PURE YEAR 1 Flashcards
what is a^m x a^n
a^(m+n)
what is (a^m)^n
a^mn
what is a^m / a^n
a^(m-n)
what is (ab)^n
a^n b^n
what is a^(1/m)
m root a
what is a^-m
1/a^m
what is a^(n/m)
m root a^n
what is a^0
1
simplify sqrt(ab)
sqrt(a) x sqrt(b)
simplify sqrt(a/b)
sqrt(a)/sqrt(b)
what is the domain
the set of possible inputs for a function
what is the range
the set of possible outputs of a function
what are the roots for a function
the values of x for which f(x)=0
how to find turning point of a quadratic graph
complete the square
f(x)= a(x+p)^2 +q
turning point (-p, q)
different cases of discriminants
b^2-4ac
if >0, 2 roots
if =0, 1 root
if <0, 0 roots
when is an inequality represented with a dotted line
if y>f(x) or y<f(x), the curve y=f(x) is not included in the region and is represented by a dotted line
when is an inequality represented with a solid line
if y≥f(x) or y≤f(x), the curve y=f(x) is included in the region and is represented by a solid line
where do the graphs y=k/x and y=k/x^2 have asymptotes
x=0
y=0
y=-f(x) transformation description
reflection of y=f(x) in x-axis
y=f(-x) transformation description
reflection of y=f(x) in y-axis
gradient formula
m= (y2-y1)/(x2-x1)
product of gradients of perpendicular lines
-1
equation of distance between 2 points
d= sqrt((x2-x1)^2 + (y2-y1)^2)
when are 2 quantities said to be in direct proportion
what is the graph of these quantities
when they increase at the same rate
a straight line through the origin
midpoint of a line segment equation
(x1+x2)/2, (y1+y2)/2
what is the perpendicular bisector of line segment AB
straight line perpendicular to AB that passes through midpoint of AB
equation of circle w centre 0,0
x^2 + y^2 = r^2
equation of circle w centre (a,b)
(x-a)^2 + (y-b)^2 = r^2
tangent to circle property
perpendicular to radius at point of intersection
perpendicular bisector of a chord property
goes through centre of circle
property of right angle in circle
if angle PRQ=90 then R lies on circle w diameter PQ
angle in semicircle is always a right angle
how to find centre of a circle given any 3 points
find the equations of the perpendicular bisectors of 2 different chords
find coordinates of intersection of the perpendicular bisectors
factor theorem of polynomials
if f(p)=0, then (x-p) is a factor of f(x)
3 methods of proof
deduction
exhaustion
counter-example
cosine rule w a^2 as subject
a^2= b^2 + c^2 - 2bc cosA
cosine rule w A as subject
A= cos^-1 ( (b^2 + c^2 - a^2)/2bc )
sine rule
sinA/ a = sinB/ b
(or reciprocal)
sin(x) is same as what
sin (180-x)
area of triangle
1/2 ab sinC
how often does y=sin(x) repeat
every 360 degrees
where does y=sin(x) cross x-axis
-180
0
180
360
how often does y=cos(x) repeat
every 360 degrees
where does y=cos(x) cross x-axis
-90
90
270
450
how often does y=tan(x) repeat
every 180 degrees
where does y=tan(x) cross x-axis
-180
0
180
360
where are asymptotes of y=tan(x)
-90
90
270
if vectors PQ=RS what is to say
equal in length
parallel
if vectors AB=-BA:
AB is equal in length, parallel and in opposite direction to BA
triangle law for vector addition
AB + BC = AC
is AB=a, BC=b, AC=c, a + b = c
what does adding vectors PQ and QP give
the zero vector 0
what can any vector parallel to vector a be written as
λa
λ is a non-zero scalar
how to multiply a column vector by a scalar
multiply each component by the scalar
λ(p q)= (λp λq)
how to add 2 column vectors
add the x- components tgt
add the y- components tgt
how is the magnitude of vector xi + yj given
sqrt(x^2 + y^2)
what is a unit vector in the direction of a
a / |a|
(a over magnitude of a)
equation for differentiating from first principles
f’(x)= lim h->0 (f(x+h)-f(x)) / h
derivative of x^n
n x^(n-1)
derivative of ax^n
an x^(n-1)
when is a function said to be increasing
if, on an interval, f’(x)>0
when is a function said to be decreasing
if, on an interval, f’(x)<0
local maximum?
f’(x)=0
f’‘(x)<0
local minimum?
f’(x)=0
f’‘(x)>0
derivative of e^x
e^x
derivative of e^kx
ke^kx
what is loga n = x equivalent to
a^x=n
loga x + loga y =
loga xy
loga x - loga y =
loga (x/y)
loga (x^k) =
k loga x
loga (1/x) =
-loga x
loga a =
1
log a 1=
0
what is the graph of y= lnx a reflection of
the graph y=e^x in the line y=x
e^lnx =
ln (e^x) = x
if y= a x^n , what is graph of log y against log x
straight line
gradient n
vertical intercept log a
if y= a b^x , what is the graph of log y against log x
straight line
gradient log b
vertical intercept log a
