FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble) Asmptotes (Where pplicble) A quick reminder from Gr The Stright Line 3 Drw onto the set of es below, the functions f() = + nd g() = 3 For grph f: For grph g: -intercept, let = 0 i.e (0 ; ) -intercept, let = 0 i.e (0 ; ) -intercept, let = 0 -intercept, let = 0 i.e + = 0 3 3 = 0 Answer the following questions: 1. For which vlues of is f() > 0?. For which vlues of is f() = g()? 1
The Prbol = +b +c or ( p) q Drw onto the set of es given below, the grphs of f ( ) 4 g( ) ( 1) 4 Use the tble function on our clcultor. Use our grphs to nswer the following questions: 1. Wht is the is of smmetr of f? =. Wht is the is of smmetr of g? We cn s tht the is of smmetr is = p 3. The minimum vlue of both grphs is = We cn s tht the minimum vlue is = q. 4. The turning point of f is ( ; ), nd the turning point of g is ( ; ). For which vlues of is g() < 0?. Give the eqution of h if h() = - f() 6 Give the eqution of m if m() = g() + 7. Write down the turning point of the function m.
SUMMARY: to sketch prbol,we find: shpe: if > 0 ; if < 0 -intercept: let = 0 -intercepts: let = 0 nd solve qudrtic eqution,.o.s 1 or b or = p minimum vlue (when > 0) or mimum vlue (when < 0) : substitute the vlue of the.o.s into the originl eqution. turning point ( ; ) or (p ; q) (from the turning point form).o.s min/m vlue Conversion of = + b + c to the form = ( p) + q Emples: 1. = 4 1. = 4 = ( 4 + ) 4 1 = [ /] = ( ) 16 = [( + 1 ) 1 /] = [( 1) 3 1 /] Turning point is ( ; 16) = ( 1) 7 Turning point is (1 ; 7) 3. = 3 + 1 Method: = 3[ + 3 3 1 ] 1. Ensure co-efficient of = 1. = 3[( + If not, tke out number. 3 + ( 1 3 ) 1 ) 9 1 3 ]. Complete the squre. = 3[( + 1 3 ) 4 9 ] 3. Blnce b subtrcting. = 3( + 3 1 ) + 3 4 4. Find the perfect squre. Turning point is ( 3 1 ; 1 3 1 ). Remove squre brckets. 3
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Finding the eqution of prbol: 1. Given the -intercepts, nd one other point: (Use 1 ) e.g 3-3 0 1 = ( + 3)( - 1) Sub (0 ; 3) 3 = (0 +3)(0 1) 3 = (-3) = -1 = -1( + -3) 3. Given the turning point, nd one other point: (Use q p ) ( ) e.g (-;6) (-1;4) O 4 1 Subst (-;6) 4 6 4 (1) 6 4 1) ( 6 6 4 4 1
Find the eqution of ech of the following prbols: 1.1. 1.. (1;) -1 O 4 O - -4 1.3. 1.4. (4;) -1 O 6 O (;0) 1.. 1.6 (- ; 4) -1 O -4 O (1;-4) 6
Eercise : Grphicl Interprettions 1. Given tht f() = - + 8 +0, nd g() = 4 + 8, nd EFG is. 1.1 If the vlue t G is = 1, find the co-ordintes of E nd F, nd hence find the length of EF. 1. Show tht the length of EF cn be epressed s L = - +4 +1. 1.3 Clculte the mimum length of EF, nd give the vlue of t which it occurs. E F O G. A prbol f intersects the -is t B nd C nd the -is t E. The is of smmetr of 7 the prbol hs eqution = 3. The line through E nd C hs eqution g ( ). B O C g f E Show tht the co-ordintes of C re (7 ; )..1 Clculte the -coordinte of B.. Determine the eqution of f in the form p q.3 Write down the eqution of the grph of h, the reflection of f in the -is..4 Write down the mimum vlue of t() if t() = - f() + 1.. Solve for if f ( ) 0. 7
The Hperbol p q Drw onto the set of es given below, the grphs of: f ( ) g( ) h( ) 4 4 4 Use the tble function on our clcultors. Mke note of the horizontl nd verticl smptote in ech cse, nd link them to the eqution. NOTE the following: if > 0, then the grph lies in Qudrnts 1 nd 3 (ie top right nd bottom left) if < 0, then the grph lies in Qudrnts nd 4 (ie top left nd bottom right) q ffects the horizontl smptote ( the movement) p ffects the verticl smptote (the movement) ech hperbol hs lines of smmetr: = + c, nd = - + k, nd these lines psses through the point (p ; q) -intercept: let = 0, nd -intercept: let = 0 When sketching n hperbol: 1. Drw in the smptotes, lbel them clerl.. Find nd plot the nd -intercepts. 3. You m wish to plot nother point on ech section to complete the sketch. (Use the tble function on our clcultor; or use crefull selected vlues nd substitute them into the eqution to find the vlues.) 8
Emple: Consider the eqution of the function f ( ) 4 the -intercept: (let = 0) 0 4... 1 (0 ; ) the -intercept: (let = 0) 0 4 4 8...( ;0) the horizontl smptote is = the verticl smptote is = plot the bove onto the es, so ou cn see which etr points to plot. complete the sketch b choosing, s = - nd = -6, find the vlues, then plot the points. A chllenge: Find the eqution of the lines (es) of smmetr of the grph of f. 9
Eercise 3: Use the grids provided, nd sketch the grphs of the following equtions: 1 3.1 f ( ) 3. f ( ) 1
Finding the eqution of n hperbol: Fill in the vlues of p nd q from the smptotes on the sketch. Substitute the given point into this eqution to find the vlue. Emple: Find the eqution of f in the digrm below: f O = (-1;-6) Solution: 0 Subst (-1;-6) 6 1-8 = - i.e. = 8 8 f ( ) Eercise 4: Find the eqution of the grphs of the following functions: 4.1 (1;) O = 11
4. (1;1) (3;) O 4.3 =1 O = - (1½ ; -4) 4.4-6 O (-;-3) 1
13 The Eponentil function q b p Drw onto the set of es given below, the grphs of: h g f 1 ) ( ) ( ) ( Use the tble function on our clcultors Drw onto the set of es given below, the grphs of: e c b ) ( 3. ) ( ) ( Use the tble function on our clcultors
NOTE the following: = q ffects the horizontl smptote ( the movement) the p vlue ffects the movement of the grph to the left or right (the movement) if 0 < b < 1, the function decreses, nd if b > 1, the function increses. Also, the lrger the numericl vlue of b, the steeper the grph; the smller b is, the shllower the grph. the vlue lso ffects the shpe: the lrger the vlue of, the steeper the grph; the smller is, the shllower the grph. Also, if < 0, the grph is reflected bout the -is. -intercept: let = 0 -intercept: let = 0, nd solve the resulting eponentil eqution. (this m require the use of logrithms, which is Gr 1 work) When sketching the Eponentil function: 1. Drw in nd lbel the horizontl smptote.. Find nd plot the -intercept. 3. Use the tble function on our clcultor, or plot selected points, to complete n Emple: ccurte grph. Drw the grph of the eqution f ( ) 3 3 o horizontl smptote: = -3 o -intercept (let = 0): 3 0 3 o -intercept (let = 0): 0 3 3 = - i.e (0 ; -) 3 3 o choose =, so 3 3 6 i.e ( ; 6) = 1 i.e (1 ; 0) 14
Eercise Use the grids provided, nd sketch the grphs of the following equtions:.1 f ( ). 1 1 1
1.3 g ( ) 1.4 m ( ).3 1 1 16
Finding the eqution of n eponentil function. When considering b p q, ou will lws be given or 3 of the vlues of, b, p or q. Fill in the q vlue. (You re usull given this.) Substitute the given point into the eqution to find the missing unknown. Emple: f In the digrm, f ( ) b q. Find the eqution of f. O 1 = - Solution: (Note: = 1) f ( ) b Sub (1 ; 0) into bove: 0 b 1 b i.e f ( ) Eercise 6 Find the equtions of the grphs of the following functions: 6.1. q 3 O 6. b q (1;4) O = 17
6.3 b q (-1;0) O = - p 6.4 q (1;6) = O 18