Numer Plane Graphs and Coordinate Geometr Now this is m kind of paraola! Chapter Contents :0 The paraola PS, PS, PS Investigation: The graphs of paraolas :0 Paraolas of the form = a + + c PS Fun Spot: Wh didn t the ald man need his kes? :0 The hperola: k = - PS :0 Eponential graphs: = a PS Fun Spot: The tower of Hanoi :0 The circle PS :0 Curves of the form = a + d PS Fun Spot: What is HIJKLMNO? :07 Miscellaneous graphs PS, PS :0 Using coordinate geometr to solve prolems PS Maths Terms, Diagnostic Test, Revision ssignment, Working Mathematicall Learning Outcomes PS PS PS Determines the midpoint, length and gradient of an interval joining two points on the numer plane and graphs linear and simple non-linear relationships from equations. Draws and interprets graphs including simple paraolas and hperolas. Draws and interprets a variet of graphs including paraolas, cuics, eponentials and circles and applies coordinate geometr techniques to solve prolems. Working Mathematicall Stages Questioning, ppling Strategies, Communicating, Reasoning, Reflecting
:0 The Paraola Outcomes PS, PS, PS Up until this point, all the graphs have een straight lines. In this section, we will look at a most famous mathematical curve, the paraola. The equations of paraolas are called quadratic equations and have as the highest power of. 9 The simplest equation of a paraola is: 7 = s with the straight line, the equation is used to find the points on the curve. Some of these are shown in the tale. = 0 9 0 9 For an accurate graph, man points would have to e plotted. From the graph we can see that the paraola has a turning point, or verte, which is the minimum value of on =. The -ais is an ais of smmetr of the curve, so the right side of the curve is a reflection of the left side. This can e seen when points on either side of the ais are compared. The paraola is concave up, which means it opens out upwards. The shape of the paraola is clearl demonstrated the water arcs of this fountain. 0 Paraolas can e happ (up) or sad (down). New Signpost Mathematics Enhanced 0..
Eercise :0 Note: graphics calculator or computer graphing software could e used in the following eercise as an alternative to plotting points. Complete the following tales and then graph all four curves on one numer plane. Hint: On the -ais, use values from 0 to. a = = 0 0 0 0 c = d = 0 0 0 0 0 For the equation = a, what is the effect on the graph of varing the value of a? Match each of the paraolas to D C D D C with the equations elow. 0 a = 0 = c = d = 0 These paraolas are all concave up. 0 0 Complete the following tales and then graph all four curves on one numer plane. Hint: On the -ais, use values from to. a = = + 0 0 c = + d = 0 0 What is the difference in the curves = and = +? Can ou see that the shape of the curve is the same in each case? For the equation = + c, what is the effect on the graph of varing the value of c? Chapter Numer Plane Graphs and Coordinate Geometr
a Complete the tale of values for = and sketch its graph. = = () 0. 0 0. Sketch the graph of =. c For = a, what does the graph look like if the value of a is negative? On the same numer plane, sketch the graphs of = and = +. Match each of the following equations with the graphs to E. C 0 a = = + c = d = e = C D E 0 E D 7 The graphs of = and = ( ) are shown on the diagram. a How are the two graphs related? Graph the paraola = ( + ). How is it related to the graph of =? c Sketch the graphs of = ( ) and = ( + ) on different numer planes. d How are the graphs of = ( h) and = ( + h) related to the graph of =? 0 = = ( ) Curve stitching New Signpost Mathematics Enhanced 0..
9 0 The graphs of = ( ) and = ( ) + are shown on the diagram. a How is the graph of = ( ) + related to the graph of = ( )? How would the graph of = ( ) e otained from the graph of = ( )? c Sketch the graph of = ( + ) +. d Use the questions in a to c to eplain the connection etween the equation of the paraola and e the coordinates of its verte. Sketch the graph of each paraola on a separate numer plane. i = ( ) + ii = ( + ) + iii = ( ) iv = ( + ) On separate numer planes, sketch the following paraolas: a = ( + ) = ( + ) c = ( + ) + Find the equation of the paraola that results from performing the following transformations on the paraola =. a moving it up units moving it down units c moving it units to the right d moving it units to the left e turning it upside down and then moving it up units f turning it upside down and then moving it down units g moving it up units and then reflecting it in the -ais h moving it units to the right and then turning it upside down i moving it up units and then moving it units to the left. j turning it upside down, moving it units to the left and then moving it down units. Match each equation with the corresponding paraola in the diagram. a = C = -- 0 c = + d = + e = f = = ( ) + = ( ) D E F Chapter Numer Plane Graphs and Coordinate Geometr 7
Match each equation with the corresponding paraola to F in the diagram. a = ( ) = ( + ) c = ( + ) + d = ( ) + e = ( + ) f = ( ) C D E F The paraolas shown are the result of translating and/or reflecting the paraola =. Find the equation of each paraola. a c d e f New Signpost Mathematics Enhanced 0..
Investigation :0 The graph of paraolas graphics calculator or computer graphing package are ecellent tools for investigating the relationship etween the equation of a paraola and its graph. Use either of the aove to investigate graphs of the following forms for varing values of a, h and k. = a = a ± k = ( ± h) = ( ± h) + k Write a report on each of the forms, eplaining how the features of the graph, such as the concavit, the position of the verte and the numer of -intercepts, are related to the values of a, h and k. Lost the plot? Use one of these! Investigating paraolas The dish of a radio-telescope is paraolic in shape. Chapter Numer Plane Graphs and Coordinate Geometr 9
:0 Paraolas of the Form Outcome PS = a + + c In the last section, we saw that all paraolas have the same asic shape. The are all concave up or concave down with a single verte or turning point. The are smmetrical aout an ais of smmetr. We looked at the connection etween the paraola s shape and its equation and at what numers in the equation influenced the steepness, the concavit and the position of the graph on the numer plane. In this section, we look at how to sketch the paraola when its equation is given in the form = a + + c. We will also look at how to find features of the paraola, such as the - and -intercepts, the ais of smmetr, the verte and the maimum or minimum value of. Finding the -intercept To find the -intercept of = +, we let e zero. = + When = 0, = The -intercept is. The curve cuts the -ais at (0, ). This one looks happ! (, 0) (, 0) Finding the -intercepts To find the -intercepts of = +, we let e zero. = + When = 0, 0 = + Solving this, 0 = ( + )( ) = or ais of The -intercepts are and. smmetr The curve cuts the -ais at (, 0) and (, 0). Note: If + = 0 was difficult to factorise, then the formula = ± ac ------------------------------------ a could have een used to find the -intercepts. 7 9 (0, ) = + 0 New Signpost Mathematics Enhanced 0..
Finding the ais of smmetr Since the paraola = + has a vertical ais of smmetr, the ais of smmetr will cut the -ais half-wa etween (, 0) and (, 0), which are the two -intercepts. The ais of + smmetr will have the equation = ----------------, ie = -- is the ais of smmetr. For the paraola = +, a =, =, c =. The ais of smmetr is = ---------- ( ) ie = The ais of smmetr of the paraola = a + + c is given the equation = ----- a -- For the paraola = a + + c, the -intercepts are found solving 0 = a + + c. ± ie ac = ------------------------------------ a The ais of smmetr will cut the -ais half-wa etween these two values. + ac ------------------------------------- ac + ------------------------------------- a a = ---------------------------------------------------------------------------------------- -------- a = -------------- = -------- a = ----- a I think I ll just use a The ais of smmetr is = ----- a Finding the verte (or turning point) and the maimum or minimum value s the verte lies on the ais of smmetr, its -coordinate will e the same as that of the ais of smmetr. The -coordinate can e found sustituting this value into the equation of the paraola. For = +, the ais of smmetr is = --. Now, when = --, = ( ) -- + ( -- ) = -- The verte of the paraola is ( --, -- ). (, ) Chapter Numer Plane Graphs and Coordinate Geometr
The minimum or maimum value of will occur at the verte. The paraola will have: a minimum value of if the paraola is concave up (when the coefficient of is positive, eg = ) a maimum value of if the paraola is concave down (when the coefficient of is negative, eg = ) Hence, on the paraola = +, the minimum value of is -- when = --. The method of completing the square can also e used to find the minimum or maimum value and the verte as shown elow. = + = ( + + -- ) -- = ( + ) -- -- s ( + -- ) is alwas greater than or equal to 0, the minimum value of will e -- when = -- and the verte is the point ( --, -- ). Worked eamples For each equation, find: a the -intercept the -intercept c the ais of smmetr d the verte (turning point) Use these results to sketch each graph. = + = + + = Solutions a For the -intercept, let = 0 = (0) + (0) The -intercept is. For the -intercepts, let = 0 0 = + ( + )( ) = 0 The -intercepts are and. c is of smmetr: = + ---------------- (midpoint of -intercepts) = is the ais of smmetr. d To find the verte, sustitute = into the equation to find the -value. = ( ) + ( ) = The verte is (, ). We now plot the aove information on a numer plane and fit the paraola to it. -intercepts = 0 (, ) verte ais: = -intercept New Signpost Mathematics Enhanced 0..
a To find the -intercept of = + +, let e zero. = (0) + (0) + The -intercept is. For the -intercepts, solve + + = 0. However, when we use the formula, we get ± = -------------------------------------------------- ± = ----------------------- This gives us a negative numer under the square root sign. You can t find the square root of a negative numer! Thus, there are no solutions, so the paraola does not cut the -ais. c = + + has a =, =, c =. is of smmetr is = -----. a ie = ---------- ( ) The ais of smmetr is =. d The verte is the turning point of the curve, and is on the ais of smmetr =. When =, = ( ) + ( ) + = The verte is (, ). reflection of (0, ) verte (, ) 0 -intercept ais of smmetr = 0 = continued Chapter Numer Plane Graphs and Coordinate Geometr
a For =, if = 0, then = 0. The curve cuts the -ais at the origin. When = 0, = 0 ( ) = 0 The -intercepts are 0 and. c is: = (midpoint of -intercepts) d When =, = () () = The verte is (, ). is: = (, ) 0 This is a sad graph ecause the coefficient of is negative. Eercise :0 For each of the graphs, find: i the -intercept ii the -intercepts iii the equation of the ais of smmetr iv the coordinates of the verte c a 0 Foundation Worksheet :0 The paraola = a + + c PS a Use the graph to find: i the -intercept ii the -intercepts iii the equation of the ais of smmetr iv the coordinates of the verte afor the paraola = + +, find: i the -intercept ii the -intercepts iii the equation of the ais of smmetr iv the coordinates of the verte New Signpost Mathematics Enhanced 0..
7 Find the -intercepts of the following paraolas. a = + = c = ( )( + ) Find the -intercepts of the following paraolas. a = = + 0 c = ( )( + 7) Find the equation of the ais of smmetr and the coordinates of the verte of the following paraolas. a = ( )( ) = ( )( + ) c = -- ( + )( ) d = + 7 e = 9 + f = Find the minimum value of on the following paraolas. a = = + c = 9 0 + Find the maimum value of on the following paraolas. a = = + 0 7 c = 7 9 For the paraola = +, find: a the -intercept the -intercepts c the ais of smmetr d the verte e hence, sketch its graph When finding the -intercepts, if ou can t factorise, then use the formula. 9 Repeat the steps in question 7 to graph the following equations, showing all the relevant features. a = + = c = 0 d = + e = + 9 f = + + g = h = i = Match each graph with one of the equations written elow the diagram. Each graph has an shape. Find the turning point. Is it happ (a is ve) or sad (a is ve)? Visualise the graph efore C ou sketch. D 0 E F a = + = + c = + d = + + e = 0 f = + Chapter Numer Plane Graphs and Coordinate Geometr
0 Sketch each set of three paraolas on the same numer plane. a i = ii = iii = + i = 9 ii = 9 iii = 0 + 9 c i = ( )( + ) ii = ( )( + ) iii = ( )( + ) d i = ii = iii = + Sketch the graph of each quadratic relationship, showing all relevant features. a = = c = ( + )( ) d = + + e = + 7 f = g = ( )(7 + ) h = i = + + 7 j = + 9 k = + l = 7 a Sketch the graphs of = + and = +. Compare the two graphs and descrie the difference etween them. Sketch the graphs of = ( )( + ) and = ( )( + ). Compare the two graphs and descrie the difference etween them. The paraola in the diagram has its verte at (, ) and it passes through the point (, ). The equation of the paraola has the form = a + + c. (, ) a Use the -intercept to show that c =. Use the equation of the ais of smmetr to show that = a and that the equation of the paraola is of the form = a + a. c Sustitute the coordinates of the verte or the point (, ) to find the value of a. d What is the equation of the paraola? (, ) Use the method of question to find the equation of each of the following paraolas. a c 0 0 0 (, ) (, ) (, ) 0 New Signpost Mathematics Enhanced 0..
d e f (, ) (, ) (, ) Fun Spot :0 Wh didn t the ald man need his kes? Work out the answer to each question and put the letter for that part in the o that is aove the correct answer. Factorise: E S O S + What is the ais of smmetr for: H = +? S = +? K =? O =? Where does each paraola elow cut the -ais? I = + H = T = What is the verte for each paraola? L = + C = + L = + (0, 0) ( )( + ) (, ) = -- ( )( + ) (0, ) = (0, ) = (, 0) ( ) (, ) = 0 ( + )( ) Chapter Numer Plane Graphs and Coordinate Geometr 7
:0 The Hperola: = -- Outcome PS Prep Quiz :0 Find the value of -- when is: -- If = --, what is the value of when is:? 7?? 9 If = --, what happens to as increases from to 0? 0 If = --, what happens to as decreases from to 0? k We need to take man points when graphing a curve like = --, as it has two separate parts. The curve of such an equation is called a hperola. = can also e written as = Use values correct to dec. pl. = k -- is a hperola if k is a constant (eg, or ). Worked eample = -- 0 0 0 0 0 7 0 7 0 Notice that there is no value for when = 0. When = 0, ecomes = -- = --. This value cannot eist as no numer can e divided 0. 0 What will happen to the values as the values get closer to 0? What will happen to the values as the values ecome larger? New Signpost Mathematics Enhanced 0..
Plotting the points in the tale gives us the graph of = --. Note: The hperola has two parts. = The parts are in opposite quadrants and are the same shape and size. The curve is smmetrical. The curve approaches the aes ut will never touch them. 0 The - and -aes are called = asmptotes of the curve. No value for eists when = 0. Eercise :0 Use our calculator to complete the tales elow, giving values for correct to two decimal places. a = -- = -- c = -- d = -- Complete the tale elow for = --. 0 0 Use a sheet of graph paper to graph the curve = --, using our tale. Use values to on oth aes. There seems to e a pattern here. Chapter Numer Plane Graphs and Coordinate Geometr 9
Graph the curve = first completing the tale elow. 0 0 0 0 -- How aout that! What does a negative value of k do to the graph? Match each of the graphs to F with the following equations. a = -- c 0 = -- = ----- d = 0 e ----- f = ----- = ----- 0 0 D E C F 0 0 0 0 C F E D 0 0 a Does the point (, ) lie on the hperola = --? If the point (, ) lies on the hperola k = --, what is the value of k? c The hperola k = -- passes through the point (0, ). What is the value of k? For each of the following, find a point that the hperola passes through and, sustituting k this in the equation = --, find the equation of the hperola. a 0 0 0 0 0 0 0 0 0 New Signpost Mathematics Enhanced 0..
c 0 d 0 0 0 0 0 0 0 :0 Eponential Graphs: = a Outcome PS Prep Quiz :0 Find the value of: 0 If =, find when is: 7 Use our calculator to find, to one decimal place, the value of: 9 0 curve whose equation is of the form = a is called an eponential curve. On the following numer plane, the graph of = has een drawn. = is an eponential curve. The curve passes through (0, ) on the -ais since 0 =. The curve rises steepl for positive values of. The curve flattens out for negative values of. The -ais is an asmptote for this part of the curve. ecause is alwas positive, the curve is totall aove the -ais. 7 = 0 Chapter Numer Plane Graphs and Coordinate Geometr
Eercise :0 a Complete the tale elow for = and graph the curve for. 0 0 0 0 0 0 7 c (In the tale, use values of correct to two significant figures.) Complete the tale elow for = and graph the curve for. Use the same diagram ou used in part a. 0 0 0 Compare the graphs of = and =. What do ou notice? The graph of = a will alwas pass through (0, ), since a 0 =. a Complete the tale of values for = and graph the curve on a numer plane. when = is ( )... That s the same as! 0 0 0 Compare our graph with =. What do ou notice? a Draw on the same numer plane the graphs of = and =. With reference to the graphs in part a, now draw the graphs of = and = on the same diagram. c What is the effect of graphing the negative relationships? The graphs of the curves =, =, = and = 0 are shown on the C numer plane. D a Match each of the curves to D with its equation. 0 What is the effect of multipling an eponential function a constant, k (ie = ka )? 0 New Signpost Mathematics Enhanced 0..
a c For the graph of = a, where would the curve cut the -ais? To which end of the -ais is the curve asmptotic? (Note: a > 0) For the graph of = a, where would the curve cut the -ais? To which end of the -ais is the curve asmptotic? Using our answers to parts a and, draw sketches of: i = ii = iii = -- iv = The quantit of caron- present after t ears is given the formula: Q = 70 where Q is the quantit of caron- present, was the amount of caron- present at the start, t is the time in ears. If 0 g of caron- were present at the start, = 0, and the formula ecomes: Q = 0 t ----------- t ----------- 70 a Find the value of Q when t = 0. Find the quantit of caron- remaining after 70 ears. (Caron- has a half-life of 70 ears.) c Find the value of Q when t is: i 0 ii 7 90 iii d Use the values found aove to sketch the t ----------- graph of Q = 0 70 for values of t from 0 to 0 000. Q 0 0 Caron- is a radioactive sustance. 0 000 0 000 t Fun Spot :0 The tower of Hanoi This famous puzzle consists of three vertical sticks and a series of discs of different radii which are placed on one stick to form a tower, as shown in the diagram. The aim of the puzzle is to move the discs so that the tower is on one of the other sticks. The rules are: onl one disc can e moved at a time to another stick a larger disc can never e placed on top of a smaller one. The puzzle can e made more difficult having more discs. Investigate the minimum numer of moves needed if there are, or discs. Can ou generalise our results? Can ou predict the minimum numer of moves needed if there are, sa, discs? (Hint: n eponential relationship can e found!) Chapter Numer Plane Graphs and Coordinate Geometr
:0 The Circle Outcome PS circle ma e defined as the set of all points that are equidistant (the same distance) from a fied point called the centre. We need to find the equation of a circle of radius r units with the origin O as its centre. If P(, ) is a point on the circle which is alwas r units from O, then, using Pthagoras theorem, + = r. This is the equation that descries all the points on the circle. O r P(, ) The equation of a circle with its centre at the origin O and a radius of r units is: + = r Worked eamples What is the equation of the circle that has its centre at the origin and a radius of units? What is the radius of the circle + =? Solutions r =, so the equation is + = + = is of the form + = r. + = is the equation of the r =, so r = circle. The radius of the circle is units. Eercise :0 What is the equation of each circle? a Foundation Worksheet :0 The circle PS For each circle, write down its i radius ii equation a 0 Sketch the circle represented the equations: a + = + = c 0 0 New Signpost Mathematics Enhanced 0..
. What is the equation of a circle with the origin as its centre if the radius is: a units? 7 units? c 0 units? d units? e units? f units? g -- units? h -- units? i units? What is the radius of these circles? a + = + = c + = 0 d + = e + = f + = g + = 9 h 9 + 9 = 0 (, ) (, ) For the circle + =, there are two points that have an value of. Sustituting = into + = we get + = = 9 = = ± For g divide through first, ie + = 9 = ± [+ or ] So (, ) and (, ) are the two points. 7 a Find the two points on the circle + = that have an value of: i ii iii Find the two points on the circle + = that have a value of: i ii iii Graph each of the circles in question on separate numer planes. Which equation in each part represents a circle? a =, + =, + = = 9, + = 9, + = c =, =, = + d = +, = + 7, = a How could it e determined whether a point was inside, outside or ling on a particular circle? State whether these points are inside, outside or on the circle + = 0. i (, ) ii (, ) iii (, ) iv (, ) v (, ) vi (, ) vii ( --, -- ) viii (, ) Find the equation of the circle with its centre at the origin that passes through the point: a (, ) (, ) c (, ) Chapter Numer Plane Graphs and Coordinate Geometr
:0 Curves of the Form Outcome PS = a + d Curves of the form = a + d are called cuics ecause of the term. The simplest cuic graph is =, which occurs when a = and d = 0. s with other graphs, a tale of values is used to produce the points on the curve. = 0 0 0 0 0 0 7 = Features of = : It is an increasing curve. s increases, the value of, and hence, ecomes large ver quickl. This means it is difficult to fit the points on a graph. When is positive,, and hence, is positive. When is negative,, and hence, is negative. When is zero, is zero. In this section, the relationship etween the curve = and the curve = a + d for various values of a and d will e investigated. 7 Eercise :0 a c d Match each of the equations elow with the graphs, and C. i = ii = -- iii = Which graph increases the fastest? (Which is the steepest?) Which graph increases the slowest? How can ou tell which graph is the steepest looking just at the equations? 0 0 0 0 C New Signpost Mathematics Enhanced 0..
Which of the curves is steeper: a = or =? = or = --? c = or =? The graphs of = -- and = -- are shown. a How are the graphs related? What is the effect on = a of the sign of a? = = From our results so far, ou should have noticed that all the curves are either decreasing or increasing. Without sketching, state whether the following are increasing or decreasing. a = = 0 c = 0 d = -- e = f a Cop and complete the tales of values for the three curves =, = + and =. 0 0 + What is the equation of curves and? c How is the graph of = + related to the graph of =? d How is the graph of = related to the graph of =? Given the graph of = -- -- ----, descrie how ou would otain the graphs of: a = -- + = -- c = -- + d = -- e = -- f = -- + = 0 9 7 7 9 0 Chapter Numer Plane Graphs and Coordinate Geometr 7
7 In each diagram, the two curves and were otained moving the other curve up or down. Give the equations of the curves and. a c 0 0 = = 0 = 0 Sketch each pair of graphs on the same numer plane. a = = c = -- = = + = -- + 9 Sketch each pair of graphs on the same numer plane. a = + = + = + = + 0 From the list of equations, write the letter or letters corresponding to the equations of the curves: = = a that can e otained moving = up or down C = -- that are the same shape as D = c that are decreasing E = + d that pass through (0, 0) F = -- e that can e otained from the curve = -- G = -- reflection in the -ais f that have the largest -intercepts H = + I = -- Give that each of the graphs is of the form = a + d, find its equation. a (, 0) (, ) This can e done in another wa. (0, 0) (0, ) For equations of the form = a + d, descrie the effect on the graph of different values of a and d. New Signpost Mathematics Enhanced 0..
Fun Spot :0 What is HIJKLMNO? Work out the answer to each question and put the letter for that part in the o that is aove the correct answer. R F U F O H O HIJKLMNO (Ignoramus Humungus) For the numer plane shown, match each graph with its correct equation elow. U F H R U M F U M What is the equation of the paraola that results if the paraola = is: O moved up units moved down units E moved units to the right R moved units to the left R turned upside down and moved units up O moved units to the right and turned upside down From the equations =, = and = --, which one is a: T paraola? L straight line? W hperola? = = ( ) = = = = = = + = = = = -- = = ( ) = ( + ) -- = + = + Chapter Numer Plane Graphs and Coordinate Geometr 9
:07 Miscellaneous Outcomes PS, PS Graphs It is important that ou e ale to identif the different graphs ou have met so far their equations. Stud the review tale elow and then attempt the following eercise. Tpe of graph Equation Graph Straight line = m + or a + + c = 0 Lines parallel to the aes = a or = a Paraola = = a + + c Circle + = r k Hperola = -- or = k r r r r Eponential curve = a or = a Cuic curve = = a + d 0 New Signpost Mathematics Enhanced 0..
Eercise :07 From the list of equations given on the right, choose those that represent: a a straight line a circle c a paraola d a hperola e an eponential curve f a cuic curve + = = C = D = E = F = G = H + = I = -- J + = K = L = -- Sketch the graphs of the following equations, showing where each one cuts the coordinate aes. a = = c + = d = e = f = g = + h = i = ( ) j = ( + )( ) k = + l = + m = n = ( + ) o = p + = q + = 00 r + = s = -- t = u = -- v = w = = = z = + = + Match each graph with its equation from the given list. a = 9 = + C = D + = E + = 0 F = G = + H = I = c d Chapter Numer Plane Graphs and Coordinate Geometr
e f g h i Determine the equation of each graph. a c d e f g h i 0 (, ) New Signpost Mathematics Enhanced 0..
j k l m n o 9 p q r :0 Using Coordinate Outcome PS Geometr to Solve Prolems Prep Quiz :0 (, ) and (, ) are shown on the diagram. Find: the length of (, ) the slope of the midpoint of 0 the -intercept of the equation of the line What is the gradient of the line: (, ) =? 7 + =? What is the equation of the line that passes through (, ) with a slope of? What is the gradient of a line that is: 9 parallel to the line =? 0 perpendicular to the line =? Chapter Numer Plane Graphs and Coordinate Geometr
In Year 9, coordinate geometr was used to investigate: the distance etween two points the midpoint of an interval the gradient (or slope) of an interval the various equations of a straight line parallel and perpendicular lines. These results can e used to investigate the properties of triangles and quadrilaterals as well as other tpes of geometrical prolems. The results are reviewed in Chapter. Worked eamples Eample triangle is formed the points O(0, 0), (, ) and (, 0). E and F are the midpoints of the sides O and. Show: a that ΔO is isosceles that EF is parallel to O Solution a O = ( 0) + ( 0) = ( ) + ( 0 ) = + 9 = + 9 = = ΔO is isosceles (two equal sides). Now, E is (, -- ) and F is (, -- ). EF is horizontal (E and F have same -coordinates). O is horizontal. EF is parallel to O. Eample W(, 0), X(, ), Y(, 0) and Z(, ) are the vertices of a quadrilateral. a Show that WXYZ is a parallelogram Show that the diagonals isect each other. Solution a Calculating the slopes of the four sides gives the following. 0 0 ( ) Slope of WX = ------------------- Slope of ZY = ------------------- ( ) ( ) = -- = -- 0 ( ) 0 Slope of WZ = ------------------------ Slope of XY = ----------- ( ) = = WX ZY (equal slopes) and WZ XY (equal slopes) WXYZ is a parallelogram (opposite sides are parallel). E (, ) O 0 F (, 0) X W Y 0 Z New Signpost Mathematics Enhanced 0..
+ ( ) Midpoint of XZ = -------------------, ------------------- + ( ) Midpoint of WY = (, 0) -- = (--, 0) ( --, 0) is the midpoint of oth diagonals. The diagonals XZ and WY isect each other. Eercise :0 a c a c Show that the triangle formed the points O(0, 0), (, ) and (, ) is isosceles. Show that the triangle formed the points (0, 0), (, ) and (7, ) is right-angled. Show that the triangle with vertices at (0, 0), (, ) and (, ) is oth right-angled and isosceles. Show that the quadrilateral with vertices at (0, ), (, 0), C(0, ) and D(, 0) is a rhomus. quadrilateral is formed joining the points O(0, 0), (, ), C(, 0) and D(, ). Show that it is a rectangle. The points (0, ), (, 0), C(0, ) and D(, 0) are joined to form a quadrilateral. Show that it is a square. a If OC is a rectangle, what are the coordinates of? Find the length of O and C. What propert of a rectangle have ou proved? c Find the midpoint of O and C. What does our answer tell ou aout the diagonals of a rectangle? The points (0, 0), (, ) and C(, ) form a triangle. a Find the midpoints of and C. Find the slope of the line joining the midpoints in a. c What is the slope of C? d What do our answers to parts and c tell ou? Foundation Worksheet :0 Coordinate geometr PS ashow that the triangle formed the points (, 0), (0, ) and C(, 0) is isosceles. ashow that the quadrilateral with its vertices at the points (, ), (, ), C(, ) and D(0, ) is a parallelogram. (, 0), (0, ), C(, 0) and D(0, ) are the vertices of a quadrilateral. Show that CD is a rhomus. The points (, 0), (, ), C(0, ) and D(0, 0) form a square. Find the slopes of D and C. What does our result sa aout the diagonals D and C? C O Chapter Numer Plane Graphs and Coordinate Geometr
right-angled triangle O is shown. a Find the coordinates of E, the midpoint of. Find the length of OE. c Find the length of E. d What can ou sa aout the distance of E from O, and? O E 7 triangle has its vertices at the points (, ), (, ) and C(, ). a Show that the triangle is isosceles. Find E, the midpoint of C. c Find the slope of the line joining E to. d Show that E is perpendicular to C. e Descrie how ou could find the area of ΔC. a c Find the midpoints of O and. Find the length of the line joining the midpoints in a. Show that our answer in is half the length of O. O 9 CD is a quadrilateral. a Find the coordinates of the midpoints of each side. Join the midpoints to form another quadrilateral. What tpe of quadrilateral do ou think it is? c How could ou prove our answer in? (, ) 0 C D(, ) 0 The points (, 0), (0, ) and C(, 0) form the vertices of an acute-angled triangle. a Find the equation of the perpendicular isectors of the sides, C and C. Find the point of intersection of the perpendicular isectors of the sides and C. c Show that the perpendicular isector of the side C passes through the point of intersection found in. 0 C New Signpost Mathematics Enhanced 0..
median is a line joining a verte of a triangle to the midpoint of the opposite side. a Find the equations of the medians. Find the point of intersection of two of the medians. c Show that the third median passes through the point of intersection of the other two. (, ) 0 (, ) C(, ) Properties of triangles are fascinating. Chapter Numer Plane Graphs and Coordinate Geometr 7
Literac in Maths Maths terms circle The equation of a circle in the numer plane with its centre at the origin is: + = r where r is the radius. cuic curve curve that contains an term as its highest power. In this chapter, the curve s equation is of the form: = a + d hperola k curve with the equation = -- where k is a constant. It has two asmptotes (the -ais and -ais), which are lines that the curve approaches ut never reaches. paraola curve with the equation = a + + c. eponential curve curve with an equation of the form = a, where a > 0. equation n algeraic statement that epresses the relationship etween the - and -coordinates of ever point (, ) on the curve. graph (of a curve) The line that results when the points that satisf a curve s equation are plotted on a numer plane. The line of smmetr of the paraola is its ais of smmetr. The equation of the ais of smmetr is = -----. a Paraolas can e concave up or concave down. The highest (or lowest) value of on the paraola is the maimum (or minimum) value. The point where the paraola turns around is its verte (or turning point). - and -intercept(s) The point(s) where a curve crosses the - or -ais. Maths terms New Signpost Mathematics Enhanced 0..
Diagnostic Test Numer Plane Graphs and Coordinate Geometr These questions reflect the important skills introduced in this chapter. Errors made will indicate an area of weakness. Each weakness should e treated going ack to the section listed. These questions can e used to assess all or parts of outcomes PS, PS and PS. On the same numer plane, sketch the graphs of: a = = c = + On the same numer plane, sketch the graph of: a = = -- c = Sketch the graphs of: a = ( ) = ( + ) c = ( ) + Find the -intercept of the paraolas: a = ( )( + ) = c = d = + Find the -intercepts for each of the paraolas in question. Find the equation of the ais of smmetr for each of the paraolas in question. 7 Find the verte of each of the paraolas in question. Sketch each of the paraolas in question. lso state the maimum or minimum value for each quadratic epression. 9 Determine the equation of each paraola. a Section :0 :0 :0 :0 :0 :0 :0 :0 :0 c 9 d 0 Sketch the graphs of: a = -- = -- c = :0 Chapter Numer Plane Graphs and Coordinate Geometr 9
Sketch, on the same numer plane, the graphs of: a = = c = a What is the equation of this circle? What is the equation of a circle that has its centre at the origin and a radius of 7 units? c What is the radius of the circle + =? 0 Section :0 :0 Sketch the graphs of: a = + = c = From the list of equations on the right, which one represents: a paraolas? straight lines? c circles? d hperolas? e eponential graphs? f cuic graphs? = 9 = C = 9 D = E = 9 F = -- G + = 9 H = + I = 9 J + = K = 9 L = -- Match each graph with its equation from the list. = = C = D = E + = F = a c :0 :07 :07 d e f 70 New Signpost Mathematics Enhanced 0..