Pure Flashcards
What is the domain
The domain of a function is the set of possible inputs
What is the range of a function
A set of possible outputs
What are roots of a functions
The values of x oh which f(x) = 0
What does
X E r(with an extra line)
Mean in functions
X - input x…
E- is a member of …
R- the set of real numbers…
What does lN mean
Natural numbers - aLl positive intergers
What does z(with a lien) mean
All intergers
What is an asymptote
A line which approaches the graph but never reaches
What is the order of polynomials
0 - constant
1 - linear
2 - quadratic
3 - cubic
4 - quartic
5 - Quintic
The number of turns is ……….. than the order
One less
What does x ——-> infinity shape
As x tends to infinity
If a>0
And the Order Is odd described the graph
It goes uphill from left to right
If a>0
And the Order Is even described the graph
The tale goes upward
Y = f(x) + a
Describe the transformation mathematically
Translation
( 0)
( a)
Y = f(x) -a
Describe the transformation mathematically
Translation
(0)
(-a)
Y = f(x+a)
Describe the transformation mathematically
Translation
(-a)
(0)
Y = f(x-a)
Describe the transformation mathematically
Translation
(a)
(0)
Y = -f(x)
Describe the transformation mathematically
Reflection in x axis
Y = f(-x)
Describe the transformation mathematically
Reflection in your axis
Y = af(x)
Describe the transformation mathematically
Stretch parallel to the y axis with a scale factor of a
Y = f(ax)
Describe the transformation mathematically
Stretch parallel to x a us with scale factor 1/a
Y = f(x) +a
Describe the transformation in words
Moves graph a. Units up
Y = f(x) -a
Describe the transformation in words
Moves graph a units down
Y= f (x+a)
Describe the transformation in words
Moves graph a units to the left
Y = f (x-a)
Describe the transformation in words
Moves graph a units to the right
Y = -f(x)
Describe the transformation in words
Flips graph over
Makes all positive y coordinates negative and vice versa
Y = f(-x)
Describe the transformation in words
Makes all positive x coordinates negative and vice versa
Y = af(x)
Describe the transformation in words
Makes graph taller and skinnier
All y values are multiplied by a with x values unchanged
Y = f(ax)
Describe the transformation in words
Squashes graph in if a is bigger than 1; stretches graph out if a is between 0 and 1. All x values are divided by a, with y values unchanged
What other way could you write the equation m= y-yi/ x- Xl,
Y-yi =m(x- xi)
The gradient of parrauel lines are?
Equal
If two lines are perpendicular the the gradient of one is the ——— of the other
Negative reciprocal
What is a vector
Magnitude and direction
What is magnitude of a vector
The length of the line
What do u get if you add 2 vectors
A resultant vector
Multiplying a vector by a non-zero scalar produced a
…………. vector
Parallel
How do you show 2 vectors are parallel
Show there scalar multiples of each other
What is a position vector
Position of a point in relation to the origin
What is the position vector of A
— >
OA
What is a unit vector
Vector with magnitude one
What are the standard unit vectors
i
j
i is rhe direction of the …. Axis (vectors)
x
j is rhe direction of the …. Axis (vectors)
Y
Xi + Yj =
As a column vector
(X)
(Y)
Vector xi + yj has a magnitude
And angle ……. With horizontal
Square root x2 +y2
Tan-1 (y/x)
How is magnitude written
|a|
How is a unit vector written
a = a/ |a|
What are the 3 equations of a line
Y-Y1 = m (x - x1)
Y =mx +c
ax +by +c = 0
What are similar about parallel lines
They have equal gradients
What are the gradients like in perpendicular lines
The negative reciprocal
How do you turn X^2 + y^2 +2fx + 2gy + c =0 into a familiar form of a circle
Complete the square by collecting like terms first
What is the equation of a circle
(X-a)2 + (y-b)2 = r2
Whatare the 3 circle theorems you need to know
1) the angle in a semi circle is a right angle
2) the perpendicular line form the centre to a chord bisects the chord
3) a radius and tangent to the same point will meet at right angles
How to find the equation of a tangent
It will be perpendicular to radius so find gradient of radius and find perpendicular gradient
Then substitute in the point on the tangent to complete
How do you find the centre of a circle with a triangle in
The triangle sides are chords so there perpendicular bisectros meet in the centre
So find perpendicular bisectors id 2 sides
Find rhe midpoint + gradient to find equation
Then find centre by simultaneous equation
What is disproof by counter example
While to prove a statement is true we need to prove every possible case. We only need to disprove one example to disprove a statement
(Known as a counter example)
What is proof by exhaustion
This is breaking down the state but into all possible smaller cases where we prove each individual case
( sometimes known as case analysis )
What is an identity
An equation that is true cal all the values of the variable
What is a conjecture
A mathematical statement that has yet to be proven
What is a theorem
A mathematical statement that has been proven
What should a proof show
All assumptions
A sequenced list of steps that logically follow and must cover all possible cases
+ make a concluding statement
What is proof by deduction
Start from known facts and reach the desired conclusions via deductive steps
What are the 3 types of proof
Proof by deduction
Proof by exhaustion
Disproof by counter example
What is rhe dividend
The thing your dividing
What is the degree
The highest power of x in the polynomial
What is the divisor
This is what your dividing by
What is a quotient
What you get when you divide by the divisor ( not inc the remainder
What is the remainder
What’s left over
……. Degrees = …….. radians
180
Pi
How do you turn radians to degrees
Divide by pi multiply by 180
How do you turn degrees to radians
Divide by 180
Multiply by pi
What is a unit circle
A circle with radius 1, centre on the origin
What can any point on a unit circle be defined as
( cos theater, sin theater )
Sector area
A = 1/2 r^2 theater
Or
Theater / 360 x Pi r ^2
Arc lenght equation
S = r theater
Or
2 pi r ^2 x theater/360
Sine rule equation
a. b. c
———. =. —— — =. — ——
SinA. SinB. SinC
Cosine equation
a2 = b2 + c2 - 2bc cos A
Rearranged cosine equation for angle
Cos A = b2 + C2 -a2
— - - ——-
2bc
What are the trig identities
Tan x = sinx / cos x
Sin^2x + cos2^x =1
When f(x) is decreases
Is f’(x) positive or negative
Negative
When f(x) is increasing
Is f’(x) positive or negative
Positive
When the gradient dy/dx of a graph is positive the value of y is ……
Increasing
What is a stationary point
Where the gradient is 0
What is the local maximum
The gradient is posotive to the left, zero at the point , and negative to the right
What is the local minimum
The gradient is negative to the left, zero at the point, and positive to the right
How can you distinguish between local max and local min
You can test values of the derivative either side of the stationary point, to see where gradient is positive or negative
What is the point of infection
Where the curve moves from convex to concave (or vide versa)
A point of infection which is a stationary point Is knwon as what
Saddle point
At a point of inflection, is the gradient postives or negative just before or after the stationary point
Just before and just after positive and positive
or
negative and negative
What is the gradient at a station point
0
If a function is increasing the gradient is …
Positive
If a function is decreasing the gradient is …
Negative
What does
d
—
dx
mean?
Derivative in respect to x
f( x) derives to
f’(x)
Y dervives to
dy
—
dx
A constant always differentiates to
0
What does differntatiating tell you
The gradients of a curve at any point
What is differentiate first principles
Lim = f(x+h)-f(x)
h—>0. (.—————)
h
How do you use differntiate first principles
Find, simplify, remove h from denominator
Find limit of the expression as h tends to 0, by h=0
How do you find the nature if a turning point
Differentiate again
f’’(x) > 0 it’s minimum
f’’(x) <0 its maximum
what is the second derivative of y
d2y
____
dx2
What is the second derivative of f(x)
f’’(x)
What do you get if you integrate dy/dx
y + c
How do you integrate
increase powers by one
Then divide constant by power
Add c
nCr = …..
n!
———
r!(n-r)!
Does the first term in a bingo expansion descend or ascend
Descend
The second term ascends
What is an increasing sequence
Each term is greater then the one before
What is a decreasing sequence
Each term is less then the other one
What is an arithmetic sequence
The differnce between one therm and the next is always the same
What is a geometric sequence
The ratio of one term to the next is always the same
What is a periodic sequence
Repeats itself at regular intervals. The number of terms before the sequence repeats is called the period
What is a series
The sun of the terms of a sequence
What 2 ways can sequences been defined
Inductively or deductively
What is a deductive definition
Gives a direct formula for the Kth term of the sequence in terms of K. The terms of the sequence can be found by substitution the numbers 1,2,3,4 for K
What is an inductive definitions
Tells you have to find a term in a sequence from the previous term - the definition must also inculde the value of the first term of the sequence, you can find the second term form the first term and the third term form the second term …
What is the formula for ak, the kth term of the sequence
ak - a + (k-1)d
Ensure you can dervie arithmetic formula for Sn
.
What is an arithmetic series
The sum of the terms of an arithmetic sequence
What is the modulus of a number
.the non negative numerical value
