The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the

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Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through

1 Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th origin ( 0,0) Domain: R; rang: R Th graphs of vn-powr functions, g y and 4 y hav a turning-point at ( 0,0) Thy show symmtry undr rflction in th y-ais th y-ais, i th lin 0 is calld th ais of symmtry Domain: R; rang: [ 0, ) 5 Th graphs of odd-powr functions, g y and y hav a stationary point of inflction at ( 0,0) Domain: R; rang: R Th graph of y (or y ) consists of two branchs, on in th first quadrant and th othr in th third quadrant Th ais of symmtry is th lin y Th function shows th following asymptotic bhaviours: As, y 0 ; as, y 0 y 0 is th horizontal asymptot of th function As 0, y ; as 0, y 0 is th vrtical asymptot Domain: R\{0}; rang: R\{0} Th graph of y (or y ) also has two branchs, on in th first quadrant and th othr in th scond quadrant Th lin 0 is th ais of symmtry Th function shows th following asymptotic bhaviours: As, y 0 ; as, y 0 y 0 is th horizontal asymptot of th function As 0, y ; as 0, y 0 is th vrtical asymptot Domain: R\{0}; rang: R Th function y can b writtn as y It is undfind for < 0 It has an nd point at ( 0,0) Domain: [ 0, ) ; 0, rang: [ ) Th function y can b prssd as y vrtical tangnt at ( 0,0) Domain: R; rang: R It has a Eponntial functions For a >, th graphs of sam y-intrcpt (,) y a, whr a R y a hav th sam shap and th 0 Asymptotic bhaviour: As, y 0, th sam horizontal asymptot y 0 for th functions a is always >0 Domain: R; rang: R Copyright itutcom 005 Functions and graphs

2 For 0 < a <, th graphs of y a hav th sam shap and th sam y-intrcpt ( 0,) Asymptotic bhaviour: As, th functions y 0, th sam horizontal asymptot y 0 for a is always >0 Domain: R; rang: R Logarithmic functions y log and y log0 Th log functions y log (dnotd as ln ( ) ) and y log0 (dnotd as log ( ) ) hav th sam shap and a Thy ar undfind for 0 Both common -intrcpt (,0) functions hav ngativ valu for 0 < < and positiv valu for > Asymptotic bhaviour: As 0, y, th sam vrtical asymptot 0 Domain: R ; rang: R Circular (trigonomtric) functions y sin, y cos Circular (trigonomtric) function y tan Th function y tan is also a priodic function It rpats itslf aftr a priodt π, i it has symmtry proprty undr a horizontal translation ofπ, tan ( a ± π ) tan a It also has symmtry proprty undr rflctions in both and y-as, tan ( a) tan a Th trm amplitud is not applicabl hr Th function is undfind at ± n π for n J (st of positiv intgrs) It shows asymptotic bhaviour as ± n π Th vrtical asymptots ar ± n π Domain: : ± n π, n J ; rang: R Both functions ar priodic functions Each rpats itslf aftr a priodt π, i ach has symmtry proprty undr a horizontal translation of π, g sin ( a ± π ) sin a, cos ( a ) cos a undr rflction in th y-ais, cos ( a) cos a ± π Othr symmtry proprtis: For cos, For sin, undr rflction in th y-ais and a horizontal translation of π, sin ( π a) sin a ; undr rflctions in both and y-as, ( a) sin a sin Th valu of ach function fluctuats btwn and inclusivly, th amplitud of ach is Domain: R;, rang: [ ] Copyright itutcom 005 Functions and graphs

3 Modulus function y y can b dfind as, y, < 0 0 or y It has symmtry proprty undr rflction in th y-ais, i y and y ar th sam, and th lin 0 is its ais of symmtry Its vrt is ( 0,0) Domain: R; rang: [ 0, ) Horizontal dilation of function f ( ) y f ( ) y f ( n), whr > 0 of f ( ) y : n For 0 < n <, th graph y is strtchd away from th y-ais to giv it a widr apparanc; for n >, it is comprssd towards th y- ais to giv it a narrowr apparanc In this transformation th dilation factor is n Eampl Compar y 0 5 and y with y Transformations Rflction of function f ( ) y f ( ) y f ( ) y in th -ais: Eampl Compar y log with y log Any of th abov functions can b transformd by on or a combination of th following function oprations Vrtical dilation of function y f ( ) : y f ( ) y Af ( ), whr [ 0, ) graph of f ( ) A For 0 A <, th y is comprssd towards th -ais to giv it a widr apparanc; for A >, it is strtchd away from th - ais to giv it a narrowr apparanc A is calld th dilation factor Eampl Compar th graphs of th transformd functions y and y with th graph of th original function y Rflction of function f ( ) y f ( ) y f ( ) Eampl Compar y y in th y-ais: with y Copyright itutcom 005 Functions and graphs

4 Vrtical translation of function f ( ) y f ( ) y f ( ) ± c, whr c > 0 y by c units Th and oprations corrspond to upward and downward translations rspctivly Eampl Compar y cos and y cos with y cos 5 Eampl Sktch ( 4) y 5 This function is a transformation of y It involvs a vrtical dilation by a factor of, and translations of 4 lft and down Th stationary point of inflction changs from ( 0,0) 4, to ( ) Horizontal translation of function f ( ) y f ( ) y f ( ± b), whr b > 0 y by b units Th and oprations corrspond to lft and right translations rspctivly π Eampl Compar y sin with y sin π Eampl Sktch y sin π 4 for 0 Eprss th function as y sin π 4 This circular function is a transformation of y sin It has π an amplitud of (not: not ) and a priod of T π Thy corrspond to a vrtical dilation by a factor of and a horizontal dilation by a factor of rspctivly Thr is a π rflction in th -ais followd by translations of right and 4 down Combination of transformations If a transformd function is th rsult of a combination of th abov transformations, it would b asir to rcognis th transformations involvd by prssing th function in th form y ± A f ± n ± b ± ( ( )) c Vrtical translation ( up) Horizontal translation ( lft) Horizontal dilation (factor /n) Rflction in th y-ais ( sign) Vrtical dilation (factor A) Rflction in th -ais ( sign) To sktch th transformd function from th original function, always carry out translations last Copyright itutcom 005 Functions and graphs 4

5 04 Eampl Sktch y ( ) This function is th transformation of y It involvs rflction in th -ais and a vrtical dilation by a factor of 04, and thn translations lft and up Th function has and y as its asymptots y 5 Linar function y Quadratic function ( ) π y 0 5 y ( 4 ) Cubic function 4 Quartic function Som polynomial functions can b changd to factorisd form Th linar factors giv th -intrcpts Som polynomial functions may not hav any linar factor(s); hnc not all polynomial functions hav -intrcpt(s) A quadratic function may hav 0, or distinct linar factors, hnc 0, or -intrcpts Eampl 4 Sktch y This function is th transformation of in th form y ± Af ( ± n( ± b) ) ± c y Firstly prss it y ( ) Th last stp is du to th symmtry proprty of y undr rflction in th y-ais Th transformation involvs th followings: Rflction in th -ais; vrtical dilation by factor, translations right and up, Th vrt is ( ) A cubic function may hav, or distinct linar factors, hnc, or -intrcpts A quartic function may hav 0,,, or 4 distinct linar factors, hnc 0,,, or 4 -intrcpts Polynomial functions A polynomial function P ( ) is a linar combination of powr n functions, whr n { 0,,,, } Eampls ar: Copyright itutcom 005 Functions and graphs 5

6 If th powr of a linar factor in a polynomial is vn, thn th corrsponding -intrcpt is a turning point If th powr of a linar factor in a polynomial is odd, thn th corrsponding -intrcpt is a stationary point of inflction Eampl Find th quation of th quartic function shown blow If y a( b) ( c) ( d ) 4 ( )( f ) 5, thn th - intrcpts at b and d ar turning points; and th - intrcpts at c and f ar stationary points of inflction Th factor is not linar and dos not corrspond to an -intrcpt If a is a positiv (ngativ) valu, th graph of a polynomial function hads upwards (downwards) in th positiv - dirction Eampl Sktch y ( ) ( ) 00 Th -intrcpt at is a turning point; at th - intrcpt is a stationary point of inflction Eampl Sktch y ( 4 )( ) Eprss th function as y ( ) 4 Th function 0 4 crosss th -ais at ; it touchs th -ais at At th function has an -intrcpt that is a stationary point of inflction; at th function crosss th -ais Hnc, y a ( ), whr a is th vrtical dilation factor to b dtrmind using furthr information, in this cas, 0,5 th y-intrcpt ( ) 5 a ( ), a, y ( ) 5 5 This quartic function can also b prssd as y ( ) ( ) ( )( ) 5, 5 i y ( )( ) 40 Th sign corrsponds to th obsrvation that th graph hadd south in th positiv -dirction Eampl 4 Find th quation of th cubic function shown in th graph blow Th cubic function has only on -intrcpt at 4, and only on linar factor Its factorisd form must b y a( 4 )( b c) Us th othr givn points to st up simultanous quations, thn solv for a, b and c 0, 4 ac 0 5 () ( ) ac (,) a ( 4 b c) (,5) 5 5a ( b c) 4 a ab ac () a ab ac () Copyright itutcom 005 Functions and graphs 6

7 Substitut q () in qs () and (), 4 a ab 05 (4) a ab 05 (5) a ab (6) Add qs (4) and (6), 6 a 5, a 0 5 (7) Sub q (7) in (5) to obtain b Sub q (7) in () to obtain c Hnc 05( 4)( ) y All quadratic polynomial functions can b changd to turning y A ± b ± by complting th squar Th point form ( ) c turning point is ( m b, ±c) Som cubic polynomial functions can b prssd in similar form y A( ± b) ± c ( m b, ±c) is th stationary point of inflction of th cubic function Eampl Factoris ( ) y ( ) and thn sktch This cubic function is th sum of two cubs: y ( ) [( ) ] ( ) [ ] [ ]( ) ( )( ) ( ) ( )( ) Thr is only on linar factor, only on -intrcpt at Not that th -intrcpt can b obtaind by ltting y 0 and solv for ( ) 0, ( ), ( ),, Th givn function is in stationary inflction point form Th, stationary point of inflction is ( ) Som quartic polynomial functions can also b prssd in 4 similar form y A( ± b) ± c ( m b, ±c) is th turning point of th quartic function Ths forms should b viwd as th transformations (discussd prviously) of th powr functions,, rspctivly and 4 Eampl Find th turning point and th -intrcpts of y 4 Sktch its graph Th function is in turning point form Th turning point is ( 0, 4) Factoris y 4 ( ) ( )( ) Th linar factor givs -intrcpt (,0); th linar factor givs -intrcpt (,0) Th y-intrcpt is obtaind by ltting 0 0, 4, ( ) Graphs of sum and diffrnc of functions Th sum (or diffrnc) of two functions f and g is dfind only for D f Dg, whr D f and D g ar th domains of f and g rspctivly Eampl Givn f ( ) y g( ) ( ) y and log, find th domain of f g ( ) f, 0, ( ) ( ) g log, > 0, <, { : } D f, { : < } D g Hnc, D f g D f Dg { : < }, i [,) If th graphs of y f ( ) and g( ) graph of y f ( ) g( ) y ar givn, thn th can b sktchd by th mthod of addition of ordinats (i by adding th y-coordinats of th two functions at svral suitabl valus in D D ) f g Copyright itutcom 005 Functions and graphs 7

8 Graph of product of functions Nw functions can b gnratd by addition (or subtraction) of functions as discussd in th prvious sction Nw functions calld products (or quotints) of functions can also b gnratd by multiplication (or division) of functions Th product (or quotint) of two functions u and v is dfind only for D u D and v 0 if v is th divisor v Eampl Us addition of ordinats to sktch y ( ) Sktch y and y on th sam as, thn add ( ) th y-coordinats of th two functions at svral suitabl - valus Not that y is undfind at, its ( ) domain is R \ {} If th graphs of y u( ) and v( ) u graph of y u( ) v( ) ( ) or ( ) v y ar givn, thn th y can b sktchd by multiplying (or dividing) th y-coordinat of on function by th y-coordinat of th othr at svral suitabl valus within D D u v Eampl y D u R and D v R, D D D R, th product of functions u ( ) and v( ) uv u v Eampl Sktch y sin by addition of ordinats Sktch y sin and y on th sam as, thn add th y-coordinats of th two functions at svral suitabl -valus Eampl y, th quotint of functions ( ) ( ) v D u R and D v R, D D D R uv u v u and Copyright itutcom 005 Functions and graphs 8

9 Eampl log y, u( ) log and v ( ) D u R and D v R \ {} 0, D uv Du Dv R Rplacing by sin in function g ( ) to obtain ( ) sin g o f g o f ( ) sin is dfind whn sin 0 i [ nπ, (n )π ], whr n 0, ±, ±, Hnc D { : nπ ( n ) π, n 0, ±, ±, } g o f AND R, Graphs of composit functions Givn two functions y f ( ) and y g( ) can b gnratd in th following ways: In y f ( ) th variabl is rplacd by g ( ) function is y f ( g( ) ) In y g( ) th variabl is rplacd by f ( ) function is y g( f ( ) ), nw functions ; th nw ; th nw Eampl Find th domain and rang, and sktch th graph y cos of ( ) y cos( ) is a composit function in th form y f o g( ) f ( g( ) ), whr f ( ) and g( ) cos( ) For y cos( ) to b dfind, cos ( ) R AND R, i R Hnc D R f o g Sinc cos( ), 0 cos( ) th composit function is R o [ 0,] Hnc th rang of f g Functions gnratd in th abov mannr ar calld composit functions Th two nw composit functions ar dnotd as f o g and g o f rspctivly, i f o g( ) f ( g( ) ) and g f ( ) g( f ( ) ) o Eampl Givn f ( ) sin and ( ) composit functions Rplacing by in function f ( ) sin ( ) sin( ) f o g g, gnrat two to obtain ( ) sin( ) f o g is dfind whn R AND 0, i 0 Hnc { : 0} D f o g Th graph of y cos( ) is also shown for comparison Th ngativ half is rflctd in th -ais Copyright itutcom 005 Functions and graphs 9

10 Eampl Find th domain and rang, and sktch th graph of y This is a composit function of th form y f o g( ) f ( g( ) ), whr f ( ) and g ( ) Th function is dfind whn 0 (i ± ) AND R Hnc D f o g R \ {, } and th function has vrtical asymptots and Th valu of th function cannot b zro, R o R \ {} 0 f g Graphs of invrs rlations A rlation is a st of points A nw st of points can b gnratd by intrchanging th and y-coordinats of ach point This nw st of points is calld th invrs of th original rlation Th quation of th invrs is obtaind by intrchanging and y in th original quation Th y-intrcpt of th original rlation bcoms th -intrcpt of th nw rlation; th -intrcpt of th original bcoms th y-intrcpt of th nw Th horizontal asymptot of th original rlation bcoms th vrtical asymptot of th nw rlation; th vrtical asymptot of th original bcoms th horizontal asymptot of th nw Th rang of th original rlation bcoms th domain of th nw rlation; th domain of th original bcoms th rang of th nw Graphically th invrs rlation and th original rlation ar rflctions of ach othr in th lin y Not that only qual scals for both as can display th rflction visually Eampl Eampl 4 Find th domain and rang, and sktch th graph of ( 4) y This is a composit function of th form y f o g( ) f ( g( ) ), whr f ( ) and g ( ) 4 It is dfind for all ral valus Hnc D f o g R Th lowst valu of th function is ( 4) 64 Hnc [ 64, ) R o f g Th function can b prssd as ( 4 ) ( ) ( ) y th -intrcpts at ± ar stationary points of inflction Eampl Copyright itutcom 005 Functions and graphs 0

11 Approimat solutions of quations by graphical mthod Eampl Us graphics calculator to solv , corrct to four dcimal placs Us graphics calculator to draw y nd calc zro to find all th -intrcpts: 0 755, 047, 708 Eampl Solv y and y simultanously, corrct to four dcimal placs Us graphics calculator to draw th two functions nd calc intrsct to find th coordinats of th intrsction: 076, y 75 Eampl Solv sin( ), corrct to dcimal placs It can b solvd in two ways using graphical mthod Do not forgt to st graphics calculator in radian mod and y nd calc intrsct to find th -coordinat(s) of th intrsction(s) First way: Draw y sin( ) Scond way: Rwrit th quation to obtain sin ( ) 0 Draw y sin ( ) nd calc zro to find th -intrcpt(s), 77 Eampl 4 Solv 5 sin( ( 05) ) 0 π, < <, corrct to dcimal placs St graphics calculator in radian mod St th window, min ma Draw 5 sin( ( 05) ) y π nd calc zro to find th - intrcpts: 0 955, 0 455, 045, 455 Not that th last two solutions can also b obtaind by adding to th first two solutions, Qth function has a priod of Copyright itutcom 005 Functions and graphs

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv