Help Your child with Higher Maths - Craigmount High School

doubt, check with your child. If still in doubt, please feel free to get in touch with usat the school.Thanks in advance for your help, and good luck!The Straight LineWhat is the gradient of a horizontalline?What is the equation of a horizontalline?So the equation of the x-axis is…?What is the gradient of a verticalline?What is the equation of a verticalline?So the equation of the y-axis is…?m 0y by 0m is undefinedx ax 0What is the gradient formula?m yx22 y1 x1How do you find the equation of astraight line?You need to know the gradient of the lineand a point on the line, then youuse y b m( x a)How do you find the gradient of astraight line if you know itsequation?How do you find the size of the anglebetween a line and the x-axis?Rearrange to the formUsem tany mx cWhat is the rule for parallel lines? m1 m2What is the rule for perpendicularlines?m1 m2 1How do you find where two linesmeet?Use simultaneous equations or usey yWhat does it mean if three points aresaid to be collinear?They lie in a straight lineHow do you show that three pointsA, B and C are collinear?Show that mAB mBCso the lines areparallel, but B is a common point so A, Band C are collinearCraigmount High School Mathematics Department

How do you find the stationary pointsof a function?How do you find where a function isincreasing or decreasing?How do you show that a function isALWAYS increasing (decreasing)?At the stationary points f ( x) 0Differentiate then solve to find the xvaluesSubstitute back in to find y valuesUse a nature table to determine the natureDifferentiate then use a nature table (orsolve the inequality)Differentiate then complete the square toshow that f ( x) 0 ( f ( x) 0) for allvalues of xHow do you find the solution to an Investigate stationary points (and endoptimisationproblem? points if necessary)What do you get if you differentiatedistance?What do you get if you differentiatespeed?SpeedAccelerationFamilies ofgraphsGiven the graph ofyou sketch…y f ( x) ky f ( x) ky f (x) , how doMove graph up k unitsMove graph down k unitsy kf (x)Stretch graph up/down by factor of ky f(x)Reflect graph in the x-axisGiven the graph ofyou sketch…y f (x k)y f (x k)y f (x) , how doMove graph k units to the leftMove graph k units to the righty y f (kx)f ( x)Compress the graph by a factor of khorizontallyReflect graph in the y-axisGiven the graph of y f (x), how doyou sketch…y f (x)(the derived graph)The x-coordinates of the stationary pointsbecome zeroes of the graph; then look atthe gradient of the curve between theseCraigmount High School Mathematics Department

Answer: ½ )cos 13212120tan 0131 3-(we saytan 90 isundefined)Radian measure:You need to be able to convertfrom degrees to radians and viceversa.Some common examples are givenopposite and should be memorised:practice asking these both ways.180 (pi radians)360 20 090230645460322What does sin x cos x alwaysequal?(sine squared x plus cos squared x)1sin xWhat does always equal? tan xcos xRecurrence relationsWhat is meant if a recurrence relationis said to be…Convergent?Divergent?What is the condition for a recurrencerelation to have a limit?How do you find the limit of arecurrence relation?There is a limitThere is no limit1 a 1 ( a is between -1 and 1)Use the formulabL 1 a(or replace un1and unby L in theoriginal expression and solve)Given three consecutive terms in a Set up two equations using pairs of valuesrecurrence relation, how can you work then solve simultaneouslyout the formula?Craigmount High School Mathematics Department

Quadratic FunctionsHow do you sketch a quadratic curve(parabola)?Completing the square:Why do we complete the square?What is the process for completing thesquare?What form must the expression be inbefore you can complete the square?The discriminant:What is the condition for…… equal roots?… two distinct real roots?… real roots?… non-real roots? (or no real roots)1. Find the shape – “happy” or“sad”?2. Find the roots (if they exist) – ie.where the curve cuts the x-axis(solve y 0 )3. Find where the curve cuts the y-axis ( x 0)4. Use symmetry to find the turningpoint (or use differentiation)To allow us to make a quick sketch of theparabola, which allows us to find theturning pointIdentify the x-coefficientHalve itSquare itAdd it on/take it awayOR: expand brackest and equatecoefficientsM ust be x 2 2 ... and not 2x etc, so takeout a common factor if you have tobbbb2222 4ac 0 4ac 0 4ac 0 4ac 0How do you show that a line is atangent to a curve?What does it mean to say that aquadratic is irreducible?Substitute the line into the curve andsolve the equation to show that there are2equal roots (or show that b 4ac 0 )It cannot be factorisedCraigmount High School Mathematics Department

PolynomialsHow do you show thatof f (x)?x a is a factor Use synthetic division (with a ) to showthat the remainder is zero, or show thatf ( a) 0How do you factorise a cubic?First find a linear factor, using syntheticdivision, then factorise the quadratic fromthe bottom row of the table.How do you sketch the graph of a 1. Find where the curve crosses the x-polynomial? axis ( y 0 ) and the y-axis ( x 0)dy2. Differentiate and solve 0 to finddxthe stationary points3. Use a nature table to determine nature4. Sketch the graphIntegrationHow do you integrate?How do you prepare for integration?Increase the power by one, then divide bythe new powerChange any roots into powersx must not be on the denominator(bottom) of any fractionAny pairs of brackets should be expandedWhen integrating an indefiniteintegral (one with no lim its), what +Cmust we always remember?Why do we integrate?What do we have to remember whenthe enclosed area is below the x-axis?To find the area under a curve,or to recover f (x)from f (x)The answer will be negative, so we explainthis fact and change the answer to positiveCraigmount High School Mathematics Department

What do we have to remember whenthe area is partly above and partlybelow the x-axis?How do we find the area between twocurves or a line and a curve?How do we find where the curvesmeet?What do we get if we integrateacceleration?What do we get if we integrate speed?We have to work out the areas separately(one above x-axis, one below) then add ( curve above curve below)dxUseSpeedDistancey y and solveCompound Angle Formulaecos(A + B) = ?cos(A – B) = ?sin(A + B) = ?sin(A – B) = ?When asked to find the exact value ofsin, cos or tan, what should you lookfor?If a right-angled triangle is notinvolved, what should you do?If you are give sin, cos or tan and toldthat the angle is acute( 0 x 90,0 x 2), how can youfind the other ratios as exact values?cosAcosB – sinAsinBcosAcosB + sinAsinBsinAcosB + cosAsinBsinAcosB – cosAsinBRight-angled trianglesTry to make an expression up whichinvolves right-angled triangles and exactvalues you know (eg 30, 45, 60 degrees)Draw a right-angled triangle, usePythagoras to find the missing side, thenuse SOHCAHTOAsin2A = ?cos2A = ? (three possible answers)2 sinAcos A2cos Asin22cos A 121 2sin A2AHow can you expand cos4A, sin4Aetc?How can you expand cos3A etc?When solving a trig equation, whattwo-step process shoud you follow?How do you recognise and solve a“straight-forward solve”?Write as (2A+2A) then expand using theformulaeWrite as (2A+A) then expand using theformulaeIs it a “straight-forward solve”?If not, then “double angle solve”sin, cos or tan appears once onlySolve to find acute angle, then use ASTCCraigmount High School Mathematics Department

How do you recognise and solve a“double-angle solve”?What should you always check at theend of a trig question?Look for a double angle and a single angle(eg 2A and A)Replace the double-angle formula with anappropriate single expression, then makeone side zero and factorise in order tosolveShould the answer be given in degrees orradians?Craigmount High School Mathematics Department

The CircleWhat kind of circle has equation…x2 2 2 y r ?( xr22 2 a) ( y b) ?2 2x y 2gx 2 fy c 0 ?How do you find the equation of acircle?Do you need to expand the brackets andtidy up your answer?How can you show that an equationdoes NOT represent a circle?How do you find where a line meets acircle?How do you show that a line is atangent to a circle?How do you show that a line does notmeet a circle at allHow do you find the equation of atangent to a circle?What is a common tangent?How do I show that two circles touchexternally?What is meant by congruent circles?What is meant by concentric circles?centre (0, 0), radius rcentre ( a,b), radius rcentre( g , f ) , radius g2 f2 cFind the centre and radius, then use22 2( x a) ( y b) rNo!!Try to find the radius – you should beleft with the square root of a negativenumber, which is impossible, or zeroRearrange the line into the form y orx (whichever is easier) then substitutethis into the circle and solveAs above – you should find equal roots,ie only one point of contact2(alternatively, show that b 4ac 0 )As above – this time show that there are2no real roots, ie b 4ac 0Find the gradient of the radiusUse m1 m2 1to find the gradient ofthe tangentThen use y b m( x a)A line which is a tangent to two circlesShow that the distance between the twocentres is equal to the sum of the tworadiiCircles that are the same sizeCircles with the same centreVectorsCraigmount High School Mathematics Department

What is the difference between avector and a scalar?What is meant by giving a vector incomponent form?How would you write this in i, j, kform?A vector has magnitude (size) anddirection, whereas a scalar only hasmagnitudeWriting the answer as a column vectora with brackets, eg b c ai b j ckHow do you find the magnitudea (length) of vector u = b? c ua2 b2 c2How do you find vector AB ?How do you show that two vectors areparallel?If point P divides AB in the ratio m:n,how do you find the coordinates of P?What are the two forms of the scalar(or dot) product?How do you find the angle betweentwo vectors?How do you show that two vectors areperpendicular?Useful rules:a a ?AB b aShow that one vector is a multiple of theotherUse the Section Formula:1p ( na mb)and then write outm nthe coordinates of P(or use ratios to create an equation andsolve)a b a b cos(for this version, remember that thevectors must NOT be “nose-to-tail”)a b a1b1 a2b2 a3b3Use the dot product and solve to find a b- or use formula cos a bShow that a b 0a2Craigmount High School Mathematics Department

a ( b c) ? a b a cCraigmount High School Mathematics Department

Further CalculusWhat do you get if you differentiate:sin x ?cos x ?What do you get if you integrate:sin x ?cos x ?What is the chain rule fordifferentiation?(How do you differentiate f ( g(x))?)What do you get if you differentiate:sin( ax b) ?cos( ax b) ?What do you get if you integrate:sin( ax b) ?cos( ax b) ?What you get if you integraten( ax b)?cos x sin x cos x csin x cf ( g(x)) g(x)(differentiate around the brackets, thenmultiply by the derivative of what isinside the brackets)a cos( ax b) asin( ax b)1 cos( ax b) ca1sin( ax b) can( ax b)a(n 1)1 cCraigmount High School Mathematics Department

The Wave FunctionHow do you express a cos x bsinx inthe form k cos( x ) or k sin( x )?How do you know which quadrant isin?Given the choice, which version of thewave function should you use?How do you find the maximum orminimum values of a wave functionExpand the bracketsEquate coefficientsSolve to find k (square and add to get2k )Solve to find (divide to get tan )Look at the signs for k cosandk sin- if both are positive then is acute,otherwise you need to do an ASTCdiagramIf it starts with cos, use k cos( x )If it starts with sin, use k sin( x )Use the version which keeps bothcoefficients positive, if possibleThink of the graph: when is cos (or sin)at a maximum or minimum, then adjustas necessaryHow do you solvea cos x bsinx c ? Put the left-hand side into a wavefunction form, then solve in the usualwayWhat if the question has 2xor 3xetc? You still solve the problems in the usualway with k and found as before – butat the end you will need to divide anyanswers to find xCraigmount High School Mathematics Department

Exponential and Logarithmic FunctionsxWhat points does the graph of y aalways pass through? (y equals a to thepower x)( 0,1) and ( 1, a)What points does the graph ofy logax always pass through? (y ( 1,0)and (a,1)equals the log of x, base a)How do you rewriteform?y logaxin poweryx aHow do you solve an equation where xxis the power? (eg 4 10 )How do you solve a log equation?Log rules:log x log y ?log x log y ?nlog x ?log a1 ?log a ?aHow do you get log to base e (thenatural log) on your calculator?How do you get log to base 10 on yourcalculator?If the graph of log y against log x is astraight line, how do you find y interms of x ?If the graph of log y against x is astraight line, how do you find y interms of x ?Take logs of both sides then use logrules to work out xExpress each side as a single log then“cancel” the logsOr, get logs to one side and numbers tothe other, then rewrite using power formlog xyxlogyn log x01ln buttonlog buttonny kxThe values of k and n can be found fromthe graphxy abThe values of a and b can be found fromthe graphCraigmount High School Mathematics Department

Revision from Credit/Intermediate 2What is the Sine Rule?asin Absin Bcsin CWhat is the Cosine Rule?a2 b2 c2 2bccos AWhat is the formula for the area of atriangle?How do you solve a quadraticequation?How do you factorise a quadratic?Area12absinCMake one side zero, then factorise the other(or use the quadratic formula)Look for:1. Common factor2. Difference of squares3. Double bracketsWhat is the quadratic formula?x b b2 4ac2aHow do you find the solution to asimple trig equation, with solutionsfrom 0 to 360 degrees?Find the acute angle then use an ASTCdiagram to find the solutions (usually two)How do you writex?x as a power of12x (x to the power a half)How do you write n xm as a power ofx?(the nth root of x to the power m)mnx0What is x ?1How do you writepower?nx with a positive1nxCraigmount High School Mathematics Department