Higher Flashcards
How do you find the equation for a perpendicular bisector
- find gradient of line from co ordinates
- m1m2=-1 (perpendicular)
- find midpoint
- sub points into y=m(x-a)
How do you find the circumcentre
Find the equation for both lines and do simultaneous equations
How do you find the equation of an altitude of a line?
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)
How do you find the equation of the median?
- 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)
how do you find the vector journey PQ?
q-p
How do you find the magnitude of a vector AB
square root of a squared + b squared
how do you prove that a vector journey is collinear?
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
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??
p= (x1+x2)/m+n , (y1+y2)/m+n
how do you know if 2 vectors are perpendicular?
a.b=0
how do you find the angle between 2 vectors?
cosx=a.b/(magn a x magn b) = a1b1+a2b2+a3b3/(mag axmag b)
what are the steps for sketching a quadratic function?
- find the shape = the coefficient of x^2
- find y intercept
- find the roots of the equation (factorise and set=0)
- find midway between the roots
- find the coordinates of the TP (complete the square and use a and b)
how do you find the coordinates of the TP of a quadratic
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)
how can you solve a quadratic equation?
quadratic equations may be solved by: -the graph -factorising -completing the square - using quadratic formula (find the x's)
how can solve a quadratic inequation?
by sketching the quadratic function
- factorise equation
- find roots by setting factorised equation to 0
- solve for x’s
how do you find the number’ if there are roots
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
how can you find unknown coefficients in a quadratic equation?
use the discriminant and solve for the unknown coefficient (e.g p)
how can you determine if a line is a tangent to a curve?
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
how do you find the equation of a tangent
- set the 2 equation equal to eachother and bring both to one side so it =0
- use discriminant to find missing coefficent
- sub number into y=mx+c
how do you find the point of intersection between a curve and a tangent
- set both equations equal to eachother
- use discriminant to find out how many points of intersection
- complete the square/factorise and find coordinates
how do you find the degree of the polynomial?
its the number of the highest power in the equation
how do you know if something is a factor of a polynomial?
if the remainder is 0
how can you factorise a polynomial equation fully
- 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
how can you find a polynomials coefficient?
- 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
how can you find the roots of a polynomial equation?
- synthetic division
- factorise fully
- set equal to 0
- solve for x
how can you find an expression for f(x) in a polynomial equation
- find k by making f(x) the y coordinate
- sub value for k into original equation and simplify
how do sketch a curve using a polynomial equation?
- find the y intercept
- use synthetic division to find the x intercepts
- find the stationary points by differentiating the original equation (find x and sub into og equation to find y (the TPs))
- do nature table using factorised differentiated equation to find max and min TPs
- sketch
how do you draw a trigonometric graph?
- start with a simple cos or sin graph
- put in the scale to match the constant,c.
- slide the graph vertically to match the constant a
how do you convert from degrees to radians?
multiply by pi/180
how do you convert from radians to degrees?
multiply by 180?
how do you know if you need to differentiate in a question?
when the question says:
- the rate off change of the function
- the gradient of the tangent to the graph of function x
how can you find the equations of the tangent in terms of differentation
- find the point of intersection
- differentiate to find the gradient
- sub all these into y-b=m(x-a)
how can you find the stationary points.
- derive the equation and st equal to 0
- find the x values and sub into the ORIGINAL equation and solve for the y values
- you now have the TP(s)
- do a nature table to determine the nature of the SPs
what are the steps for curve sketching in terms of differentiating
- 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 - Stationary points
a) where is the curve stationary (f’(x)=0
b) what is the nature of the stationary points - behaviour
a) what happens for large +/- values of x
what happens to a graph when y=f(x)+a and y=f(x)-a
all points move up by ‘a’ units if a is positive and down if a is negative
what happens to a graph when y=f(x+a) and y=f(x-a)
all points are moved to the left by ‘a’ units if a is positive and to the right if a is negative
what happens to a graph when y=-f(x) and y=f(-x)
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
what happens to the graph y=ksinx
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
what happens to the graph y=fsinkx
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
what happens to a graph of an inverse function
the graph of an inverse function is found by reflecting the in the line y=x
how do you find the inverse of a function algebraically
- rewrite the function, but this time replace the term ‘f(x)’ with ‘y’
- rearrange the formula using the algebriac manipulation to make x the subject (i.e obtain x by itself)
- interchange the letters x and y (swap x for y and y for x)
- replace y with f^-1(x)
what is the sin addition fomula
sin(a+b)=sinacosb+cosasinb
what is the cos addition formula
cos(a+b)=cosacosb-sinasinb
what does sin2a equal
2sinacosa
what does cos2a equal
cos^2a-sin^2a
cos^2a-1 (if cosa appears in eqation)
sin^2a-1 (if sina appears in the equation)
what can cos^2a-1 be rearranged to be
1/2(1+cos2a)
what can sin^2a-1 be rearranged to be
1/2(1+sin2a)
if y=a^x then…
x=loga(y)
what are the properties of the Log function?
- since every exponential function f(x)=a^x passes theough (0,1), then the corresponding log function must pass through the point (1,0)
- 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
what is the formula for intergration?
ax^n dx =( ax^n+1)/(n+1) +c
area between 2 curves
upper - lower
formula for distance between 2 points
square root((x2-x1)+(y2-y1))
what is the general equation of a circle?
x^2+y^2+2gx+2fy+c=0
how can you find the circles centre and the radius
centre(-g,-f )
r= root(g^2+f^2-c)
how can you prove that something is a tangent to a circle?
you can either solve the equation or use the discriminant
whats the equation for finding a limit for a recurrence relation
L=b/(1-a)
what happens when you differentiate sinx
it turns into cosx
differentiate cosx
-sinx
what is the formula for the chain rule if y=(ax+b)^n
= an(ax+b)^n-1
what is the formula for the chain rule if y=(x+b)^n
=n(x+b)^n-1`
integrate (ax+b)^n
=(ax+b)^n/a(n+1)
integrate cos(ax+b)
=1/a sin(ax+b)+c
integrate sin(ax+b)
1/a -cos(ax+b)+c
