SIMPLE NONLINEAR GRAPHS

S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u

Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle Liner euse their grphs re stright lines. If the power is higher thn, like = + or = then the equtions re not liner or nonliner. This ooks will show how to rw these tpes of grphs. Answer these questions, efore working through the hpter. I use to think: Whih of these grphs is liner: = or =? Wh? How is qurti eqution ifferent to liner eqution? Whih metho n lws e use to rw n grph for n eqution? Answer these questions, fter working through the hpter. But now I think: Whih of these grphs is liner: = or =? Wh? How is qurti eqution ifferent to liner eqution? Whih metho n lws e use to rw n grph for n eqution? Wht o I know now tht I in t know efore? 00% Simple Nonliner Grphs Mthletis 00%

Bsis Wht is Liner Grph? A liner eqution is n eqution where the highest powers of oth n re. A liner grph is se on liner funtion like = + or =- +. In eh se, the powers of n re oth. Using Tle to Plot Grph There re mn methos tht n e use to rw the grph of n funtion (liner or nonliner) ut metho tht never fils, for n funtion (liner or otherwise) uses tle to plot points ^, h on oorinte plne. Complete this tle of vlues for = + n use it to plot the grph on the gri elow - - 0-0 6 Now plot the points ^, h: ^-, -h, ^-, 0h, ^0, h, ^, h, ^6, h n join the ots. 7 6 5-5 - - - - 0-5 - - An of ourse the stright line goes through eh point. The tle metho works for n funtion, not onl liner funtions s ove. This hpter will show how the tle metho n e use to rw: A Prol. This is the grph of n eqution where the power of is. For emple =. A Hperol. This is the grph of n eqution in the form = k where k is n numer. For emple =. An Eponentil. This is the grph of n eqution whih hs s the ine of numer. For emple =. 00% Simple Nonliner Grphs Mthletis 00%

Questions Bsis. Ientif whih of these reltionships is liner. = + = + =- + + = 6 e = f =. Complete the tles for these equtions. = + = - - 0 - - 0 =- =-- - - 0 - - 0. Use the tle in the previous question to plot the line =-- on the oorinte plne elow. 5-5 - - - - 0 5 - - - - -5 00% Simple Nonliner Grphs Mthletis 00%

Knowing More Wht if the Highest Power is? In the eqution =, (where n e n numer) the highest power of is. The eqution is not liner, ut is lle qurti. The grph of qurti eqution oes not form stright line, ut shpe lle prol. Using Tle to Plot Prol Rememer, the tle metho n e use for ANY reltionship etween n. Here s n emple with =, so =. Complete this tle of vlues for = n use it to plot the grph on the gri elow - - - - 0 6 9 0 9 6 Now plot the points ^, h: ^-, 6h, ^-, 9h, ^-, h, ^-, h, ^00, h, (,,) (,, ) ( 9,, ) ( 6, ) Join the ots with smooth ontinuous urve, not stright lines 8 7 6 5 0 9 8 7 6 5-0 -9-8 -7-6 -5 - - - - 0-5 6 7 8 9 0 - This shpe is lle 'Prol'. 00% Simple Nonliner Grphs Mthletis 00%

Knowing More The Prol Fes Downwrs if is Negtive Let's s is negtive, then the prol will e 'upsie own'. Here s n emple with =-. Complete this tle of vlues for =- n use it to plot the grph on the gri elow - - -8 - - 0-0 - - -8 0-5 - - - - - 5 - - - -5-6 -7-8 -9 The onvit of prol is the iretion it fes. It oul e fing up or fing own. If is positive ( 0) onve up If is negtive ( 0) onve own Sometimes prols whih re onve upwrs re lle 'hpp prols' n prols whih re onve ownwrs re lle 's prols'. 00% Simple Nonliner Grphs Mthletis 00% 5

Knowing More Prols of the Form = + Wht out prol for the funtions like = + or = -? Complete the tles for these funtions = - = = + - - 0 6 0-0 6 - - 0 8 0 8 - - 0 0 0 Use the tles to grph eh funtion on the sme set of es = - = = + 0 9 8 7 6 5-5 - - - - 0 5 - - The prol for = + is the sme s =, just shifte units upwrs. The prol for = - is the sme s =, just shifte units ownwrs. The rule for these prols is: = + is shifte units upwrs of =. = - is shifte units ownwrs of =. 6 00% Simple Nonliner Grphs Mthletis 00%

Questions Knowing More. Ientif whether these reltionships re liner or qurti. =- + = = - = +. Complete the tle n rw the grph of =-. - - 0 0-5 - - - - 5 - - - - -5-6 -7-8 -9-0 - - - Is this prol onve upwrs or onve ownwrs? 00% Simple Nonliner Grphs Mthletis 00% 7

Questions Knowing More. Answer these questions se on the funtion =. Complete this tle se on the funtion. - - - 0 Use the spe elow to rw our own set of es n the grph of this funtion. Is this prol onve upwrs or onve ownwrs?. Ientif if the prol for these funtions is onve upwrs or onve ownwrs (without rwing the grphs). =-7 = 8 00% Simple Nonliner Grphs Mthletis 00%

Questions Knowing More 5. = is qurti funtion. Complete this tle for this funtion. - - 0 Complete the tle for the funtion = +. - - 0 Complete the tle for the funtion = -. - - 0 Use the tles plot ll grphs on this sstem of es. 5-6 -5 - - - - 0 5 6 - - - e Wht o ou notie out the grphs of = + n = - in funtion to =? 00% Simple Nonliner Grphs Mthletis 00% 9

Questions Knowing More 6. The grph on the right represents =. On the sme set of es, rw the grph of = + without using tle. On the sme set of es, rw the grph of = - without using tle. Are these grphs onve upwrs or onve ownwrs? Does ing or sutrting onstnt hnge the onvit? - - - 5 0 9 8 7 6 5 - - - 7. The grph on the right represents =- +. On the sme set of es, rw the grph of =- without using tle. On the sme set of es, rw the grph of =- - 6 without using tle. Are these grphs onve upwrs or onve ownwrs? Does ing or sutrting onstnt hnge the onvit? - - - - - - - - -5-6 -7-8 -9-0 - - - - 0 00% Simple Nonliner Grphs Mthletis 00%

Using Our Knowlege The Turning Point The turning point of the prol is the oorintes of the point where the prol 'turns roun'. Conve upwrs Conve ownwrs Turning Point Turning Point Another nme for the turning point is the 'verte'. Here re some emples. Fin the turning points of the following = - =- + 5 Turning Point 0 - - - 0 - - - - - - - - - - - -5 Turning Point -5-6 -7-8 The turning point or verte of this prol is ^0, -h The turning point or verte of this prol is ^0, h The verte of onve upwrs prol is t the lowest point n so it is lle minimum turning point. The verte of onve ownwrs prol is t the highest point n so it is lle mimum turning point. This mens tht: If 0 (positive), then = + will hve minimum turning point. If 0 (negtive), then = + will hve mimum turning point. 00% Simple Nonliner Grphs Mthletis 00%

Using Our Knowlege The Ais of Smmetr The 'is of smmetr' of prol is the vertil line whih psses through the turning point. Ais of Smmetr Mimum turning point Minimum turning point Ais of Smmetr Sine the is of smmetr is vertil line, it will hve the eqution = -vlue of turning point Fin the is of smmetr of eh of these prols =- + = ( -) - - 0-0 0 - - 0 0 - - - 0 - - - - - - 0 - The turning point is ^0, h The turning point is ^0, h -vlue -vlue Ais of smmetr is = 0 Ais of smmetr is = Some importnt points out these emples In the is of smmetr is the -is or = 0. In the is of smmetr is the vertil line =. hs mimum turning point, n mimum -vlue of. hs minumum turning point, n minumum -vlue of 0. 00% Simple Nonliner Grphs Mthletis 00%

Questions Using Our Knowlege. Wh o some prols hve mimum turning points n others hve minimum turning points?. Use tle to rw this prol: = - on the es provie.(hint: = - is the sme s =- + ) - - 0 6 Wht is the verte of this prol? 5 Is the verte mimum or minimum? Wht is the is of smmetr? - - - -. Without rwing nthing, stte whether eh of the following will hve minimum or mimum turning point. = - =- + 7 = 5+ 6 = 5-5 00% Simple Nonliner Grphs Mthletis 00%

Questions Using Our Knowlege. Mth these equtions to the prols elow. = =- = + = - 0 9 8 7 6 5 - - - 0-9 8 7 6 5 - - 0 - - - - 0 - - - - -5-6 - - - 0 - - - - -5-6 -7-8 -9 Ientif whih eqution hs mimum turning point t ^00, h. Ientif whih eqution hs minimum turning point t ^00, h. Wht is the is of smmetr of ll these funtions? e Ientif whih eqution hs mimum -vlue of. f Ientif whih eqution hs minimum -vlue of. 00% Simple Nonliner Grphs Mthletis 00%

Thinking More There re two more importnt nonliner grphs lle the 'hperol' n 'eponentil'. The tle metho is use for oth. Using Tle to Plot the Hperol A hperol is the grph of n eqution = k where k n e n numer like = or = -. Rememer, the tle metho n e use for ANY funtion etween n. Here's n emple: Complete this tle of vlues for = n use it to plot the grph on the gri elow - - - - 0 - - - - UD - - - - 0 - - - - Four Importnt Things to Notie Oviousl n never e 0 euse nthing ivie 0 is unefine. You n see tht in the tle. The n is re smptotes. This is fn w of sing tht the grph gets loser n loser towrs the n is, ut never touhes them. The hperol is split into two seprte rnhes in opposite qurnts ( st n r ). It is not ontinuous. The rnhes re the sme size n smmetril out the line =-. 00% Simple Nonliner Grphs Mthletis 00% 5

Thinking More Wht if k is Negtive? Let's s k is negtive ^k 0h. The Hperol will e in the other qurnts. Here's n emple with k =-. Complete this tle of vlues for = - n use it to plot the grph on the gri elow - - - - 0 UD - - - - - - - - 0 - - - - As ou n see, when k 0 the rnhes of the hperol re in the n n th qurnts When k is negtive the sme properties hol s when k is positive. The onl ifferene is the qurnts the rnhes re in. Oviousl STILL n never e 0 euse nthing ivie 0 is unefine. The n is re still smptotes. The hperol never touhes them. The hperol is split into two seprte rnhes in opposite qurnts ( n n th ). It is not ontinuous. The rnhes re the sme size n smmetril out the line =. 6 00% Simple Nonliner Grphs Mthletis 00%

Thinking More Using Tle to Plot n Eponentil In n eponentil funtion the is in the eponent, =. is n positive numer. Some emples re = or = -. As usul, here is n emple whih uses the tle metho to plot n eponentil grph. Complete this tle of vlues for = n use it to plot the grph on the gri elow - - - 0 8 8 9 8 7 6 5 -interept - - - - 0 - Importnt things to notie For n vlue of in =, the -interept is ALWAYS t ^0, h. The -is is n smptote. The grph gets loser ut oes not touh it. The -is is not n smptote, euse the grph psses through it. The -vlue inreses s the -vlue inreses. The -vlue ereses s the -vlue ereses. 00% Simple Nonliner Grphs Mthletis 00% 7

Questions Thinking More. = n e written s =. Use the tle n es elow to rw the grph of =. -6 - - - - 6 0 9 8 7 6 5 - - -0-9 -8-7 -6-5 - - - - 0-5 6 7 8 9 0 - - - -5-6 -7-8 -9-0 - - Will the either of the rnhes ever touh the or is? Whih qurnts re the rnhes of the hperol in? 8 00% Simple Nonliner Grphs Mthletis 00%

Questions Thinking More Use the tle n es elow to rw the grph of =-. -6 - - - - 6 0 9 8 7 6 5 - - -0-9 -8-7 -6-5 - - - - 0-5 6 7 8 9 0 - - - -5-6 -7-8 -9-0 - - e Wht is ifferent in the ove hperol from the hperol on the previous pge? f Highlight the smptotes on eh of the ove grphs. 00% Simple Nonliner Grphs Mthletis 00% 9

Questions Thinking More. The eponentil eqution is n eqution of the form = where is n positive numer. Wht is the vlue of 5 0? Wht is the vlue of 0 where is n positive numer? Wht is the -interept of n eponentil eqution =? Wh?. Fin the missing vlues on the -is, if the grph elow is of the funtion =. - - - - 0 0 00% Simple Nonliner Grphs Mthletis 00%

Questions Thinking More. The tle elow hs een omplete for = n the grph plotte unerneth. - - - 0 8 8 9 8 7 6 5 -interept - - - - 0 - Complete the tle elow for = -. - - - 0 Plot the grph of = - on the sme set of es ove. In the grph of =, inreses s inreses. Is this the sme in = -? Highlight the smptotes for oth grphs. 00% Simple Nonliner Grphs Mthletis 00%

Answers Bsis: Knowing More:. Liner Nonliner. Liner Qurti Liner Liner Qurti Qurti e Liner f Nonliner. - - 0. = + - - 0 - - - - 0 5 = - - 0 - - 0 =- - - 0 0 - - =- - - - 0 Prol onve ownwr (s) - - - - -5. - - - 0..5 0.5 0 0.5.5 Prol onve upwrs (hpp) 00% Simple Nonliner Grphs Mthletis 00%

Answers Knowing More: Knowing More:. =- 7 onve ownwr (s) 6. n = onve upwrs (hpp) 5. - - 0 0 - - 0 7 7 - - 0 - - - All three grphs re onve upwrs (hpp) No. Aing or sutrting onstnt oes not hnge onvit 7. n e When is e the urve moves units up the -is. Similrl, when is sutrte the urve moves units own the -is. All three grphs re onve ownwrs (s) No. Aing or sutrting onstnt oes not hnge onvit 00% Simple Nonliner Grphs Mthletis 00%

Answers Using Our Knowlege:. Conve upwr prols (where 0). hve turning point tht is minimum. While onve ownwr prols (where 0) hve turning point tht is mimum. Using Our Knowlege: =. - - 0 0 0 =- = + (0, ) Mimum. Conve ownwrs (s) prols lws hve verte tht is mimum. = 0 = -. Minimum t (0, -) Mimum t (0, 7) Minimum t (0, 5 ) Mimum t (0, -5) =- = = 0 e = - f = + 00% Simple Nonliner Grphs Mthletis 00%

Answers Thinking More: Thinking More:.. -6 - - - - 6 - - - -6-6 -6 - - - - 6 6 - -6 - - - No. Both the n es re smptotes. The hperol hs rnh in eh of the st n r qurnts. e This hperol hs rnh in eh of the n n th qurnts (rther thn st n r) f The smptotes for oth of these hperols is the n es. This mens tht the grph will get ver lose to these smptotes ut will never touh these. 00% Simple Nonliner Grphs Mthletis 00% 5

Answers Thinking More:. The vlue of 5 0 = The vlue of 0 = The -interept is lws for n eponentil eqution of the form =. This is euse n numer rise to the power of zero will e one.. (0, ) ` = (, ) ` = (, 6) ` = 6 (, 6) ` = 6. - - - 0 8 8 No. As inreses, ereses. The eqution = - is shown in the ove grph using the she line. The smptote for oth grphs is the sme. It is the -is or the line = 0. In other wors, while oth the urves = n = - get ver lose to the -is, the never touh it. 6 00% Simple Nonliner Grphs Mthletis 00%

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