56 litres of petrol in the tank. For ever time increase of 1 minute, L increases b 15 litres. After 1:5 minutes there are 3:5 litres of petrol in the tank. There are 50 litres of petrol in the tank after about :7 minutes.

9 COORDINATE GEOMETRY (Chapter 1) 53 EXERCISE 1C 1 Each week a department store emploee receives a basic salar of $300. In addition, she is paid a bonus of $0 for each new member she signs up to the store s rewards club. Let I be the emploee s income and m be the number of members she signed up to the rewards club. a What are the independent and dependent variables? b Construct a table and draw a graph of income I against members m where m =0, 1,, 3,..., 10. c Is the relationship linear? d e Is it sensible to join the points with a straight line? For each new member signed, what is the emploee s increase in income? Mangoes can be bought for $3:50 each. a Cop and complete the table: Number of mangoes, n Cost, $C b Plot the graph of C against n. c Identif the independent and dependent variables. d e Is the relationship between C and n linear? Is it sensible to join the points with a straight line? f For each etra mango bought, what is the change in C? g Find the cost of 5 mangoes. h How man mangoes could be bought for $:50? 3 Simon is filling a large container with sports drink for his football team. It alread contains 3 litres when he starts to fill it up using L bottles. a Make a table of values for the volume of sports drink S in the container after Simon has emptied n full bottles of sports drink into it. Consider n =0, 1,,..., 8. b What are the independent and dependent variables? c Plot the graph of S against n. d e Is the relationship between S and n linear? Is it sensible to join the points graphed with a straight line? f For each full bottle of sports drink added, what is the change in S? g What volume of sports drink is in the container after Simon has emptied 1 bottles into it? h How man full bottles must be emptied into the container so it contains 15 L in total?

10 5 COORDINATE GEOMETRY (Chapter 1) Adrian has just bought an old cookbook. He notices that the cooking temperatures are given in degrees Fahrenheit ( ± F), whereas he is familiar with degrees Celsius ( ± C). a Draw a set of aes as shown. b Identif the independent and dependent variables. 300 temperature ( C) c There is a linear relationship between ± F and ± C. The boiling point of water is ± F or 100 ± C. The freezing point of water is 3 ± For0 ± 150 C. Mark these points 100 on our graph and join them with a straight line. 50 temperature ( F) d Find the point where the number of degrees -100 Celsius equals the same number of degrees Fahrenheit. What is the temperature? -100 e Help Adrian b converting these temperatures into ± C: i a cake must be cooked at 350 ± F ii a roast must reach an internal temperature of 15 ± F. f Use our graph to complete the following table: Temperature in ± F Temperature in ± C PRINTABLE GRID D PLOTTING GRAPHS OF LINEAR EQUATIONS The equation = 1 1 describes a relationship between two variables and. For an given value of, we can use the equation to find the value of. These values form the coordinates (, ) of a point on the graph of the equation. The value of depends on the value of, so the independent variable is and the dependent variable is. When =, = 1 ( ) 1 = 1 1 = From calculations like these we construct a table of values: When =, = 1 () 1 =1 1 = The points ( 3, 1 ), (, ), ( 1, 1 1 ), (0, 1),... all satisf = 1 1 and lie on its graph.

11 COORDINATE GEOMETRY (Chapter 1) 55 Notice also that if =1:8 then = 1 (1:8) 1 =0:9 1 = 0:1 ) (1:8, 0:1) also satisfies = 1 1. = Qw -1 In fact, there are infinitel man points which satisf = 1 1. The points make up a continuous straight line which continues indefinitel in both directions. We indicate this using arrow heads (1.8, -0.1) Eample Self Tutor Consider the equation =. a Construct a table of values using = 3,, 1, 0, 1,, and 3. b Draw the graph of =. a b EXERCISE 1D 1 For each of the following equations: i construct a table of values using = 3,, 1, 0, 1,, 3 ii plot the graph. a = b = c = d = e = 1 f = 1 g = 1 + h = 1 + i = j =3 k =1 l = 1:5 1

12 56 COORDINATE GEOMETRY (Chapter 1) Eample 5 Self Tutor Use technolog to draw the graph of =1. TI-8 Plus Casio f-9860g Plus GRAPHICS CALCULATOR INSTRUCTIONS Click on the icon for help. Use technolog to draw the graph of: a = + b = 1 c =7 d = 3 + Discussion Eamine the graphs ou have drawn, and the corresponding equations. equation do ou think controls: ² the steepness of the line ² whether the graph slopes upwards or downwards ² where the graph cuts the -ais? What part of the E HORIZONTAL AND VERTICAL LINES Consider the line with equation =. At first it ma be unclear how we should complete our table of values, because is not mentioned in the equation. However, the equation means that no matter what the value of is, the value of is alwas :

13 COORDINATE GEOMETRY (Chapter 1) 57 We can plot these points on a number plane. The result is a horizontal line. It includes all points with -coordinate, such as (:5, ) and ( 1:85, ). 6 (-1.85, ) (.5, ) = - - All horizontal lines have equations of the form = k. Similarl, the line with equation = consists of all points with -coordinate. (, 3), (, 0:), and (, :3) are eamples of points which lie on this line. The result is a vertical line. - = (, 3) (, 0.) - (, -.3) - All vertical lines have equations of the form = k. Eample 6 Self Tutor Draw the graph of: a = b = 3 a The line = consists of all points with -coordinate. It is a horizontal line. b The line = 3 consists of all points with -coordinate 3. It is a vertical line. =-3 =

14 58 COORDINATE GEOMETRY (Chapter 1) EXERCISE 1E 1 Draw the graph of: a =1 b =3 c = d = e =1:5 f = 1 g =0 h =0 a On the same set of aes, draw the graphs of = and = 3. b Write down the coordinates of the point where the lines meet. F POINTS ON LINES If we know the equation of a line, we can easil check to see whether a given point lies on the line. We simpl replace b the -coordinate of the point being tested, and see if the equation produces the -coordinate. Eample 7 Self Tutor Determine whether (, 11) and ( 3, 3) lie on the line with equation =3 +5. If =, =3 +5 =6+5 =11 ) (, 11) lies on the line. If = 3, =3 ( 3) + 5 = 9+5 = Since 6= 3, ( 3, 3) does not lie on the line. EXERCISE 1F 1 a Determine whether the following points lie on the line = 3: i A(, 1) ii B( 1, 5) iii C(, 7) b Check our answers b drawing the graph of = 3, and plotting the points A, B, and C. Consider the graph of =3 shown alongside. a State the coordinates of P, Q, and R. b Verif that: i Q and R lie on the line ii P does not lie on the line. c State the coordinates of the point with -coordinate which lies on the line =3. P 3 Q =3-3 R

15 COORDINATE GEOMETRY (Chapter 1) 59 3 Determine whether the point: a (, 6) lies on the line =3 6 b (3, ) lies on the line = 5 c ( 1, 5) lies on the line =6 d (0, ) lies on the line = e (:5, 1:5) lies on the line =5 3 f (6, ) lies on the line = g ( 5, ) lies on the line = 5. a Show that (, 5) lies on the line = +1. b Show that (, 5) also lies on the line =7. c d Use a and b to cop and complete: The graphs of = +1 and =7... at the point (, 5). Graph = +1 and =7 on the same set of aes to illustrate our findings. G USING GRAPHS TO SOLVE EQUATIONS In Chapter 7, we used algebraic methods to solve linear equations. However, we can also solve linear equations graphicall. In question of the previous eercise, we saw that the graphs of = +1 and =7 meet at the point (, 5). We sa that (, 5) is the point of intersection of the lines. =+1 When =, + 1 = () + 1 = 5 and 7 =7 =5 also. This means that = is the solution to the equation +1=7. 5 (, 5) =7- So, we can solve equations b graphing the LHS and the RHS of the equation on the same set of aes. The -coordinate of the point of intersection gives us the solution to the equation. When =, +1 and 7 are equal.

16 60 COORDINATE GEOMETRY (Chapter 1) Eample 8 Use graphical methods to solve the equation 3 5= +1. Self Tutor Alwas check our answer. =3 5: =+1 (3, ) = +1: The graphs intersect at the point (3, ). So, the solution is = =3-5 Check: LHS = 3(3) 5= RHS =3+1= X - EXERCISE 1G.1 1 a On the same set of aes, draw the graphs of = + and =. b Find the point where the graphs intersect. c Hence, solve the equation +=. Solve the equation 7=5 : a graphicall b using algebra. 3 Use graphical methods to solve: a 3= +1 b 6= c 5=10 d 3 += e 1 += 1 f 1 = 1 +5 Eample 9 Self Tutor Use graphical methods to solve for : 1 3 = We draw the graphs of =1 3 and the horizontal line = on the same set of aes. =1-3 The intersection point is (1, ). So, the solution is =1. Check: LHS =1 3(1) = =RHS X (1, -) =-

17 Use graphical methods to solve: COORDINATE GEOMETRY (Chapter 1) 61 a 3 =8 b = c +3= 1 d +5=0 Discussion Do two straight lines alwas intersect at eactl one point? USING TECHNOLOGY Finding where two graphs meet can be difficult if the coordinates of the intersection point are not whole numbers. In these situations, we can use our graphics calculator to find the point of intersection. Instructions for doing this can be found b clicking on the icon. GRAPHING PACKAGE Alternativel, ou can use the graphing package. Eample 10 GRAPHICS CALCULATOR INSTRUCTIONS Self Tutor Use technolog to solve for : 3 =9 We use technolog to graph =3 and =9, and find their intersection point: TI-8 Plus Casio f-9860g Plus The graphs intersect at (:6, 3:8). So, the solution is =:6. EXERCISE 1G. 1 Solve the equation 5 = +1: a using technolog b algebraicall. Use technolog to solve the following equations: a 3 5= +6 b 3= 11 c 1 3 += 1 d =0 +3 e +1= 1 f 1=3 5

18 6 COORDINATE GEOMETRY (Chapter 1) Discussion What are the advantages and disadvantages of solving equations graphicall rather than algebraicall? Investigation 1 Zoo prices Tickets to the zoo normall cost $0 per visit. However, for $30 per ear ou can become a friend of the zoo. This means ou can bu tickets for onl $10 per visit. Alternativel, for $100 per ear ou can become a member of the zoo. This means ou can visit the zoo as man times as ou like for no additional cost. Suppose C dollars is the cost of visiting the zoo n times. What to do: 1 Cop and complete the table of values below, for someone who bus ordinar tickets: Number of visits (n) Cost (C dollars) Create a similar table of values for someone who is: a a friend of the zoo b a member of the zoo. 3 Plot the three sets of points from 1 and on a single set of aes like the one shown C (dollars) alongside. Use a different colour for each set of points. 150 Which two options give the same cost for visiting the zoo 7 times in a ear? 5 Which option is: 100 a the most epensive for visiting the zoo twice in a ear b the cheapest for visiting the zoo 6 times in a ear? 50 6 Monica is wondering which option is best for her. What information can ou give her? 6 8 n

19 COORDINATE GEOMETRY (Chapter 1) 63 Investigation The graph of = = is an eample of a non-linear relationship. When we graph the relationship, the result is not a straight line. GRAPHING PACKAGE This investigation is best attempted using the graphing package or a graphics calculator. What to do: 1 a Draw the graph of = from a table of values. 9 b Use technolog to check our graph. c Eplain wh no part of the graph is below the -ais. a On the same set of aes, graph = and =. b What is the relationship between the two graphs? 3 a On the same set of aes, graph =, = +3, and = 3. b Do the graphs have the same basic shape? c What is the relationship between the three graphs? a On the same set of aes, plot the graphs of =, =( 3), and =( +3). b Do the graphs have the same basic shape? c What is the relationship between the three graphs? Review set 1 1 State the coordinates of the points A, B, C, D, and E. B A C D E On a set of aes, illustrate all the points which have equal and -coordinates. 3 For the following tables of values, plot the points on a number plane: a b

20 6 COORDINATE GEOMETRY (Chapter 1) For the equation = 3: a construct a table of values with = 3,, 1, 0, 1,, 3 b plot the graph. 5 Draw the graphs of: a =1 b =3 c = :5 d = 1 6 Use technolog to draw the graph of = Tina measured the rainfall at her house each da for 5 das: Da number () Rainfall ( mm) a b Plot these values on a set of aes. Do the points form a straight line? 8 The graphs of = 5, = 1 +1, and = are shown alongside. Use the graphs to solve the following equations: =-- =-5 a 5= 1 +1 b = 5 c 1 +1= = Qw +1 9 a Determine whether the following points lie on the line with equation =3 : i P(3, 7) ii Q(, ) iii R( 1, 5) b Check our answers b drawing the graph of =3, and plotting the points P, Q, and R. 10 Use technolog to solve the equation 7 +0=36 3. Practice test 1A Click on the icon to obtain this printable test. Multiple Choice PRINTABLE TEST

21 COORDINATE GEOMETRY (Chapter 1) 65 Practice test 1B Short response 1 Plot the points P(, 5), Q(3, 3), R(, 0), and S(, ) on a set of aes. On a set of aes show all points with a -coordinate equal to 5. 3 Use a table of values to draw the graphs of: a =3 b = c = 1 + Use technolog to draw the graphs of: a = 1 b =9 :5 5 a Complete the table of values for the equation = 5: b Use our table of values to draw the graph of = 5. 6 Determine whether the point (3, 7) lies on the line = Use graphical methods to solve +3= Use graphical methods to solve 1 3= 1. 9 Use technolog to solve the equation 1 = Ben recorded the number of drinks sold at his café for one hour: Number of people at the table () Number of drinks sold () a Plot these values on a set of aes. b Do the points lie in a straight line? Practice test 1C Etended response 1 An accountant charges a $90 consultation fee and then $0 per hour thereafter. a Identif the independent and dependent variables. b Make a table of values for the cost $C of an appointment with the accountant for t hours where t =0, 1,, 3,. c Draw a graph of C against t. d Is the relationship between C and t linear? e Is it sensible to join the points graphed with a straight line? Give a reason for our answer. f For ever increase in t of 1 hour, what is the change in C?

22 66 COORDINATE GEOMETRY (Chapter 1) A tank contains 00 litres of water. The tap is left on and 0 litres of water escape per minute. Suppose V is the volume of water remaining in the tank t minutes after the tap is turned on. a What are the independent and dependent variables? b Make a table of values for V and t, for t =0,,, 6, 8, 10 minutes. c Draw the graph of V against t. d Is the relationship between V and t linear? e Is it sensible to join the points graphed with a straight line? Give a reason for our answer. f g h i For ever increase of 1 minute for t, what is the change in V? How much water is left in the tank after 1 minutes? How long will it take for the volume of water to fall to 310 litres? How long will it take for the tank to be empt? 3 Tomatoes can be bought at a local market for $:50 a kilogram. a Cop and complete the table of values: Mass of tomatoes (m kg) b Identif the independent and dependent variables. Cost ($C) c Is the relationship between C and m linear? d Is it sensible to join the points with a straight line? Eplain our answer. e How man kilograms of tomatoes could be bought for $17:10? Consider the line with equation =3 5. a Construct a table of values for this equation, using = 3,, 1, 0, 1,, 3. b Hence draw the graph of =3 5. c Do the following points lie on the line? Eplain our answer in each case. i (5, 10) ii (, 10) d Draw the line =7 on the same set of aes. i Find the coordinates of the point of intersection of =3 5 and =7. ii Hence find the solution to the equation 3 5= a To solve the equation = 1 +3 graph? graphicall, which two lines do we need to b Construct a table of values for each of the lines in a, using =, 0,. c Hence, or otherwise, graph the two lines on the same set of aes. d What do ou notice about our two lines? e Use our answer to d to eplain wh +8 = 1 +3 has no solution.