Distance Formula Straight Lines Distance ( ) + ( ) between points (, ) and (, ) Gradients m between the points (, ) and (, ) where Positive gradients, negative gradients, zero gradients, undefined gradients eg 4 eg Lines with the sam e gradient are parallel eg The line parallel to + 5 has gradient m since + 5 + 5 5 + (m ust be in the form m + c ) Perpendicular lines have gradients such that m mperp. eg if m then mperp. m tan positive direction Equation of a Straight Line is the angle that the line m akes with the positive direction of the -ais The line passing through ( a, b ) with gradient m has equation: b m( a) Medians M C M is the m idpoint of C, ie M + +, M is not usuall perpendicular to C, so m m cannot be used To work out the gradient of M, use the gradient form ula Page HSN44
ltitudes D is not usuall the m idpoint of C D C D is perpendicular to C, so m m can be used to work out the gradient of D Perpendicular isectors C C D D CD passes through m idpoint of C CD is perpendicular to, so m m can be used to find the gradient of CD Perpendicular bisectors do not necessaril have to appear within a triangle the can occur with straight lines Composite Functions Eample Functions and Graphs If f ( ) and g( ), find a form ula for ( ) ( ) (a) h( ) f g( ) (b) k ( ) g f ( ) and state a suitable dom ain for each. (a) ( ) h( ) f g( ) f ( ) ( ) Dom ain: { :, } (b) ( ) k ( ) g f ( ) ( ) g { ± } Dom ain: :, You will probabl onl be asked for a dom ain if the function involved a fraction or an even root. Rem em ber that in a fraction the denom inator cannot be zero and an num ber being square rooted cannot be negative eg f ( ) + could have dom ain: { :, } Page HSN44
Graphs of Inverses To draw the graph of an inverse function, reflect the graph of the function in the line g( ) g ( ) Eponential and Logarithmic Functions a, a > Eponential a, < a < Logarithmic log a Trigonometric Functions (, a) sin cos tan (, a) ( a,) 8 6 Period 6 m plitude Graph Transformations 8 6 Period 6 m plitude Period 8 m plitude is undefined The net page shows the effect of transform ations on the two graphs shown below. (, ) g( ) 8 6 8 6 Page HSN44
Function Effect Effect on f ( ) Effect on sin f ( ) + a Shifts the graph a up the -ais g( ) + (, ) (,) (,) sin + 8 6 f ( + a) Shifts the graph a along the -ais g( + ) (, ) sin ( 9) 8 6 f ( ) Reflects the graph in the -ais g( ) (, ) sin 8 6 f ( ) Reflects the graph in the -ais (, ) g( ) sin( ) 6 8 kf ( ) Scales the graph verticall (, 4) g( ) sin Stretches if k > Com presses if k < 8 6 f ( k ) Scales the graph horizontall Com presses if k > Stretches if k < (, ) g( ) sin 8 6 Page 4 HSN44
The rea under a Curve If F( ) is the integral of ( ) f ( ) b f, then ( ) ( ) ( ) a f d F b F a a Rem em ber that areas split b the -ais m ust be calculated separatel and an negative signs ignored; these just show that the area is under the ais. The rea between two Curves b The area between the graphs of f ( ) and g( ) is defined as ( ) ( ) a f g d g( ) b a b f ( ) If the lim its are not given, f ( ) and g( ) should be equated to find a and b ackground Knowledge Trigonometr You should know how to use all of the inform ation below: SH CH T sin tan cos sin + cos The sine rule: a b c sin sin sin C The cosine rule: a b c bc cos + or b + c a cos bc Page 9 HSN44
The area of a triangle, ab sin C CST diagram s Eact values: 45 45 6 Radians You should know how to convert between radians and degrees: 6 9 45 4 8 6 6 Degrees Radians 8 8 Radians Degrees eg 5 5 8 6 5 6 Trigonometric Equations Look at the restrictions on the dom ain, eg < 6, or < e aware of whether the answer is required in degrees or radians Rem em ber a CST diagram whenever ou are asked to solve Eamples. Solve sin where < 6. sin ( sin ) ( sin ) sin ± sin ± ( ) 5. 8 5. 44.7 S T C Solution set { 5., 44.7, 5., 4.7 } 8 + 5. 5. 6 5. 4.7 Page HSN44
. Solve sin, <. sin sin sin sin ( ) S T C 5 6 + 9 95 5 5 8 6 8 5 75 6 +8 5 55 75 75 8 5 6 5 5 7 { } Solutions set,,, Compound ngle Formulae cos( ± ) cos cos sin sin sin ( ± ) sin cos ± cos sin These are given on the form ula sheet 95 95 8 9 6 55 55 8 5 6 7 Double ngle Formulae sin sin cos cos cos sin sin cos These are given on the form ula sheet Page HSN44