Pure Flashcards
∆
∆ > 0
2 real distinct roots
∆ = 0
1 repeated root
∆ < 0
no real roots
the natural numbers
the integers
the rationals
the real numbers
A is a subset of B
A ⊂ B
0 < x ≤ 10 in interval notation
(0,10]
The set of integers between 1 and 100 inclusive
{1,2,3,…,100}
Is a member of
∈
The number of members in set A is 5
n(A) = 5
Universal set
ξ
Complement of A
A’
Union of A and B
A ∪ B
Empty set
Ø
A is not a subset of B
A ∉ B
x is a real number and is less than 10
{x ∣ x∈N, x<10}
Modulus function
∣x∣
one-one
if every y value corresponds to only one x value
many-one
if there is at least one y value that comes from more than one x value
domain
the set of allowed input values to a function
range
the set of all possible outputs of a function
one-many
if there is at least one x value that gives more than one y value
a mapping is a function if…
every input value maps to a single output value
how can you test whether a mapping is a function?
use the vertical line test
mapping
takes numbers from a given set and assigns each of them one or more output values
image
the output of a given input
y = f(x) + c
translation c units up
y = f(x+d)
Translation d units to the left
y = af(x)
Vertical stretch by scale factor a
y = f(ax)
Horizatal stretch by 1/a
y = -f(x)
reflection in the x-axis
y = f(-x)
reflection in the y-axis
order of operation for vertical transformations (e.g. y = af(x) +c)
normal order of operations
order of operations for horizontal transformations (e.g. y = f(ax + c))
reverse of normal order of operations
order of operations for one horizontal and one vertical transformations (e.g. y = f(ax) + b)
doesn’t matter
equation of a straight line
y = mx + c
gradient of normal
(a) (a,b)
(b) c
If the hypotenuse of a triangle within a circle is the diameter…
It is a right angled triangle
the radius of a circle at a given point on its circumference is…
perpendicular to the tangent to the circle at that point
increasing sequence
a sequence where each term is larger than the previous one
decreasing sequence
a sequence where each term is smaller than the previous one
periodic sequence
one where the terms start repeating after a while
finite sequence
has a finite number of terms
infinite sequence
continues forever (infinite number of terms)
term-to-term rule
e.g.
position-to-term rule
formula for the nth term
e.g.
converge
tend to a certain value as n increases
diverge
increase/decrease without limit
series
the sum of a sequence up to a certain point (S(n))
Write S(n) in sigma notation
arithmetic sequences
sequences in which there is a common difference between each term (e.g. +4)
the nth term of an arithmetic sequence =
sum of the first n terms for an arithmetic sequence (a & d) =
geometric sequence
a sequence in which there is a common ration between each term (e.g. x2)
the nth term of a geometric sequence =
sequence notation - what does a mean?
the first term
sequence notation - what does d mean?
the common difference
sum of the first n terms for a geometric sequence =
sequence notation - what does r mean?
common ratio
sum to infinity of a geometric series
under what conditions does the sum to infinity condition apply?
|r| < 1 (series converges)
if |r| > 1 …
the geometric series diverges
sequence notation - what does L mean?
the last term in a sequence
sum of the first n terms for an arithmetic sequence (a & L) =
0 (for any base a>0)
1 (for any base a>0)
Direction of a vector
The angle the vector makes with the horizontal
Displacement vector
The vector representing the translation from one point to another
Direct proportion
A relationship between two variables in which their quotient is constant
Congruent expressions
Expressions connected by an identity symbol (they are equal for all values of the variable)
Coefficient
A constant in front of a variable
Collinear
Points that lie along the same straight line
Chord
The line segment between two points on a curve
Exponential decay
A relationship of the form y=e-x
Exponential equation
An equation with the ‘unknown’ variable in the power
Exponential growth
A relationship of the form y=ex, where a>1
Factorial
The product of integers from 1 to n, denoted by n!
Identity
A relation which is true for all values of the unknown
Interval notation
A form of notation to represent all numbers in a range - uses () if point is not included and [] if it is
Inverse proportion
A relationship between two variables in which their product is constant
Lead coefficient
The coefficient of the highest degree term in a polynomial
a < x < b
‘P implies Q’ or ‘If P is true then Q is true’
‘P is equivalent to Q’ or ‘Q is true is and only if P is true’
What is the degree of a polynomial?
The highest power of the unknown (x) occuring in the function
For a quadratic function how many…
(a) x-intercepts
(b) turning points
(a) 0-2
(b) 1
For a cubic function how many…
(a) x-intercepts
(b) turning points
(a) 1-3
(b) 0 or 2
For a quartic function how many…
(a) x-intercepts
(b) turning points
(a) 0-4
(b) 1 or 3
Direct proportion
Ratio of the two quantities is constant
What is the relationship between y and x2 if they are directly proportional?
What is the relationship between y and x2 if they are inversely proportional?
What are the 3 key circle angle rules?
1) The angle in a semi circle is a right angle
2) A tangeant to the circle is perprendicular to the radius at the point of contact
3) The radios perpendicular to the chord bisects the chord
What is the distance between the points (x1, y1) and (x2, y2)?
How can you tell whether two circles intersect, are disjoint ot tangeant to each other?
Compare the difference of the radii of the circles to the distancce betweeen their centres
What is a=bc in log form?
c=logba
Taking a logorithm of which numbers produce non-real numbers?
A negative number or zero
What is log10x written as?
log x
What is logex written as?
ln x
loga(ax)
x
x
What is true of all graphs of the form y = ax ?
1) The y-intercept is (0,1)
2) The graph of the function lies entirely above the x-axis
3) The x-axis is an asymptote
What is the gradient of ex ?
ex
What is the gradiant of e<em>kx</em> ?
k ekx
Graph of y = ln(x)
What is true of the graph y = ln(x) ?
1) Passes through the point (0,1)
2) The y-axis is a vertical asymptote
What is true of the function of the form y = Aekt ?
1) The initial value (when t=0) is A
2) The rate of change is ky
If y = kbx then…
log y = log k + x log b
For equation log y = log k + x log b, the graph of log y against x is a straight line with what
(a) gradient?
(b) y-intercept?
(a) log b
(b) log k
2 key angle identities
1) tan x = sin x / cos x
2) sin2 x + cos2 x = 1
Sine rule
Cosine rule
What do you have to look out for when using the sine rule?
There may be two possible answers (A and 180-A)
Area of a triangle
What is true if vectors a and b are parallel?
b = ta
Unit vector
A vector with magnitude 1
Position vector
Gives the vector from the origin to that point (its coordinates)
How can vectors be used to prove geometric shapes?
1) If a shape is a parallelogram then the vectors corresponding to the opposite sides are equal
2) If a shape is a rhombus then the vectors corresponding to all four sides have equal magnitudes
The midpoint of a line joining points with position vectors a and b had position vector…
1/2 (a + b)
How can the compisite function applying g to x and then f be written?
How is the inverse of a function written?
How do you write the derivative of a function?
f’(x)
How does the graph of y=f-1(x) relate to y=f(x)?
y=f-1(x) is a reflection of the graph y=f-(x) in the line y=x
What can you say about the range domain of f-1(x) and f(x)?
1) The domaine of f-1(x) is the same as the range of f(x)
2) The range of f-1(x) is the same as the domain of f(x)
What order are the transformations done for the function y = p f(x) + c ?
Stretch is performed before the translation
What order are the transformations done for the function y = f(qx + d) ?
The translation is performed before the stretch
Rational function
A fraction where both the denominator and numerator are polynomials
What is the relationship between degrees and radians?
180° = π radians
Converting from degrees to radians
Divide by 180º and multiply by π
Converting from radians to degrees
Divide by π and multiply by 180º
For the function y = a sin (bx) and y = a cos (bx) what is the
(a) period
(b) amplitude
in radians?
(a) 2π/|b|
(b) a
For the function y = a sin (b(x+c)) + d and y = a cos (b(x+c)) what is the
(a) period
(b) amplitude
(c) central value
(d) minimum value
(e) maximum value
in radians?
(a) 2π/|b|
(b) |a|
(c) d
(d) d - |a|
(e) d + |a|
Arc length in radians
Arc length in degrees
Area of a sector in degrees
Area of a sector in radians
Segment on a diagram
Sector on a diagram
Double angle identity for sin
Double angle identity for cos
Double angle identity for tan
Compound angle identity for sin
Compound angle identity for cos
Compound angle identity for tan
Harmonic form
sec x
1/cos x
cosec x
1/sin x
cot x
1/tan x
Identities for reciprocal trigonometric functions
Differentiation of sinx, cosx and tanx
Differentiation of ln x
Differentiation of trigonometric functions
What is the product rule?
Differentiate akx
Differentiate cos(kx)
Differentiate ekx
Differentiate sin(kx)
Differentiate tan(kx)
Implicit differentiation
Which two special cases of integration by substitution should be remembered?
Integration by parts
