Coordinate geometry. In this chapter. Areas of study. Units 3 & 4 Functions, relations and graphs Algebra

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1 Coordinte geometr VCEcoverge Ares of stud Units & Functions, reltions nd grphs Alger In this chpter A Sketch grphs of = m + n + c where m = or nd n = or B Reciprocl grphs C Grphs of circles nd ellipses D Grphs of hperols E Prtil frctions F Sketch grphs using prtil frctions

2 Mths Quest Specilist Mthemtics Coordinte geometr In this chpter we shll eplore the grphs of severl functions tht incorporte rtionl prts. We shll focus on reciprocl grphs, ellipses nd hperols, nd see how prtil frctions cn help in grphing some functions. It is ssumed tht ou lred know how to sketch liner grphs, prolic grphs nd hperols. Summr of sic sketch grphs = m + c = ( + ) + c + c c c m c + (, c) c = m + c = ( + ) + c m = grdient Turning point is (, c). c = -intercept -intercept is (, + c). c c --- = -intercept -intercepts = ± -- m = + c + d Asmptote c = c + d = ( + c) + d c + d d c Asmptote = d = c d = d c d = + = ( + c) + d -intercept is (, -- + d ) -intercept is (, d ) c c c c Verticl smptote: = -- Verticl smptote: = -- Horizontl smptote: = d Horizontl smptote: = d + cd -intercept = d Reminders. To work out the -intercept, let = nd solve for.. To work out the -intercept, let = nd solve for. d d. To work out the turning points for function, work out nd solve =. d d. Asmptotes cn e worked out either lgeric resoning (oes) or rule.

3 Chpter Coordinte geometr Sketch grphs of = m + n + c where m = or nd n = or In this section we shll e sketching grphs of functions of the form = m + n + c where m = or nd n = or. Emples of such functions re:. = +,, =, =, c =, m =, n =. = + +,, =, =, c =, m =, n =. =,, =, =, c =, m =, n =. =,, 6, =, c =, m =. = ,, =, = 7, c = 6, m =, n = Verticl smptote The verticl smptote occurs for the -vlue tht mkes the denomintor of the frctionl prt of the function. When the denomintor of frction is, the frction, nd hence the whole function, will e undefined. Generl lgeric resoning. For grphs of the form = m c, s, n, nd hence Therefore, n nd thus the eqution of the verticl smptote is = the vlue tht mkes the denomintor.. For grphs of the tpe = m + n + c, the verticl smptote will lws e = or the -is, s the denomintor of the frctionl prt is lws. Horizontl, olique (sloping) or curved smptote To determine the non-verticl smptote, cover up onl the frctionl prt with our hnd; the prt of the function ou cn still see is the eqution of this smptote. When finding horizontl, olique or curved smptotes we consider ver lrge vlues of. The reciprocl of ver lrge vlues is ver smll, so the frctionl prt is close to nd contriutes ver little to the vlue of. The vlue of will ecome ver close to the vlue of the function without the frctionl prt hence the method for finding the smptote. Note: An olique line is n line tht is neither horizontl nor verticl. n Generl lgeric resoning For grphs of the form = m c, s, n, nd hence n ---- nd so ecomes insignificnt in size. Thus m + c, mking n = m + c n smptote. Note: If = then the smptote will e horizontl; otherwise, it will e olique or curved.

4 Mths Quest Specilist Mthemtics Find the smptotes for the function with the eqution = WORKED Emple Find the verticl smptote considering vlues of tht mke the denomintor. = ---- As, ----, nd hence. Verticl smptote: = Find the horizontl or olique smptote As, ----, nd hence. considering lrge vlues of or covering the frctionl prt of the eqution. Olique smptote: = Sketching the grphs of rtionl functions This process involves severl steps. Step. Brek the given function into two seprte, simpler functions. Step. Sketch the grph of ech of the seprte functions on the one set of es. Do this in pencil nd use either colour or some distinguishing feture (for emple dotted nd dshed lines) for ech grph. Step. Determine the smptotes of the originl function nd pencil in how the grph of the function pproches these smptotes. You should rememer to consider:. lrge positive vlues for. lrge negtive vlues for. vlues ver close to nd either side of the verticl smptote. Step. Work out the -intercept(s), -intercept nd turning points for the given function to give greter ccurc. Sketch the grph of the function WORKED Emple = + -- Divide = + -- into two functions: = nd = --. Consider =. () This is n upright prol. We need to stte the verte nd find severl points either side of the verte to give n ide of the shpe., (do not include the turning points). Let = nd = --. The grph = is n upright prol with verte t (, ). Let =, = () = The points (, ) nd smmetr (, ), re on the prol.

5 Chpter Coordinte geometr () Sketch =. = Consider = --. For = -- : () This grph is hperol. We need Verticl smptote is =. to stte the smptotes nd points Horizontl smptote is =. on the curve either side of the Let =, = -- =, verticl smptote. so (, ) is on the hperol. Let =, = =, so (, ) is on the hperol. () Sketch = -- on the sme es s =. = = Consider the grph of = +. For = : () Let = to clculte the -intercepts. At -intercept, = = () Multipl oth sides = (c) Solve for. + = = -- (d) Tke the cue root of oth sides to otin. (e) Let = to determine the -intercept. -- = = No -intercept s the -is is verticl smptote. Continued over pge

6 6 Mths Quest Specilist Mthemtics Write down the equtions of the smptotes. Asmptotes re = nd =. 6 Consider ehviour ner smptotes. () For >, oth nd re positive As, nd +. nd s, nd. () Therefore, from ove the So from ove the grph of. grph of. (c) For <, > nd < nd s As, nd., nd. (d) Therefore, from elow the grph of. So from elow the grph of (s is negtive). (e) Consider vlues of either side of the verticl smptote. As +, nd is positive nd. So from ove the grph. As, nd is positive nd. So from ove the grph of. 7 Sketch the grph of = + sme es s nd. -- on the Approch from ove Approch from ove Approch from elow = +, =.6 6 = = Approch from ove Sttionr points (turning points) When we re working with turning points, we need to rememer the following.. The derivtive of functions of the form = n d is = n n d d. To determine locl mimum or locl minimum we solve = for. d. Distinguish etween the mimum nd minimum vlues emining the sign of d round the -vlue(s) found in the step ove. The tle elow will help ou. d d If = when =, the following pplies. d Locl minimum Locl mimum < = > < = > d < d slope is negtive d = d slope is zero d > d slope is positive d > d slope is positive d = d slope is zero d < d slope is negtive

7 Chpter Coordinte geometr 7 WORKED Emple Find the locl mimum nd minimum vlues of, for = To find the derivtive, epress the rule in inde form. Differentite. At locl mimum nd minimum d points, =. d = + -- = + d = 8 d d For =, d 8 = To solve, multipl oth sides. 8 ( ) ( ) = ( ) 8 = Rerrnge. 8 = = Tke the cue root of oth sides. = Find the corresponding -vlue sustituting = into = = Evlute. = + = Stte the turning point. = ( --, ) is the turning point. Determine if ( --, ) is mimum or < -- = -- > minimum. d d < = For < -- : s = is verticl d d smptote we need to pick vlue of ( ) (7) etween nd --, s = --, nd d evlute d For > -- : pick vlue greter thn --, d s =, nd evlute d Stte the nture of the turning point ( --, ) is locl minimum. -- d > d

8 8 Mths Quest Specilist Mthemtics WORKED Emple Sketch the grph of the function = ----, showing sttionr points (mimum nd minimum), intercepts nd smptotes. Divide = ---- into two functions: = nd = Let = nd = Consider =. For = : () This is stright line pssing through (, ) with grdient of. The grdient (m) =, nd the -intercept (c) =. The line psses through origin (, ). () Sketch =. = Consider = For = ---- : () The grph is hperol. We need verticl smptote is =. to find the smptotes nd points on horizontl smptote is =. the curve either side of the verticl smptote. Let =, = ( ) = So (, ) is on the hperol. Let =, = ( ) = So (, ) is on the hperol. () Sketch = ---- on the sme es s =. = Consider the grph of the function = For = ---- : () Clculte the -intercept. Let = = =

9 Chpter Coordinte geometr 9 () Multipl oth sides nd simplif. = = (c) Solve for. = The -intercept is (, ). (d) There is no -intercept s. No -intercept. Find the sttionr points. () Epress in inde form. = ---- = d () Differentite to find d d d = ( ) d d = + d d (c) Set = nd solve. Let =. d d + = (d) Multipl oth sides. ( ) + ( ) = ( ) (e) Solve for. + = = = -- =.9 (f) Find the vlue of sustitution. = (g) Simplif. -- ( -- ) (.9) 6 Determine whether we hve mimum or minimum < = > d > d d = d d < d slope is positive slope is zero slope is negtive (.9,.8) is locl mimum. Continued over pge

10 Mths Quest Specilist Mthemtics Stte the smptotes for = Verticl smptote = (-is). Olique smptote =. Look t the smptotic ehviour. () For >, > nd < ; s, nd from negtives. () For <, > nd < ; s, nd from negtives. (c) Consider the ehviour ner the verticl smptote ( = ). Sketch the grph of = sme es s nd on the As, from elow the grph of, s is negtive. As, from elow the grph of, s is negtive. As +, nd is positive nd. So from ove the grph. As, nd is negtive nd. So from elow the grph of. =, (.9,.8) =.6 Approch from elow = = Approch from elow Approch from ove Approch from elow rememer rememer To sketch grphs of functions of the form = m + n + c where m = or nd n = or : Step. Brek the given function into two seprte simpler functions. Step. Sketch the grph of ech of the seprte functions creted, on the one set of es. Step. Determine the smptotes of the originl function nd pencil in how the grph of the function pproches these smptotes. Consider: () lrge positive vlues for () lrge negtive vlues for (c) vlues ver close to, nd either side of, the verticl smptote. Step. Work out the -intercept(s), -intercept nd turning point for the given function to give greter ccurc.

11 Chpter Coordinte geometr WORKED Emple Sketch grphs of = m + n + c where m = or nd n = or Find the smptotes for ech of the following. = -- + = -- + c = d = -- e = f = g A = -- + h = multiple choice For ech of the following equtions, choose the lterntive tht gives the correct smptotes. 6 = -- + A = 6 nd = B = nd = C = nd = D = nd = E = = -- + A = + nd = B = nd = C = + nd = D = nd = E = nd = c 6 = + -- A = nd = B = nd = 6 C = nd = 6 D = nd = E =, = + nd = d 7 = A = 9 nd = B = 9 nd = 7 C = 9 + nd = D = nd = E = 9 nd = WORKED Emple WORKED Emple Sketch the grph of ech function given in question. (Do not include the turning points.) For ech of the functions given in question, find the locl mimum nd/or minimum vlues of. multiple choice For ech of the following equtions, choose the lterntive tht gives the correct grph. = 6 -- A B C D E Sketch grphs Addition of functions EXCEL Single grph plotter Mthcd Mthcd Spredsheet

12 Mths Quest Specilist Mthemtics A B C D E c = ---- = ---- A B C D E d = + -- A B C D E WORKED Emple 6 Sketch the grphs of ech of the following, showing sttionr points (m. nd min.), intercepts nd smptotes. = = -- c = -- + d = + -- e -- 8 = + f = g = h = -- (Hint: When finding -intercepts, use the fctor theorem to find fctor.) 7 The volume of solid clinder is 8π cm. Show tht the totl surfce re, A cm, is A = πr 6π where r >. r Sketch the grph of A ginst r. Stte the eqution of n smptotes nd the coordintes of the sttionr point. c Hence, find the ect minimum totl surfce re. 8 A o with volume of cm hs the shpe of rectngulr prism. It hs fied height of cm, length of cm nd width of cm. If A cm is the totl surfce re: Epress A in terms of. cm Sketch the grph of A ginst. cm c Find the minimum totl surfce re of the o nd the dimensions in this cse. cm

13 Chpter Coordinte geometr Reciprocl grphs This technique involves sketching the grph of = from the grph of = f (). f( ). When f (), = , the grph of = pproches the verticl f( ) f( ) smptote(s).. Therefore, the grph of = will hve verticl smptotes t the f( ) -intercepts of = f ().. When f (), , the grph of = pproches the horizontl f( ) f( ) smptote (the -is in this cse).. These grphs lso hve common points: () When f () = ±, = ±. The grphs re in the sme qudrnt. f( ) () When f () <, <. f( ) (c) When f () >, >. f( ) Note: If = then: f( ). for f () =, = nd for f () =, =. nd for f () <, > nd for f () >, <. WORKED Emple Sketch the grph of the function the given grph of = 9. Sketch the grph of the function = 9 s given. = , ± from 9 6 = = 9 Continued over pge

14 Mths Quest Specilist Mthemtics Work out the smptotes for = The -intercepts of = 9 re = ±. The verticl smptotes for = re = ±. 9 The horizontl smptote is =. For the -intercept, =. The -intercept is (, -- ). Drw digrm sed on the informtion gthered so fr. 9 9 As, pproches from the positive direction ( + ) As pproches from the negtive direction ( ), 9 6 As +, As +, 9 7 As + +, As, + 9

15 Chpter Coordinte geometr 8 To determine the shpe of the grph ner the -intercept, evlute the vlue of when is ± nd ±. 9 Sketch the grph of = = 6 8 (, ) 9 = = 9 The following emples show different pproch to sketching reciprocl functions. WORKED Emple 6 Sketch the grphs of f () nd g() on the sme set of es where f () = nd g() = ,,. The grph of f () is n upright prol, s =. Clculte the -intercept. -intercepts: = so ( )( + ) = so = or + = nd = or = Stte the coordintes of the -intercepts. The -intercepts re (, ) nd (, ). Clculte the -intercept. The verte or turning point -coordinte is hlf-w etween the -intercepts. -intercept: f () = The -intercept is (, ). Turning point: + = = Continued over pge

16 6 Mths Quest Specilist Mthemtics 6 Sustitute to find the -vlue of the turning point. f ( ) = ( ) ( ) = = The turning point is (, ). 7 Sketch the grph of f () =. (See elow.) 8 9 Use the ove to determine importnt fetures for g() = Verticl smptotes occur when f () hs its -intercepts. For g ( ) = : ( ) = ( ) ( + ) Verticl smptotes: = nd =, , nd so g() + Find the horizontl smptotes. The horizontl smptote is g () =. The reciprocl of the turning point for The reciprocl of the turning point (, ) f () is turning point for g(). is (, -- ). Wherever f () = or, g() = or. As g() = , the grphs of f () nd f( ) g() re in the sme qudrnts. Sketch the grph of g() on the sme es s f (). g() = (, ) f() = = (, ) = WORKED Emple 7 Sketch the grphs of f () nd g() on the sme set of es where: f () = ( + ) nd g() = ,. ( + ) Work out importnt fetures for f () = ( + ). This is n inverted prol, s =. For f () = ( + ) :

17 Chpter Coordinte geometr 7 Clculte the -intercept(s). -intercepts: ( + ) = ( + ) = + = = Stte the coordintes of the -intercepts. The -intercept is (, ). Clculte the -intercept. The -intercept: f () = ( + ) = (9) = 9 The -intercept is (, 9). As the grph touches the -is t The turning point is (, ). (, ), it must lso turn t this point. Hence, (, ) is the turning point. 6 Sketch the grph of f (). (See elow.) Use the ove to determine importnt fetures for g() = For g() = : ( + ) ( + ) 7 Verticl smptotes occur when f() hs its -intercepts. Verticl smptote: = 8 Find the horizontl smptotes., ( + ), nd so g() 9 We cnnot tke the reciprocl of the turning The horizontl smptote is g() =. point for f () s the reciprocl of is not defined it ws worked out in step 7 ove tht this ws the verticl smptote. The -intercept of g() is the reciprocl of the -intercept of f(). Since g() = then g() = or f( ) when f () = or. As g() = , the grphs of f () nd f( ) g() re in the sme qudrnts. Sketch the grph of g() on the sme es s f (). 9 The -intercept is (, -- ) (, g() = ) 9 ( + ) f() = ( + ) =

18 8 Mths Quest Specilist Mthemtics Sketch the grphs of f () nd g() on the sme set of es where: f () = + + nd g() = Work out importnt fetures for f() = + +. For f () = + + : This is n upright prol, s =. Find the -vlue of the turning point WORKED Emple d solving = or f () =. d f () = + For f () =, + = = = Evlute f () when =. f ( ) = ( ) + ( ) + = 8 + = The turning point is (, ). As the prol is upright nd turns t There is no -intercept. (, ) it is completel ove the -is nd hence there is no -intercept. Clculte the -intercept. 8 -intercept: f () = The -intercept is (, ). B smmetr (, ) is lso on the curve. Sketch the grph of f (). (See elow.) Use the ove to determine importnt fetures for g() = For g() = : Since there re no -intercepts for f (), g() There re no verticl smptotes. hs no verticl smptotes. Find the horizontl smptote., , nd so g() The horizontl smptote is f () =. The verte of g() is the reciprocl of the The verte is (, ). verte of f (). The -intercept for g() is the reciprocl of The -intercept is (, -- ). the -intercept of f (). Since g() = the grphs of f () nd f( ) 6 g() re in the sme qudrnts. f() = + + Sketch the grph of g() on the sme es s f (). g() = + + (, ) (, )

19 Chpter Coordinte geometr 9 rememer rememer. To sketch the grph of = from the grph of = f (): f( ) () find the verticl smptote t the -intercepts of f () () the horizontl smptote is the -is (c) find the common points when f () = ±.. The grphs re in the sme qudrnt, tht is, f () <, < nd f () >, f( ) >. f( ) Note: If = then f () =, = nd f () =, = nd lso f () <, f( ) > nd f () >, <. B Reciprocl grphs WORKED Emple Sketch the grph of ech of the following functions from the given grph. = , ± = , + 6 = = + c = ,, d = ,, + + = = 6 + (, ) (, ) (, ) Reciprocl functions EXCEL Single grph plotter Mthcd Spredsheet

20 Mths Quest Specilist Mthemtics WORKED Emple 6 WORKED Emple 7 WORKED Emple 8 e = , -- f = ,, + + = + (, ) ( (, ), ) = g = , ± h,, = = 9 = + (, ) (, ) (, ) 8 ( 9, ) 6 8 Sketch the grph of ech of the following functions, f () nd g(), on the sme set of es. f () =, g ( ) = , f () =, g ( ) = ,, c f () =, g ( ) = , d f () = + +, g ( ) = ,, + + e f () = +, g ( ) = ,, + f f () = 8, g ( ) = ,, -- 8 g f () = ( ), g ( ) = , ( ) h f () = ( + ), g ( ) = , ( + ) i f () = +, g ( ) = , + j f () = , g ( ) =, k f () = +, g ( ) = l f () = + +, g ( ) =

21 Chpter Coordinte geometr multiple choice Consider the function f () = f () hs smptotes with equtions: c A =, = nd = B =, = nd = C =, = nd = D =, nd = onl E =, nd = -- The -intercept nd turning point re respectivel: A (, ) nd (, ) B (, ) nd (, ) C (, -- ) nd (, ) D (, -- ) nd (, ) E (, -- ) nd (, ) The grph of f () is est represented : A B C D E A o in the shpe of rectngulr prism hs se of length cm nd width ( ) cm. Epress the re of the se, A cm, in terms of. c If the volume of the o is fied t cm, epress the height, h cm, in terms of. Determine the height of the o when the length of the se is.9 cm. d Sketch the grph of h ginst. e Find the minimum height of the o nd the dimensions in this cse.

22 Mths Quest Specilist Mthemtics Grphs of circles All points P(, ) which stisf the reltion + = r lie on circle with centre (, ) nd rdius r. r P (, ) If the points P(, ) re trnslted (h, k) units then the reltion ecomes ( h) + ( k) = r This reltion represents circle with centre (h, k) nd rdius r. (h, k) r P ( + h, + k) r P (, ) Assumes h, k > WORKED Emple Sketch the grph of the circle with centre (, ) nd rdius. Write the Crtesin eqution of this circle. 9 Write the eqution in the form ( h) + ( k) = r where h =, k = nd r =. Sketch its grph. The eqution is ( + ) + ( ) = (, ) 6

23 Chpter Coordinte geometr WORKED Emple Sketch the grph of =. Stte the coordintes of the centre nd the rdius. Complete the squre in nd. ( + ) 9 + ( + ) = Epress the eqution in stndrd circle form. ( + ) + ( + ) = 6 Recognise tht this is circle nd stte the centre nd the rdius. This represents circle with centre (, ) nd rdius. Sketch its grph. 7 6 (, ) 6 Prmetric equtions of circles The rule for reltion cn sometimes e epressed in terms of third vrile clled prmeter. For the Crtesin eqution of circle + = r, the vriles nd cn e epressed in terms of prmeter t, so tht the prmetric equtions re: = r cos t nd = r sin t, where t [, π] so + = r cos t + r sin t = r (cos t + sin t) = r (since cos t + sin t = ) Note: If t [, π] full circle is otined; if t [, π], semicircle is otined. For the Crtesin eqution of circle ( h) + ( k) = r, the prmetric equtions re: = h + r cos t nd = k + r sin t, The domin () nd rnge () of the Crtesin eqution cn e determined from the rnge of these respective prmetric equtions.

24 Mths Quest Specilist Mthemtics WORKED Emple Find the Crtesin eqution of the circle with prmetric equtions = + cos θ nd = + sin θ, θ [, π]. Stte the domin nd rnge of the circle. Rewrite the prmeters isolting cos θ = cos θ nd = sin θ nd sin θ. Squre oth sides of ech eqution then dd ( ) ( ) them = Epress the reltion in stndrd circle form. ( ) + ( ) = The domin is the rnge of the prmetric Domin is [, + ] = [, ] eqution = + cos θ. The rnge is the rnge of the prmetric eqution = + sin θ. Rnge is [, + ] = [, ] Grphs of ellipses If circle with Crtesin eqution + = is dilted fctor from the -is nd fctor of from the -is then ll points P(, ) on the circle ecome the points P (, ) s shown t right. The sic eqution of n ellipse is: = Its grph is shped like n elongted circle see grph t right. This ellipse:. is centred t (, ). hs vertices t (, ), (, ) (found letting = nd solving), nd (, ) nd (, ) (found letting = nd solving). If this curve were shifted h units to the right nd k units up, then the centre would move to (h, k) nd its eqution would ecome: Note: If = then the eqution ecomes nd cn e rerrnged to ( h) + ( k) = ( multipling oth sides ). This is the eqution of circle. ( h) ( k) = P(, ) ( h) ( k) = P(, )

25 Chpter Coordinte geometr ( h) For n ellipse in the form ( k) = we cn deduce the following, which will help us to sketch the ellipse:. (h, k) re the coordintes of the centre of the ellipse.. The vertices re ( + h, k), ( + h, k), (h, + k), (h, + k). Notes:. is hlf the length of the mjor is (is of smmetr prllel to the -is if > ), (is of smmetr prllel to the -is if < ).. is hlf the length of the minor is (is of smmetr prllel to the -is if > ), (is of smmetr prllel to the -is if < ).., re lengths nd so re positive vlues. WORKED Emple ( ) Sketch the grph of the function ( ) =. 9 ( ) Compre ( ) = 9 ( h) ( k) =. with h =, k = nd so the centre is (, ). = = 9 = = The mjor is is prllel to the -is s >. The etreme points (vertices) prllel to the -is for the ellipse re: ( + h, k) ( + h, k) The etreme points (vertices) prllel to the -is for the ellipse re: (h, + k) (h, + k) Vertices re: ( +, ) ( +, ) = (, ) = (6, ) nd (, + ) (, + ) = (, ) = (, ) Sketch the grph of the ellipse. 6 (, ) ( ) + ( ) 9 = (, ) (, ) (6, ) 6 6 (, )

26 6 Mths Quest Specilist Mthemtics WORKED Emple ( ) Sketch the grph of the function ( + ) =. 9 6 ( ) Compre ( + ) = with ( h) ( k) =. The mjor is is prllel to the -is s >. The etreme points (vertices) prllel to the -is for the ellipse re: ( + h, k) ( + h, k) The etreme points (vertices) prllel to the -is for the ellipse re: (h, + k) (h, + k) Sketch the grph of the ellipse. h =, k = So the centre is (, ). = 9 = 6 = = Vertices re: ( +, ) ( +, ) = (, ) = (, ) nd (, ) (, ) = (, 8) = (, ) (, ) 6 (, ) (, ) (, ) 6 8 (, 8) ( ) 9 + ( + ) 6 = Sketch the grph of the function + 9( ) =. Rerrnge nd simplif dividing oth + 9( ) = sides to mke the RHS =. 9( ) = Simplif cncelling. ( ) = 9 h =, k = nd so the centre is (, ). Compre ( ) = with 9 = 9 = s, > ( h) ( k) = = =. WORKED Emple Mjor is is prllel to the -is s >. The etreme points (vertices) prllel to the -is for the ellipse re: ( + h, k) ( + h, k) Vertices re: ( +, ) ( +, ) = (, ) = (, )

27 Chpter Coordinte geometr 7 6 The etreme points (vertices) prllel to the nd (, + ) (, + ) -is for the ellipse re: (h, + k) (h, + k) or (, ) (, + ) (,.) (,.) 7 Sketch the grph of the ellipse. 6 (, + ) + 9( ) = (, ) (, ) (, ) (, ) Sketch the grph of the reltion descried the rule: =. Tke the coefficient of out s fctor ( + 6) + ( ) + 9 = for the terms nd the coefficient of out s fctor for the terms. Complete the squre in oth nd. [( + ) 9] + [( ) ] + 9 = Epnd the squre rckets. ( + ) + ( ) + 9 = Simplif. ( + ) + ( ) = Add to oth sides. ( + ) + ( ) = 6 Divide oth sides. ( + ) ( ) = Epress the reltion in the stndrd ( + ) ( ) form of n ellipse. = 8 Stte the coordinte of the centre nd the vlues of nd. The ellipse hs its centre t (, ), with = nd =. 9 Stte the coordintes of the vertices. Vertices re (, ), (, ) nd (, ), (, 6). Sketch the grph of the ellipse. WORKED Emple (, 6) 6 (, ) (, ) (, ) (, )

28 8 Mths Quest Specilist Mthemtics Grphics Clcultor tip! Grphing ellipses (nd hperols) using function grphs CASIO Grphing ellipses (nd hperols). To grph n ellipse with n eqution (s) of ( ) =, first rerrnge the eqution to mke the suject. The eqution ecomes = ± ( ).. Enter Y = ( X ) nd enter Y = Y (see figure on left, elow). (Rememer, to enter Y, press VARS nd select Y VARS, :Function nd :Y.). Press ZOOM nd select :ZDeciml to get good jumps for trcing (djust the WINDOW settings for Ymin nd Ym lter if necessr) nd TRACE s required. Note tht ZDeciml lso gives squre or true proportion window. (Zoom in or out s necessr to get complete view. See centre figure, elow.). If ou use our own WINDOW settings, ou m not get good view. The screen elow hs window [, ] [, ]. The grph is incomplete (poor jumps) nd is not in true proportion (see figure on right). Prmetric equtions of ellipses The prmetric equtions of n ellipse with Crtesin eqution = cos t nd = sin t, where t [, π]: cos t + sin t = ut cos t = -- nd sin t = -- so = The prmetric equtions of n ellipse with Crtesin eqution re = h + cos t nd = k + sin t. WORKED Emple = re ( h) ( k) = Determine the Crtesin eqution of the curve with prmetric equtions = + sin t nd = cos t where t R. Descrie the grph nd stte its domin nd rnge. Rewrite the prmeters isolting cos t = sin t, nd = cos t nd sin t. Squre oth sides of ech eqution then ( ) ( ) dd = sin t + cos t 9 =

29 Chpter Coordinte geometr 9 Descrie the reltion. This represents n ellipse with centre (, ). The domin is the rnge of the prmetric Domin is [, + ] = [, ] eqution = + sin t. The rnge is the rnge of the prmetric eqution = cos t. Rnge is [, + ] = [, ] rememer rememer For circle with eqution ( h) + ( k) = r :. The centre is (h, k) nd the rdius is r.. The prmetric equtions re = h + rcos t nd = k + rsin t. ( h) For n ellipse with eqution ( k) = :. The centre of the ellipse is (h, k).. The vertices re ( + h, k), ( + h, k), (h, + k), (h, + k).. The prmetric equtions re = h + cos t nd = k + sin t. The domin () nd rnge () of the Crtesin eqution cn e determined from the rnges, respectivel, of these prmetric equtions. C Grphs of circles nd ellipses WORKED Emple 9 WORKED Emple WORKED Emple Write the eqution of the circle with the following centres nd rdii. Centre (, ) nd rdius Centre (, ) nd rdius 6 c Centre (, ) nd rdius d Centre (, ) nd rdius Find the coordintes of the centre nd the rdii of the circles with the following equtions. Sketch the grph in ech cse = + + = c + = d = e = f = Find the Crtesin equtions of the circles with the following prmetric equtions. Stte the domin nd rnge of ech. = cos t, = sin t, t [, π] = cos t, = sin t, t [, π] c = cos t, = + sin t, t [, π] d = sin t, = cos t, t R multiple choice For ech of the equtions elow, choose the correct lterntive for the vlues of,, h, nd k. (This question is like worked emple, ut don t drw the grph.) ( ) ( + 9) = A = 9, = 8, h =, k = 9 B = 8, = 6, h =, k = 9 C = 8, = 9, h =, k = 9 D = 9, = 8, h =, k = 9 E = 9, = 8, h =, k = 9 EXCEL Spredsheet Ellipses Ellipses Mthcd

30 Mths Quest Specilist Mthemtics WORKED Emple WORKED Emple c d Sketch the grph of the following functions. ( ) ( ) = c ( + ) = d e ( ) ( ) = f g ( + ) ( ) = h i ( + ) ( + ) = j k ( ) = l 6 Sketch the grph of the following functions. ( ) ( + ) = 9 c = d 9 ( ) e ( + ) = f g = h 6 ( ) i ( + ) = j 8 9 ( + ) k = l 7 ( ) = A = 6, =, h =, k = B = 6, =, h =, k = C = 6, =, h =, k = D = 6, =, h =, k = E =, = 6, h =, k = ( + ) ( + ) = A = 9, =, h =, k = B = 8, =, h =, k = C = 9, =, h =, k = D = 8, =, h =, k = E = 8, =, h =, k = ( ) = A =, =, h =, k = B =, =, h =, k = C = 6, =, h =, k = D = 6, =, h =, k = E =, =, h =, k = ( ) ( + ) = ( ) = ( ) ( + ) = ( ) ( ) = multiple choice ( + ) Consider the ellipse with the eqution ( ) =. The centre of the ellipse is: 6 A (, ) B (, ) C (, ) D (, ) E (, ) = ( ) ( + ) = 6 ( + ) = ( ) = ( ) ( + ) = ( ) ( ) = ( ) ( ) = ( + ) ( + ) =

31 Chpter Coordinte geometr WORKED Emple WORKED Emple WORKED Emple 6 The mimum nd minimum points on the ellipse re respectivel: A (, ) nd (, ) B (, 7) nd (, ) C (, 6) nd (, ) D (, ) nd (, 8) E (, ) nd (, 6) 8 Sketch the grphs of the following functions. 9( ) + 6( + ) = 6 + = c 6( ) + = d ( ) + 6( + ) = 6 e 6( ) + 9( + ) = f ( ) + 9( + ) = 9 Sketch the grphs of the following functions = = c = d = Determine the Crtesin eqution of ech of the curves with the following prmetric equtions. Descrie the grph nd stte its domin nd rnge. = sin t, = cos t, t [, π]. = + cos t, = sin t, t [, π]. π c = cos t, = + sin t, t [, -- ]. Give pir of prmetric equtions which correspond to the following Crtesin equtions. + = = 9 c ( ) + ( + ) ( + ) ( ) = 6 d = 9 Grphs of hperols Hperols hve the following importnt chrcteristics.. The sic eqution of hperol centred t (, ) is. = = =. If this curve were shifted h units to the right nd k units up, then the centre (, ) (, ) would move to (h, k) nd its eqution ( h) would ecome ( k) (, ) =.. The sic form of hperol centred t (, ) is shown t right. The vertices for this curve re t (, ) nd (, ) nd the two smptotes re given = -- nd = --. When the hperol is not centred t (, ): ( h). For the curve of the function ( k) =, the points on = re moved h units to the right nd k units up (or hs een replced with ( h) nd replced with ( k)).. Therefore, the vertices re ( + h, k), ( + h, k) nd the centre is t (h, k).. The smptotes re t k = -- ( h) nd k = -- ( h) or = -- ( h) + k nd = -- ( h) + k.

32 Mths Quest Specilist Mthemtics To drw sketches of hperolic reltions we simpl:. Rerrnge the eqution into the pproprite generl form nd determine the vlues of nd.. Write down the coordintes of the centre.. Stte the coordintes of the vertices.. Write down the equtions of the smptotes.. Sketch hperolic grph which fits the ove informtion. WORKED Emple 7 Sketch the grph of the hperol with eqution =. 9 The eqution is in the correct form, so red off the vlues of,, h nd k. As h =, k =, there re no trnsltions. = 9 = = = Write the coordintes of the centre. The centre is t (, ). Write the coordintes of the vertices. The vertices re (, ) nd (, ). Write the equtions of the smptotes. The smptotes re = -- nd = --. Drw the smptotes, plot the vertices nd = = centre, nd then sketch the hperol. (, ) (, ) (, ) = WORKED Emple 8 ( ) Sketch the grph of the hperol with the eqution ( ) =. 6 9 The eqution is in the correct form, so red off the vlues of,, h nd k. h =, k = = 6 = 9 = = Write the coordintes of the centre. The centre is (, ). Write the coordintes of the vertices. The vertices re ( +, ) nd ( +, ) or (, ) nd (7, ).

33 Chpter Coordinte geometr Write the equtions of the smptotes. 6 For ech smptote find the - nd -intercepts. The - nd -intercepts for = re too close to ech other so use one of these points, s ( --, ), nd the centre to sketch this line s oth smptotes intersect here. Plot the vertices nd centre nd then sketch the hperol. The smptotes: = -- ( ) = -- ( ) ( ) = ( ) ( ) = ( ) 8 = = 9 + = 7 = For + = 7 For = =, = 7 =, = 7 = = -- 7 (, ) (, -- ) =, = 7 =, = 7 7 = = -- (-----, ) (--, ) 7 Sketch the grph of the hperol with eqution 6 9( ) =. Rerrnge the eqution dividing oth sides to mke the RHS =. Simplif cncelling. Red off the vlues of h nd k Work out vlues of nd. WORKED Emple 9 6 9( ) = 6 9( ) = (, ) (, ) (7, ) + = 7 = ( ) 6 ( ) 9 = ( ) = 9 6 h =, k =, trnsltion of units up = 9 = 6 = = 6 s, > Write the coordintes of the centre. The centre is t (, ). Write the coordintes of the vertices. The vertices re:( +, ) nd ( +, ) or (, ) nd (, ) Continued over pge

34 Mths Quest Specilist Mthemtics Write the equtions of the smptotes. 6 The smptotes re: 6 6 = ( ) nd = ( ) = 6 ( ) = 6 6 = 6 6 = = 6 6 = 6 7 Write the - nd -intercepts for the Intercepts for + 6 = 6 re ( 6, ) nd (, ). smptotes. Intercepts for 6 = 6 re ( 6, ) nd (, ). 8 Drw the smptotes, plot the vertices nd centre, nd then sketch the 6 = 6 hperol. (, ) (, ) (, ) + 6 = 6 6 9( ) = Prmetric equtions of hperols The prmetric equtions of hperol with Crtesin eqution = re: = sec t nd = tn t = sec t The grph of the prmetric eqution = sec t shows how it ffects the domin of the hperol. For t π --,, [, ), which represents the right rnch of hperol. -- π For nd t π --, π, (, ], which represents the left rnch of hperol. = tn t The grph of the prmetric eqution = tn t shows how it ffects the rnge of the hperol. For t π --, or t, R. -- π π --, π Generll, the domin of the hperol with centre (, ) is (, ] [, ) nd the rnge is R. π π π = sec t π π π π π = tn t π ππ t t

35 Chpter Coordinte geometr Verifing the prmetric equtions of hperol We cn verif the prmetric equtions of hperol with eqution follows. Since: + tn t = sec t = s sec t tn t = ut sec t = nd tn -- t = -- (from the prmetric equtions) = [the Crtesin eqution of hperol with centre (, )] Similrl, the prmetric equtions of hperol with Crtesin eqution ( h) ( k) = re: WORKED Emple = h + sec t nd = k + tn t. Determine the Crtesin eqution of the curve with prmetric equtions = sec t nd = tn t, where t π --, π. Descrie the grph nd stte its domin nd rnge. -- Rewrite the prmeters isolting sec t nd tn t. -- = sec t, nd -- = tn t Squre oth sides of ech eqution then = sec t tn t sutrct. 9 = Descrie the reltion. This represents hperol with centre (, ). The domin is the right rnch of the Domin is [, ) hperol [, ). The rnge is R. Rnge is R. rememer rememer ( h) For hperol with eqution ( k) = :. The vertices re ( + h, k), ( + h, k).. The centre is t (h, k).. The smptotes re t k = -- ( h) nd k = -- ( h).. The prmetric equtions re = h + sec t nd = k + tn t.

36 6 Mths Quest Specilist Mthemtics D Grphs of hperols EXCEL Spredsheet Mthcd Grphs of hperols Grphs of hperols multiple choice For ech of the equtions elow, choose the correct lterntive for the equtions of the smptotes nd the coordintes of the vertices = A = -- 9, = -- 9, ( 9, ), (9, ) B = --, = --, ( 9, ), (9, ) C = --, = --, ( 8, ), (8, ) D = --, = --, ( 8, ), (8, ) 8 9 E = --, = --, ( 9, 8), (9, 8) c d ( ) = 6 A = + 6, = 6, ( 6, ), (6, ) B = +, = ( 6, ), (6, ) C = +, = (, ), (, ) D = + 6, = 6, (, ), (, ) E = +, = (, ), (, ) ( + ) = 9 A = 9, = 9, ( 6, ), (, ) B + 6 =, + 6 =, (, ), (, ) C = 9, = 9, (, ), (, ) D = 9, = 9, (, ), (, ) E + 6 =, + 6 =, ( 6, ), (, ) ( ) ( ) = 6 A =, =, (, ), (, ) B =, = (, ), (6, ) C =, = (, ), (6, ) D =, =, (, ), (, ) E =, = (, ), (, ) WORKED Emple 7 WORKED Emple 8 Sketch the grphs of the hperols with the following equtions. c e g i k = = = ( ) ( ) = ( ) ( ) = ( ) ( ) = d f h j l = = = 6 ( ) ( ) = ( ) = ( ) ( + ) = 9

37 Chpter Coordinte geometr 7 multiple choice The rule representing the grph shown t right is: A = B ---- = ( + ) C = D ---- = E = multiple choice The grph which est represents the reltion A C 6 (, ) (, ) (8, ) 6 8 D ( ) ( ) = 9 B is: 8 6 ( 8, ) 8 (, ) 6 8 (, ) 6 (, ) (, ) (6, ) 6 (, ) (, ) (7, ) E (, ) (, ) 6 6 WORKED ORKED Emple 9 Sketch the grph of the hperol with the eqution: 6 = 9 6 = c = d 9 = e 6( ) 9( + ) = f ( ) 9( + ) = g 6( ) ( ) = h 9 6( + ) = i 9( + ) 7 = 6 j ( ) 9( ) = 6.

38 8 Mths Quest Specilist Mthemtics WorkSHEET. WORKED Emple 6 Determine the Crtesin eqution of ech of the curves with the following prmetric equtions. Descrie the grph nd stte its domin nd rnge. = sec t nd = tn t, where t R. = sec t nd = tn t, where t R. π π c = sec t nd = + tn t, where t (--, ) π π d = + sec t nd = tn t, where t ( --, -- ) 7 Give pir of prmetric equtions which correspond to the following Crtesin equtions. ( ) ( + ) ( + ) ( ) = = 9 6 Prtil frctions Adding frctions to otin single frction is fmilir nd sic process. For emple, + -- = = or = ( ) + ( + ) ( + ) ( ) = = The reverse of this process is to split frction into the sum of simpler frctions. These simpler frctions re referred to s prtil frctions. For emple, using the ove emples: = Frctions Prtil frctions = Splitting rtionl epression into prtil frctions is useful when ntidifferentiting rtionl epressions, s we will see in chpter 6. It is lso useful for sketching grphs of some rtionl epressions, s will e seen in the net section. For now, we concentrte on the methods for splitting frction with qudrtic denomintor into prtil frctions. Equting polnomils We regulrl use the smol, which mens is identicll equl to, to indicte tht two epressions re equl for ll llowle vlues of. Two polnomils of degree n re equl if the coincide for more thn n vlues of. For emple, suppose ( + ) + ( ).

39 Chpter Coordinte geometr 9 If vlues of nd re found so tht the polnomils (degree ) on ech side of this identit re equl for two sustituted vlues of, then the must e equl for ll vlues of. The most convenient vlues of to sustitute into the ove identit re = nd = s the llow nd to e solved direct sustitution. This is demonstrted in the following worked emple. WORKED Emple Determine the vlues of nd in the following identities: ( + ) + ( ) + ( ) + 9. Sustitute = to eliminte from the identit. Solve for. Sustitute = to eliminte from the identit. ( + ) + ( ) + Let = 7 = + 7 = 7 = Let =. 7 = + 7 = Solve for. = Stte the solution. The solution is = nd =. Sustitute = -- to eliminte from ( ) + 9 the identit. Let = --. Solve for. = ( -- ) 9 Sustitute = nd =, s (or n vlue of other thn -- ), since cnnot e eliminted from the identit. Solve for. = 6 9 = Let =. = 9 = 6 = Stte the solution. The solution is = nd =. Creting prtil frctions P ( ) The generl procedure for splitting rtionl epression , where Q() is Q ( ) qudrtic epression, into prtil frctions is outlined in the following steps. Step. If the degree of P() is greter thn or equl to the degree of Q() then divide Q() into P() nd split the rtionl prt into prtil frctions using the following steps. Step. Fctorise the denomintor Q(). P ( ) Step. Equte where R() nd S() re fctors of Q() nd Q ( ) R ( ) + S ( ) re usull liner.

40 Mths Quest Specilist Mthemtics Step. Epress the right-hnd side of the identit in the sme form s the left-hnd side, with the sme denomintors: P ( ) Q ( ) S( ) + R( ) Q ( ) Step. Equte the numertors: P() S() + R(). Step 6. Solve for nd sustitution. Step 7. Sustitute nd into the prtil frctions. Notes. If Q() is perfect squre then steps to will e similr to ut not ectl like those prescried ove. This will e demonstrted lter in cse.. The solution cn e quite esil checked dding the prtil frctions which should equl the originl rtionl epression. Cse : Fctorised denomintor If f (), g() nd h() re liner functions, then: f( ) g ( )h ( ) g ( ) + h ( ) WORKED Emple Epress s prtil frctions. ( + ) ( ) Equte the rtionl epression to ( + ) ( ) Epress the right-hnd side of the identit into the sme form s the left-hnd side. ( ) + ( + ) ( + ) ( ) Equte the numertors. ( ) + ( + ) Sustitute = to eliminte. Let =. Solve for. = = 6 Sustitute = to eliminte. Let =. 8 = Solve for. = 7 8 Epress the originl rtionl epression s prtil frctions ( + ) ( ) = Cse : Perfect squre denomintor If f () nd g() re liner, then: f( ) [ g ( )] g ( ) [ g ( )]

41 Chpter Coordinte geometr WORKED Emple Write the rtionl epression s prtil frctions. ( 7) Equte the rtionl epression to ( 7) ( 7) 7 + ( ) ( 7) Epress the right-hnd side of the identit into the sme form s the left-hnd side ( 7) + ( 7) Equte the numertors. ( 7) Sustitute = -- to eliminte. Let = = Solve for. = -- 6 Sustitute nother vlue for, s =, nd Let =. =. -- = Solve for. 7 7 = -- 8 Epress the rtionl epression s prtil = frctions. ( 7) ( 7) or = ( 7) ( 7) Cse : Denomintor is not fctorised If the denomintor of rtionl epression is not fctorised, then fctorise it first efore splitting it into prtil frctions. WORKED Emple B first fctorising the denomintor, epress s prtil frctions. 9 = The denomintor fctorises s difference of perfect squres. 9 Equte the rtionl epression to = ( + ) ( ) Epress the right-hnd side of the identit into ( ) + ( + ) the sme form s the left. 9 Equte the numertors. ( )( + ) Continued over pge

42 Mths Quest Specilist Mthemtics Sustitute = to eliminte. Let =. = 6 6 Solve for. 6 = 8 = 7 Sustitute = to eliminte. Let =. = 6 8 Solve for. 6 = = 9 Epress the rtionl epression s prtil = frctions. 9 + Cse : The degree of the numertor the degree of the denomintor If the degree of the numertor of rtionl epression is greter thn or equl to the degree of the denomintor, then divide the denomintor into the numertor first efore splitting the frctionl prt into prtil frctions. WORKED Emple Epress ech of the following s prtil frctions ( ) ( + ) The degrees of the numertor nd denomintor re oth. Degrees re oth. Epnd the denomintor so tht it cn e divided into the numertor. + ( ) ( + ) Use long division to divide Epress the originl rtionl epression in terms of the quotient nd reminder. + ( ) ( + ) ( ) ( + ) Equte the frctionl prt to ( ) ( + ) + 6 Epress the right-hnd side of te identit in ( + ) + ( ) the sme form s the left-hnd side. ( ) ( + ) 7 Equte the numertors. ( + ) + ( ) 8 Sustitute = to eliminte. Let =. = 6 Solve for. = 9

43 Chpter Coordinte geometr Sustitute = to eliminte. Let =. = 6 Solve for. = Epress the frction s prtil frctions. So ( ) ( + ) = Rewrite the originl epression s + prtil frctions ( ) ( + ) = The degree of the numertor () is greter thn the degree of the denomintor (). Use long division to divide the denomintor into the numertor Epress the originl rtionl epression = in terms of the quotient nd reminder Fctorise the denomintor of the = frction prt. ( + ) ( ) Equte the frctionl prt to Epress the right-hnd side of the ( ) + ( + ) identit in the sme form s the left-hnd ( + ) ( ) 7 side. Equte the numertors. + ( ) + ( + ) 8 Sustitute = to eliminte. Let =. 8 + = 9 Solve for. = 9 = Sustitute = to eliminte. Let =. + = Solve for. = = Epress the frction s prtil frctions. + So ( + ) ( ) Rewrite the originl epression s = prtil frctions

44 Mths Quest Specilist Mthemtics rememer rememer. If f (), g() nd h() re liner functions, then: () f( ) g ( )h ( ) g ( ) + h ( ) f( ) () [ g ( )] g ( ) + [ g ( )]. If the denomintor of rtionl epression is not fctorised, then fctorise it first efore splitting it into prtil frctions.. If the degree of the numertor of rtionl epression is greter thn or equl to the degree of the denomintor, then divide the denomintor into the numertor first efore splitting the frctionl prt into prtil frctions. E Prtil frctions WORKED Emple Evlute the vlues of nd in the following identities. ( ) + ( + ) ( + ) + ( ) + 9 c ( + ) + + d ( ) + ( ) + e ( + ) + ( ) 9 f ( + ) + ( ) 6 g ( ) + ( + 7) 8 h ( + ) + ( + ) 9 8 Mthcd Prtil frctions WORKED Emple Epress the following s prtil frctions ( + ) ( + ) ( ) ( + ) c d e ( ) ( ) ( + ) ( ) f g h ( + ) ( ) ( + ) ( 6 ) i + j ( ) ( + ) ( ) ( + ) ( ) ( + ) ( + 7) ( ) multiple choice The respective vlues of nd in the identit + ( ) + re: A, B, C, D, E, multiple choice + The respective vlues of nd in the identit re: ( + ) ( ) A, B, C, D, E,

45 Chpter Coordinte geometr WORKED Emple Write ech of the following rtionl epressions s prtil frctions c ( + ) ( ) ( + ) d e f g ( ) ( ) ( + ) h ( ) ( + ) WORKED Emple 6 B first fctorising the denomintor, epress ech of the following functions s prtil frctions c d e f g h i WORKED Emple 7 Epress ech of the following functions s prtil frctions ( + ) ( ) ( ) ( + ) c d e f g h ( ) ( + ) multiple choice Consider the rtionl epression After long division the epression simplifies to: + 7 A B C D E When epressed s prtil frction, the epression simplifies to: A B C D E Mthcd To esil find the prtil frctions for n epression or to check our nswers, the Mthcd file Prtil frctions found on the Mths Quest CD-ROM cn e used. Prtil frctions

46 6 Mths Quest Specilist Mthemtics The screen shows two emples. Note tht the second emple pplies to cses with repeted fctors in the denomintor. Sketch grphs using prtil frctions In this section we investigte how prtil frctions cn e used to ssist in the grphing of rtionl functions. Emphsis is plced on locting smptotes nd the ddition-ofordintes method of grphing. Addition of ordintes The grph of function tht involves the ddition of two (or more) simpler, fmilir functions cn e otined grphing the two simpler functions on the sme set of es nd then dding the ordintes (-vlues). For emple, consider the grph of the function = Sketch the grphs of = nd = --, with roken lines, on the sme set of es.. For severl vlues of dd the -vlues, nd, to otin. Some emples re: () When =, = nd =, so = + =, giving the point (, ). (, ) = (, ) =

47 Chpter Coordinte geometr 7 () When =, = nd =, so = =, giving the point (, ). (c) When =, = nd = --, so = + -- = --, giving the point (, -- ).. Repet until the shpe of the function is deduced. The grph of the function = f () + g() cn e otined grphing = f () nd = g() on the sme set of es nd then dding the ordintes. (, ) = (, ) (, ) = (, ) = + Grphs of the tpe = c g ( ) hve smp- As we sw erlier in this chpter, grphs of the form c totes = -- nd = g( ). = c g ( ) The grph of the function = c g ( ) cn e otined :. sketching the grphs of = nd = g() (n smptote) on the sme + c es. dding the two the ddition-of-ordintes method. Notes. If g() = d, constnt, then the grph of = cn e sketched c d recognising tht it is hperol. Assuming tht,, c nd d re greter thn, the sketch is: = c d c,, c, d > = + c + d = d. Wherever possile, verif grphs using grphics clcultor.

48 8 Mths Quest Specilist Mthemtics WORKED Emple Sketch the grphs of ech of the following, showing n smptotes nd il intercepts. = = The grph is hperol with smptotes = nd = (-is). Asmptotes: = There is no -intercept ( = is n smptote). = No -intercept Sustitute = into the eqution to When =, find the -intercept. Sketch the grph. 6 = -- = (, ) Divide the denomintor into the numertor. Epress the rtionl function s prtil frctions. The grph is hperol with smptotes = -- nd =. Susitute = into the originl eqution nd solve for to determine the -intercept. Asmptotes: = -- = =, + = Sustitute = nd solve for to =, = determine the -intercept. = -- 6 Sketch the grph = = = =

49 Chpter Coordinte geometr 9 Sketch the grph of the function = Divide the denomintor into the numertor. Epress the rtionl function s prtil frctions. Sketch the grphs of = (smptote) nd = on the sme es. WORKED Emple = = = Determine n -intercepts. =, + 6 = ( )( ) = = nd = Determine the -intercept. 6 =, = Add the two grphs ddition of ordintes to otin the grph of + 6 = = -- = = = (, ) = + = Grphs of the tpe = f( ) g( ) c Grphs of the form = c, where, nd c R nd f () nd g() re f( ) g ( ) liner epressions, hve two verticl smptotes nd the line = c ehves s horizontl smptote everwhere ecept for the point where =. f( ) g ( )

50 Mths Quest Specilist Mthemtics The grph of = is otined : f( ) g( ) c. sketching the grphs of = nd on the sme es f( ) = g( ) c. solving = to find where = c f( ) g( ). dding the two grphs the ddition-of-ordintes method. WORKED Emple 8 Sketch the grph of the function = first epressing it s prtil frctions. Stte the equtions of ll verticl smptotes nd determine n il intercepts. Fctorise the denomintor of the rtionl epression ( + ) ( ) Epress the rtionl epression s prtil frctions in the form ( ) + ( + ) ( + ) ( ) ( ) + ( + ) Let =. 6 = 6 = Let =. = 6 = So or = Sketch the grphs of = = on the sme is. nd = + =

51 Chpter Coordinte geometr Determine the - nd -intercepts. =, = = =, = Solve the eqution = to + determine where the horizontl smptote, =, is crossed. Add the two grphs the ddition-ofordintes method to otin the grph of the rtionl function. = When =, = + ( ) ( + ) = = = = So horizontl smptote is crossed t (, ). (, ) = = = 7 Stte the equtions of the verticl smptotes. Verticl smptotes re = nd =. WORKED Emple 9 Sketch the grph of the function f( ) smptotes. = , clerl indicting ll + 6 Divide the denomintor of the rtionl epression into the numertor Epress f () in terms of the quotient nd 8 f( ) = reminder = ( + ) ( ) Continued over pge

52 Mths Quest Specilist Mthemtics Epress the frction s prtil frctions ( + ) ( ) ( ) + ( + ) ( + ) ( ) 8 ( ) + ( + ) Let =. = = Let =. = = Rewrite f () in prtil frction form. So f( ) = Stte the equtions of the verticl smptotes. Verticl smptotes re = nd =. 6 Sketch the grphs = nd + = + = + on the sme is. 7 8 Solve = to find where the + horizontl smptote, =, is crossed. Add the two grphs the ddition-ofordintes method to otin the grph of f (). = + When =, = + ( ) ( + ) = 6 = 8 = = 8 So horizontl smptote is crossed t ( 8, ). ( 8, ) = + 8 = + + = +

53 Chpter Coordinte geometr You cn verif the grphs otined using grphics clcultor. The following screens show the eqution entered, the WINDOW settings used nd the grph produced for worked emple. rememer rememer. The grph of = cn e otined : c g ( ) () sketching the grphs of = nd (n smptote) on the + c = g ( ) sme es () dding the two the ddition-of-ordintes method.. The grph of = c is otined : f( ) g ( ) () sketching the grphs of = nd on the sme es f( ) = c g ( ) () dding the two grphs the ddition-of-ordintes method.. Use our grphics clcultor to check our grphs. F Sketch grphs using prtil frctions WORKED Emple 6 WORKED Emple 6 Use grphics clcultor wherever possile to verif the grphs otined in the following eercise. Sketch the grphs of ech of the following functions. = = c = d = e = f = g = Single function grpher Single function grpher Mthcd EXCEL Spredsheet

54 Mths Quest Specilist Mthemtics WORKED Emple Sketch the grph of the functions. 7 = = c d = e = = WORKED Emple 8 Sketch the grph of the following functions first epressing them s prtil frctions. Stte the equtions of ll verticl smptotes in ech cse. = = c = ( ) ( + ) ( + ) ( ) d = e = multiple choice Mthcd Prtil frctions + Consider the function f( ) = As prtil frction, f() is equl to: A B C D E The grph tht est represents f() is: A B C D E

55 Chpter Coordinte geometr 6 multiple choice The grph of the function f( ) + 7 = hs smptotes with equtions: A =, =, = B = 6, =, = C =, =, = D = nd = E =, =, = + multiple choice If g ( ) = then its grph hs smptotes descried the equtions: A =, =, = B =, = C =, = nd = D =, =, = E = nd = + WORKED Emple 9 7 Sketch the grph of ech of the following, clerl indicting ll smptotes. f( ) + = f( ) + = c g ( ) = d g ( ) = e f( ) = WorkSHEET. Further grphing nd prtil frctions Grphs of the tpe = f( ) nd, g ( ) = f( ) [ g ( )] g ( ) + h ( ) where f (), g() nd h() re liner epressions, will now e eplored. Sketch the grph for ech of the following. For question nd, find the turning points. = = = = = =

56 6 Mths Quest Specilist Mthemtics summr Sketch grphs of = m + n + c where m = or nd n = or Step. Brek the given function into seprte simpler functions. Step. Sketch the grph of ech of the seprte functions creted, on the one set of es. Step. Determine the smptotes of the originl function nd pencil in how the grph of the function pproches these smptotes. Consider:. lrge positive vlues for. lrge negtive vlues for. vlues ver close to nd either side of the verticl smptote. Step. Work out the -intercept(s), -intercept nd turning point for the given function to give greter ccurc. Reciprocl grphs To sketch the grph of = from the grph of = f (): f( ). find the verticl smptote t the -intercepts of f (). the horizontl smptote is the -is. find the common points when f () = ±. The grphs re in the sme qudrnt:. f () <, < f( ). f () >, >. f( ) Note: If = , then: f( ). f () =, = nd f () =, =. lso, f () <, > nd f () >, <. Grphs of circles For circle with eqution ( h) + ( k) = r :. the centre is (h, k) nd rdius r. the prmetric equtions re = h + r cos t nd = k + r sin t. Grphs of ellipses ( h) For ( k) = :. the centre of the ellipse is (h, k). the vertices re ( + h, k), ( + h, k), (h, + k), (h, + k). the prmetric equtions re = h + cos t nd = k + sin t.

57 Chpter Coordinte geometr 7 Grphs of hperols ( h) For ( k) = :. vertices re ( + h, k), ( + h, k). centre is t (h, k). Asmptotes re t k = -- ( h) nd k = -- ( h). The prmetric equtions re = h + sec t nd = k + tn t. Prtil frctions If f (), g() nd h() re liner functions, then: f( ) g ( )h ( ) g ( ) + h ( ) f( ) [ g ( )] g ( ) + [ g ( )] If the denomintor of rtionl epression is not fctorised, then fctorise it first efore splitting it into prtil frctions. If the degree of the numertor of rtionl epression is greter thn or equl to the degree of the denomintor, then divide the denomintor into the numertor first efore splitting the frctionl prt into prtil frctions. Sketch grphs using prtil frctions The grph of = g ( ) cn e otined : + c. sketching the grphs of = nd (n smptote) on the sme es + c = g ( ). dding the two grphs the ddition-of-ordintes method. The grph of = c is otined : f( ) g ( ). sketching the grphs of = nd on the sme es f( ) = c g ( ). dding the two grphs the ddition-of-ordintes method. Use our grphics clcultor to check grphs.

58 8 Mths Quest Specilist Mthemtics CHAPTER review Multiple choice A The grph t right could e descried the rule: A = -- + B = C = -- + D = + E = -- (, ) (, ) (, ) A The grph tht represents the function f( ) = is: A B C D E B The grph representing the function = is: 9 A B (, ) 9 (, ) 9

59 Chpter Coordinte geometr 9 C D (, ) 9 (, ) 9 E (, ) 9 The grph of the rtionl function = hs smptotes: + A = nd = onl B =, = -- nd = C = -- nd = onl D =, = -- nd = E =, = -- nd = ( + ) Consider the ellipse with eqution ( ) = for questions nd 6. 9 The mimum nd minimum points re respectivel: A (, ) nd (, ) B (, ) nd (, ) C (, ) nd (, ) D (, ) nd (, ) E (, 6) nd (, ) 6 The grph representing this eqution is: A B C B C C (, ) (, ) D (, ) E (, ) (, )

60 6 Mths Quest Specilist Mthemtics C 7 The rule tht descries the ellipse t right is: A ( ) = B ( ) = (, ) C ( ) = D ( ) = E ( + ) = 6 ( ) Consider the hperol ( + ) = for questions 8 nd 9. 9 D D 8 The hperol hs vertices given : A (, ) nd (9, ) B (, ) nd (, 7) C (, ) nd (, ) D (, ) nd (, ) E ( 6, ) nd (, ) 9 The grph of the hperol hs smptotes with equtions: A = ± -- ( ) B = ± -- ( + ) + C = ± -- ( ) + 9 D = ± -- ( ) E = ± ( ) + D The rule tht descries the hperol shown elow is: ( + ) A ( + ) = 9 6 = B C ( ) ( + ) = ( + ) ( + ) = (, ) (, ) (, ) D ( + ) ( + ) = = E ( + ) ( ) = E + When epressed s prtil frctions, is equl to: A B C ( + ) + + ( ) ( ) D E

61 Chpter Coordinte geometr 6 The rtionl epression is equl to: 6 A B C D E E The grph of the rtionl function + = is: A B C F (, ) (, ) D E (, ) (, ) The rule descriing the function grphed t right is: A C E = -- B = = -- + D = = (, ) (, ) (, ) F A pir of prmetric equtions which correspond to the Crtesin eqution + = 9 is: A = 7 sec t nd = 7 tn t B = 7 cos t nd = 7 sin t C = 7 cos t nd = 9 sin t D = 7 cos t nd = 7 sin t E = 7 sin t nd = 7 cos t

62 6 Mths Quest Specilist Mthemtics Short nswer A Write down the eqution of the smptotes for = A The function = ---- is roken into the functions = ---- nd =, which pper on the grph shown. Descrie the ehviour of the function = ---- ner the smptotes. Without n further clcultions, sketch the grph of the function. 6 = 6 = A Clculte the ect vlue of the turning point for the grph of the function = --. Clculte the -intercept of the grph of the function = --. c Sketch the grph of the function = --, showing intercepts nd the turning point. A Sketch the grph of the function = ----, showing intercepts nd the turning point. B The grph of the function f () = is shown elow. f() = 6 6 Sketch the grph of the function = f( ) B 6 Sketch the grph of the function = 7 + 6, showing the turning point nd intercepts. Hence, on the sme set of es, sketch the grph of the function = Sketch the grph of ech of the following. C ( + ) ( ) = ( + 6) = 6 6 c 7( ) + ( + ) = 8 d D ( ) = 9 ( + ) e = f ( + ) 9 = g ( ) + ( ) = h = i (, ) = j = 6

63 Chpter Coordinte geometr 6 8 Determine the Crtesin eqution from the following prmetric equtions. = + cos θ, = + sin θ, θ [, π] = + cos t, = sin t, t R c = + sec t, = + tn t, t R 9 Epress ech of the following rtionl functions in prtil frction form. 9 + = = c = Sketch the grph of ech rtionl function in question 9. C, D E F Anlsis A drinking trough with semicirculr ends is to e mde from pressed metl. The volume of the trough is to e litres. If r is the rdius (in cm) of the semicirculr end, show tht the surfce re of the trough is Sr ( ) = πr, r >. r c Sketch the grph of surfce re [S(r)] versus r. (Use tle of vlues.) Ignoring n etr metl required to mke the joins, find the minimum surfce re of the trough nd the corresponding vlue of r. An open o is to e mde from roll of steel m wide (see the digrm t right). Write down the epression for the length nd width of the o. If the o is to enclose volume of 8 m, write rule linking nd. c Write n epression for the totl surfce re of the o nd then use sustitution to mke this function of onl. d Show tht the minimum surfce re occurs when the o is. m high nd then find the minimum surfce re tht will enclose the given volume. (Hint: Epress the frctionl prt of the surfce re s prtil frctions efore differentiting.) e Clculte the length tht needs to e cut off the roll of steel to mke the o. A fmil goes to the ech, nd one of the os tkes lilo nd goes pddling. He is m from the shore, mesured t right ngles to the shore, nd m from where the rest of the fmil re ling on the ech. His fther clls him to come ck to shore, nd he needs to get ck in the quickest possile time. He pddles t m/s nd runs long the ech t m/s. A digrm of his trip is shown t right. Find n epression for the distnce pddled through the wter s function of. Using the eqution distnce trvelled speed = , time tken c m m Fmil find the time he tkes to pddle to the shore in terms of. Write n epression for the distnce trvelled long the ech in terms of nd lso the time tken trvelling long the ech. Bo

64 6 Mths Quest Specilist Mthemtics d Write n epression for the totl time he tkes to get ck to his fmil. Use this epression to find the vlue of tht gives minimum vlue for the time of trvel. e Wht is the minimum time of trvel? Sketch the grphs of = + nd = -- on the sme set of es. Hence, sketch the grph of = on the sme set of es s in. Find the coordintes of the sttionr points, the - nd -intercepts nd the equtions of n smptotes. + c On seprte set of es sketch the grph of = d Eplin lgericll how the grph in prt c is otined from the grph in prt. Hence, descrie the trnsformtion required. ( ) e On the sme set of es s the grph in prt c, sketch the grph of = (Use grphics clcultor to ssist.) + test ourself CHAPTER