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Higher Maths > Higher > Flashcards

Higher Flashcards

1
Q

How do you find the equation for a perpendicular bisector

A
  • find gradient of line from co ordinates
  • m1m2=-1 (perpendicular)
  • find midpoint
  • sub points into y=m(x-a)
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2
Q

How do you find the circumcentre

A

Find the equation for both lines and do simultaneous equations

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3
Q

How do you find the equation of an altitude of a line?

A

find the gradient of the line that the altitude is perpendicular to
m1m2=-1
sub the point and the gradient into y-b=m(x-a)

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4
Q

How do you find the equation of the median?

A
  • find the midpoint of the side it bisects
  • calculate the gradient of the median
  • substitute the gradient and either point on the line into y-b=m(x-a)
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5
Q

how do you find the vector journey PQ?

A

q-p

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6
Q

How do you find the magnitude of a vector AB

A

square root of a squared + b squared

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7
Q

how do you prove that a vector journey is collinear?

A

find vector journey A to B
then B to C
if AB=2BC(or anything like that) it means AB is parallel to BC
B is a common point to both AB and BC so A, B and C are collinear

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8
Q

if P is the position vector of the point p that divides AB into the ratio m:n then what is the formula to fine the point p??

A

p= (x1+x2)/m+n , (y1+y2)/m+n

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9
Q

how do you know if 2 vectors are perpendicular?

A

a.b=0

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10
Q

how do you find the angle between 2 vectors?

A

cosx=a.b/(magn a x magn b) = a1b1+a2b2+a3b3/(mag axmag b)

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11
Q

what are the steps for sketching a quadratic function?

A
  1. find the shape = the coefficient of x^2
  2. find y intercept
  3. find the roots of the equation (factorise and set=0)
  4. find midway between the roots
  5. find the coordinates of the TP (complete the square and use a and b)
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12
Q

how do you find the coordinates of the TP of a quadratic

A

complete the square

y=ax^2+bx+c can be written in the form y=a(x+p)^2+q. so the TP is (-p,q)

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13
Q

how can you solve a quadratic equation?

A 
quadratic equations may be solved by: 
-the graph
-factorising 
-completing the square
- using quadratic formula 
(find the x's)
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14
Q

how can solve a quadratic inequation?

A

by sketching the quadratic function

  • factorise equation
  • find roots by setting factorised equation to 0
  • solve for x’s
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15
Q

how do you find the number’ if there are roots

A

use the discriminant (b^2-4ac)
if b^2-4ac=0 then the roots are real and equal
if b^2-4ac<0 there are no real roots
if b^2-4ac>0 the roots are real and unequal

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16
Q

how can you find unknown coefficients in a quadratic equation?

A

use the discriminant and solve for the unknown coefficient (e.g p)

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17
Q

how can you determine if a line is a tangent to a curve?

A

use the discriminant
if the b^2-4ac=0 then there is only one point of intersection
-if b^2-4ac>0 there are 2 distinct points of intersection
-if b^2-4ac<0 the line does not intersect with the curve

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18
Q

how do you find the equation of a tangent

A
  1. set the 2 equation equal to eachother and bring both to one side so it =0
  2. use discriminant to find missing coefficent
  3. sub number into y=mx+c
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19
Q

how do you find the point of intersection between a curve and a tangent

A
  1. set both equations equal to eachother
  2. use discriminant to find out how many points of intersection
  3. complete the square/factorise and find coordinates
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20
Q

how do you find the degree of the polynomial?

A

its the number of the highest power in the equation

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21
Q

how do you know if something is a factor of a polynomial?

A

if the remainder is 0

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22
Q

how can you factorise a polynomial equation fully

A
  • use synthetic division to fine the factor(s) of the equation
  • use the new numbers you found from the synthetic division and put it into brackets along with your factor
  • factorise the new equation
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23
Q

how can you find a polynomials coefficient?

A
  • use synthetic division and solve through
  • since you know the factor you know the remainder must be 0
  • set your last number to 0 and solve for your coefficient (e.g q)
  • if you are given more 2 factors or have 2 unknown coefficient then solve through and use simultaneous equations
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24
Q

how can you find the roots of a polynomial equation?

A
  • synthetic division
  • factorise fully
  • set equal to 0
  • solve for x
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25
Q

how can you find an expression for f(x) in a polynomial equation

A
  • find k by making f(x) the y coordinate

- sub value for k into original equation and simplify

26
Q

how do sketch a curve using a polynomial equation?

A
  1. find the y intercept
  2. use synthetic division to find the x intercepts
  3. find the stationary points by differentiating the original equation (find x and sub into og equation to find y (the TPs))
  4. do nature table using factorised differentiated equation to find max and min TPs
  5. sketch
27
Q

how do you draw a trigonometric graph?

A
  1. start with a simple cos or sin graph
  2. put in the scale to match the constant,c.
  3. slide the graph vertically to match the constant a
28
Q

how do you convert from degrees to radians?

A

multiply by pi/180

29
Q

how do you convert from radians to degrees?

A

multiply by 180?

30
Q

how do you know if you need to differentiate in a question?

A

when the question says:

  • the rate off change of the function
  • the gradient of the tangent to the graph of function x
31
Q

how can you find the equations of the tangent in terms of differentation

A
  1. find the point of intersection
  2. differentiate to find the gradient
  3. sub all these into y-b=m(x-a)
32
Q

how can you find the stationary points.

A
  1. derive the equation and st equal to 0
  2. find the x values and sub into the ORIGINAL equation and solve for the y values
  3. you now have the TP(s)
  4. do a nature table to determine the nature of the SPs
33
Q

what are the steps for curve sketching in terms of differentiating

A
  1. Intercepts
    a) where does the curve cut the x axis when y=0
    b) where does the curve cut the y axis when x=0
  2. Stationary points
    a) where is the curve stationary (f’(x)=0
    b) what is the nature of the stationary points
  3. behaviour
    a) what happens for large +/- values of x
34
Q

what happens to a graph when y=f(x)+a and y=f(x)-a

A

all points move up by ‘a’ units if a is positive and down if a is negative

35
Q

what happens to a graph when y=f(x+a) and y=f(x-a)

A

all points are moved to the left by ‘a’ units if a is positive and to the right if a is negative

36
Q

what happens to a graph when y=-f(x) and y=f(-x)

A

for y=-f(x) the y coordinates become negative. the graph is reflected in the x axis
for y=f(-x) the x coordinates become negative. the graph is reflected in the y axis

37
Q

what happens to the graph y=ksinx

A

the graph is stretched vertically by a factor of k when k>1.

when k<1 the graph is compressed vertically by a factor of k

38
Q

what happens to the graph y=fsinkx

A

the graph is compressed horizontally by a factor of k when k<1
the graph is stretched horizontally by a factor of k when k>1

39
Q

what happens to a graph of an inverse function

A

the graph of an inverse function is found by reflecting the in the line y=x

40
Q

how do you find the inverse of a function algebraically

A
  1. rewrite the function, but this time replace the term ‘f(x)’ with ‘y’
  2. rearrange the formula using the algebriac manipulation to make x the subject (i.e obtain x by itself)
  3. interchange the letters x and y (swap x for y and y for x)
  4. replace y with f^-1(x)
41
Q

what is the sin addition fomula

A

sin(a+b)=sinacosb+cosasinb

42
Q

what is the cos addition formula

A

cos(a+b)=cosacosb-sinasinb

43
Q

what does sin2a equal

A

2sinacosa

44
Q

what does cos2a equal

A

cos^2a-sin^2a
cos^2a-1 (if cosa appears in eqation)
sin^2a-1 (if sina appears in the equation)

45
Q

what can cos^2a-1 be rearranged to be

A

1/2(1+cos2a)

46
Q

what can sin^2a-1 be rearranged to be

A

1/2(1+sin2a)

47
Q

if y=a^x then…

A

x=loga(y)

48
Q

what are the properties of the Log function?

A
  1. since every exponential function f(x)=a^x passes theough (0,1), then the corresponding log function must pass through the point (1,0)
  2. since every ‘basic’ exponential function tends towards 0 as x–>-infinity, then every basic log function will tend towards -infinity as x tends towards 0
49
Q

what is the formula for intergration?

A

ax^n dx =( ax^n+1)/(n+1) +c

50
Q

area between 2 curves

A

upper - lower

51
Q

formula for distance between 2 points

A

square root((x2-x1)+(y2-y1))

52
Q

what is the general equation of a circle?

A

x^2+y^2+2gx+2fy+c=0

53
Q

how can you find the circles centre and the radius

A

centre(-g,-f )

r= root(g^2+f^2-c)

54
Q

how can you prove that something is a tangent to a circle?

A

you can either solve the equation or use the discriminant

55
Q

whats the equation for finding a limit for a recurrence relation

A

L=b/(1-a)

56
Q

what happens when you differentiate sinx

A

it turns into cosx

57
Q

differentiate cosx

A

-sinx

58
Q

what is the formula for the chain rule if y=(ax+b)^n

A

= an(ax+b)^n-1

59
Q

what is the formula for the chain rule if y=(x+b)^n

A

=n(x+b)^n-1`

60
Q

integrate (ax+b)^n

A

=(ax+b)^n/a(n+1)

61
Q

integrate cos(ax+b)

A

=1/a sin(ax+b)+c

62
Q

integrate sin(ax+b)

A

1/a -cos(ax+b)+c

Higher Maths (1 decks)

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