Straight Lines. Distance Formula. Gradients. positive direction. Equation of a Straight Line. Medians. hsn.uk.net

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1 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 (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

2 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

3 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( ) Page HSN44

4 Function Effect Effect on f ( ) Effect on sin f ( ) + a Shifts the graph a up the -ais g( ) + (, ) (,) (,) sin 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

5 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

6 The area of a triangle, ab sin C CST diagram s Eact values: Radians You should know how to convert between radians and degrees: Degrees Radians 8 8 Radians Degrees eg 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 ± ( ) S T C Solution set { 5., 44.7, 5., 4.7 } Page HSN44

7 . Solve sin, <. sin sin sin sin ( ) S T C { } Solutions set,,, Compound ngle Formulae cos( ± ) cos cos sin sin sin ( ± ) sin cos ± cos sin These are given on the form ula sheet Double ngle Formulae sin sin cos cos cos sin sin cos These are given on the form ula sheet Page HSN44

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