Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs of lies. We will coclude the chapter with the study a few methods ad techiques which are useful for graphig curves of the form y x, polyomials ad y x. We will also study the equatio of a circle ad the itersectio betwee a curve ad a lie..0 Geeralities: coordiates its two coordiates. I a D Cartesia system of coordiates as show i Figure., the positio of a poit P( x, y ) is give by its two x ad y. For example, the positio of the poit P (3,) show i this figure is fully determied by Meaig of plottig, sketchig ad drawig: Plottig (a lie or a curve) meas markig the importat poits of a graph ad joiig them as accurately as you ca. You are expected to do this o graph paper ad to be prepared to read iformatio from the graph. Sketchig meas markig the poits i approximately the right positio ad the joiig them up i the correct geeral shape. You are ot expected to do this o graph paper or to read detailed iformatio from the graph. You are expected though to mark importat poits o the graph, such as the x ad y itercepts ad poits where the curve chages directio (local maxima or local miima). Drawig mea that you use a level of accuracy appropriate to the circumstaces, which could be aythig from a rough sketch to a very accurately plotted graph.
. The lie:..a The gradiet of a lie: I mathematics, by lie we uderstad a straight lie (as opposed to everyday Eglish s usage). Note : If we kow the coordiates of ay two poits of the lie, the we ca draw the lie ad fid its equatio. Note : The graph of a lie is characterized by the mai property that its gradiet (or slope) determied by ay two poits o the lie remais costat. Cosider the lie show i Figure.. Its gradiet is defied as: 7 4 3 m 6 4. The gradiet (slope) of a lie tells us how steep the lie is, or how quickly it icreases / decreases for a fixed chage i x. formula is () rise m (betwee ay two fixed poits o the lie). ru I geeral the slope of a lie is deoted by m ad a easy way to remember its A x y B x y is: I geeral, the slope of a lie determied by two fixed arbitrary poits, ad, () rise y y m ru x x. Whe the same scale is used for both axes, we ca see (from Figure.) that (3) represets the agle betwee the lie ad the x axis. m ta, where
Exercise : What are the slopes of the 4 lies show i Figure.3 below? Note 3: Two lies are parallel if ad oly if their slopes m ad m are equal: (3) m = m. Two lies are perpedicular if ad oly if : (4) m m. Memorize equatios (), (3) ad (4) above as we will use them ofte. Activity : Prove (4) by cosiderig m =ta ad m =ta ad the fact that 90 o (why?)... B. The distace betwee two poits ad the mid-poit of a lie: The distace betwee two poits: Figure.4: Usig Pythagoras theorem, calculate the distace betwee the poits A ad B. 3
I geeral, as i Figure.5 below, Figure.5: The distace betwee the poits A( x, y) ad Bx, y is: (5) AB y y x x The midpoit of a segmet AB : Look at the poits A, ad B 8,5 show below: Figure.6: The midpoit of the segmet AB is M (5,3). Note that 8 5 5 ad 3=. I geeral, for ay two poits (, ) ad, two poits is: (6) x x, y M y. A x y B x y, the midpoit of the segmet which coects these 4
Example. i the textbook: Cosider the poits A,5 ad B 6,3. Fid: a) The gradiet of the lie AB ; b) The legth of the segmet AB ; c) The midpoit of AB ; d) The gradiet of the lie perpedicular o AB. Example.: Usig two differet methods, show that the lies joiig P(,7), Q3, ad R 0,5 form a right triagle. Draw these poits ad the triagle formed as part of your solutio (ote: a mere drawig is ot a proof that the triagle is right ). From Exercise Set A, do exercises, 3, 4, 5 ad 0... C. The equatio of a lie: C. Drawig a lie, give its equatio: Example.3: Fid the equatio of a straight lie with gradiet which passes through the poit 0, 5. Note 4: The equatio of a lie is either (7) y m x, or, more geerally: (8) p x q y r 0. To draw the lie which has oe of these equatios, it is geerally eough to fid two distict poits o the lie, ad the to draw the lie which passes through these two poits. Some of the equatios i Figure.7 below are more particular tha (7) ad (8) ad the graphs of these lies are show accordigly: 5
Figure.8: Differet forms of the equatio of a lie (ote that e) is the most geeral ad the rest ca be see as particular cases of e) ) ad their graphs. Note 5: Note that a) ad b) above are equatios (ad graphs) for lies parallel to the axes. c) If the equatio is of the form (7) y m x, this lie has a y- itercept of (0,) ad a gradiet of m. To draw its graph draw oe poit o the lie (say (0,) ad oe more poit (say (,+m). e) If the equatio is of the form (8) p x q y r 0, (say x3 y6 0 ), you ca either rearrage it to the form (7) y m x (i this case y x ad graph as i c) or you ca directly fid the 3 coordiates of two poits o the lie. I these example ad i geeral, typical poits are the x ad the y itercepts: here for x=0: y=, so that we have a y itercept of (0,), ad if y=0, the x=3, so that we have a x itercept of (3,0). Example.4: Sketch the lies x=5, y=0 ad y=x o the same axes. Example.5: Draw y=x- ad 3x+4y=4 o the same axes. From Exercise set B do problems i), ii), Iii), v), xvii), xviii) ad xx) ad i), ii), x) ad xi). 6
C. Fidig the equatio of a lie: Cases:. Give its gradiet m ad the coordiates, x y of a poit i the lie:. Give two poits x, y ad, x y o the lie: Example: Do Ex.0 o page 53 ad Ex C: i) to v) o the same page. The, from past paper Jue 0: paper, do problem 4 ad from past paper: Jue 0: paper : do problem 9. C3. The itersectio of two lies: Two lies, with respect to itersectio, ca: Itersect at oe poit (they are secat to oe aother); Not itersect (they are parallel); Itersect at a ifiity umber of poits (they are idetical). The cases above ca be see either geometrically or algebraically. Example: Do Examples. ad. from pages 56, 57 of the book, The from Exercise set D: Exercise, 3, 7, ad 9 through 4... The circle: The circle is defied as the set of poits i plae which are at a fixed distace from a give fixed poit i plae (called the ceter of the circle). The equatio of the circle of radius r with ceter at O 0,0 is: (9) x y r The graph of the circle with ceter at O 0,0 ad radius 3 is show below. 7
Figure.9: The circle with ceter at (0,0) ad radius 3..3 Drawig curves:.3.a Drawig curves of the form: y x with,,3 ad 4 The curves y x with,,3 ad 4 look like i the graphs below: 8
.3.B Drawig polyomials P x a x a x a x a To draw polyomials of the form (0) A polyomial of order as the polyomial show i formula (0) has:... 0 it is importat to kow that: At most roots (which represet the x itercepts o the graph). The roots, ad therefore the x itercepts are foud typically by fully factorig the polyomial; At most - turig poits. A turig poit o the graph of a polyomial, also called a statioary poit, is a poit of local maximum or local miimum for the graph; Example:. Graph above; y x ad idetify the umber of roots ad the umber of turig poits to check the statemet. Graph y x x statemet above; ad idetify the umber of roots ad the umber of turig poits to check the 3 3. Graph y x x. To graph this polyomial, fid all its roots first ad the the behavior of y(x) at ad at ; 4. Graph 4 y x x. Note here that a double root for a polyomial will always be a turig poit. Behavior of a polyomial at ad at : We ca show that for a polyomial of the form (0): P ad P if a 0 ad P ad P if a 0 ad This is because for a polyomial of order, the domiat term for large values of x (ad therefore at ad at ) is ad at. Steps for graphig a polyomial: ax ad therefore the value P ad P is give oly by: ax at Fid all its x itercepts ad the y itercept; Get the behavior of the polyomial at ad at ; Be careful with the double roots which are turig poits; Graph the polyomial ad check the umber of the roots ad the umber of the turig poits. Example: Do example.3 ad from Exercise Set E: exercises,, 5 ad 7. 9
.3.C Drawig curves of the form y with ad : x The curves y with ad look as i the graphs below: x Memorize the shape of the curves show i.3 A ad.3 C..4 The itersectio of a lie ad a curve: A lie ca itersect a curve i: No poit, whe they do ot meet, as i the graph below: Oe poit: this icludes the case whe the lie is taget to the curve, but the itersectio poit does ot have to be a poit of tagecy: see the graph below as a example: 0
I more tha oe poit. To fid the poits of itersectio betwee a lie ad a curve, we solve the system of equatios which give the equatio of the lie ad of the curve algebraically. We ca also check the solutio graphically. Example.4 page 7: Fid the coordiates of the two poits where the lie y3x itersects the curve: Example.5 (page 7): y x : a) Fid the value of k for which the lie y x k is taget to the curve y x ; b) For the value of k foud i a), fid the poit of tagecy. From Exercise set F (page 73) do problems 3, 5, 8 ad 9 ad from the Practice Book do problems 6 ad 7 (pages 4-43). From paper (Jue 0) solve problem 5.