UCAS Exam Key Points - Pure Maths Flashcards
Transformation: f(x)+d
Vertical shift up
Transformation: f(x)-d
Vertical shift down
Transformation: f(x+d)
Horizontal shift left
Transformation: f(x-d)
Horizontal shift right
Transformation: -f(x)
Reflection in x axis
Transformation: f(-x)
Reflection in y axis
Transformation: af(x), a>1
Vertical stretch
Transformation: af(x) a<1
Vertical compression
Transformation: f(ax) a>1
Horizontal compression
Transformation: f(ax) a<1
Horizontal stretch
Cosine rule missing side
a^2= b^2+c^2-2bcCosA
Cosine rule missing angle
CosA= (b^2+c^2-a^2)/2bc
Sine rule missing side
a/SinA = b/SinB = c/SinC
Sine rule missing angle
SinA/a = SinB/b = SinC/c
Trig area of a triangle
1/2 abSinC
How many degrees in 1 radian
180/pi
How many radian in 1 degree
Pi/180
180° in radian
Pi
360° in radian
2pi
Arc length in radian
l = rΘ
Arc length in degrees
l = (Θ/360) x pi x d
Area of a sector in radian
A = 1/2 x r^2 x Θ
Área of a sector in degrees
A = (Θ/360) x pi x r^2
Small angle approximation of sine
SinΘ ~ Θ
Small angle approximation of tan
TanΘ ~ Θ
Small angle approximation of Cos
CosΘ ~ 1- (1/2 x Θ^2)
Perpendicular gradients
M1 x M2 = -1
Equation of a line formula
y-y1 = m(x-x1)
What is the discriminant
b^2 - 4ac
2 real roots
Discriminant > 0
1 real root
Discriminant = 0
No real roots
Discriminant < 0
First principles of differentiation
Lim. (f(x+h) - f(x))/h
h—>0
When is a function increasing
F’(x) ≥ 0
When is a function decreasing
F’(x) ≤ 0
When is a value a maximum
F’’(x) < 0
When is a value a minimum
F’’(x) > 0
Differentiate e^x
e^x
Differentiate e^kx
ke ^kx
Sin^2(x) + cos^2(x)
1
Sin(x)/cos(x)
Tan(x)
Midpoint of coordinates (a,b) and (c,d)
((a+c)/2, (b+d)/2)
Equation of a circle centre (a,b) radius r
(x-a)^2+(y-b)^2=r^2
Trapezium rule
1/2width(y0+2(y1+y2+y3+y….)+yn)
Inequalities on a number line: filled circle
≤ / ≥
Inequalities on a number line: non-filled circle
< / >
Inequalities on the Cartesian plane: solid line
≤ / ≥
Inequalities on the Cartesian plane: dotted line
< / >
Inequalities: set notation
{x: …inequality…}
Using U and n when needed
U (union)
OR
n (intersect)
AND
Inequalities interval notation: ()
< / >
Using U/n when needed
Inequalities interval notation: []
≤ / ≥
Using U/n when needed
Inequalities interval notation: when LB or UB not specified
Used +/- infinity
Inequalities interval notation: when LB or UB not specified
Used +/- infinity
Factor theorem
If f(p) = 0
Then (x-p) is a factor of f(x)
nCr formula
n!/r!(n-r)!
How many decimal places is a binomial approximation accurate to
Number of terms the expansion is completed to
Modelling with quadratics - max height
Use the term outside the brackets of completed square form
Modelling with quadratics - time at max height
Inside brackets of completed square = 0
Eg (t-1.5)^2
t-1.5=0
t=1.5
Turning point coordinates from completed square
a(x+p)^2+q
(-p,q)
