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if two lines are perpendicular, what must their gradients equal to when multiplied together?
-1
What is the equation to calculate the gradient of a line?
y₂-y₁/x₂-x₁
What is the equation to calculate the length of a line?
√(y₂−y₁)^2 + (x₂-x₁)^2
What is the equation to calculate the midpoint of a line?
(x₂+x₁) over 2 , (y₂+y₁) over 2
What is the equation of a circle?
x²+y²=r²
How do you find the midpoint of a circle from the equation?
(x-a)²+(y-b)²=r²
If a factor of a line equation is (x-2)², or the line equation is just y=(x-2)², how many solutions does (x-2)² give and what does this look like on a graph.
This gives just one solution, that x=2. This means that the line just touches the x axis at (2,0).
What does a x⁴ and a -x⁴ graph look like?
An x⁴ looks like a W A -x⁴ looks like an upside down W
What does an x³ and a -x³ graph look like?
An x³ graph appears to travel from bottom left to top right with two curves A -x³ graph appears to travel from bottom right to top left with also two curves
How do you find the min/max point of a quadratic?
Complete the square; the x in the bracket will be the x coordinate and the number out the bracket will be the y coordinate
How do you establish that (x-3) for example, is a factor of an equation?
Sub in 3 for example. If it is a factor, there will be no remainder and it will equal 0.
If no factors of an equation are given and you are asked to factorise, what should you do?
Try small integers using substitution
How do you quickly find a remainder of an equation given a factor?
Substitute in the factor, and this will equal the remainder
What change does the a in f(x) + a cause to a graph?
It causes the graph to move in the direction of the y axis
What change does the a in f(x+a) cause to a graph?
It causes the graph to move in the direction of the x axis
How do you change the factors of a graph that has been translated by f(x+a)?
If it was translated by moving to the right by 3, you would add 3 to each factor bracket. e.g (x+3) would change to (x+6)
What can pascals triangle be used for during binomial expansion?
Pascals triangle shows the coefficients of the expansion. It also shows a pattern of the a²,a³ b²,b³ etc.
What is the pattern of the a² etc and b² etc?
The a's decrease as you go across the triangle. The b's increase as you go across the triangle
How else can you find a coefficient of an expansion?
Using ⁿCr
What does the n and r stand for in nCr?
n is the number it is to the power of. r is the number of term which you want to find
What does r start at?
It starts from 0 and then progresses
What is nC0 always equal to no matter what n is?
1
For example, what does 5! mean?
5x4x3x2x1
How do you plot a quadratic inequality on a graph?
Factorise the quadratic to find x values. Plot a graph and determining on 0> or <0, solve.
How do you simplify a surd in a fraction?
Multiply the denominator by DOTS of the surd clever number 1
What is the rule of x² × x⁷?
Add 2 and 7 to get x⁹
What is x⁻⁸?
1/x⁸
How would you write x to the power of ¾?
(4th root of x)³
What are natural numbers?
Postitive integers including 0
What does Z mean?
All the positive and negative integers
What does Q mean?
Integers that can be written as fractions
What does R mean?
Rational and irrational numbers (all numbers)
What does the symbol → mean?
Leads to or implies
How else can you also write → as?
∴ (three dots)
What does the symbol ← mean?
Follows from or implied by
When can you write the ↔ symbol and what does it mean?
You can write this when both←→ are true. This means they are equivalent
What is the equation to calculate the area of a triangle?
A = ½abSinA
What is the cosine equation?
CosA = b²+c²−a² / 2bc
How do calculate the stationary points of a line?
dy/dx of the equation. Make this equal to 0 and find the values of x
How do you decide whether the stationary points are minimum points, maximum points or points of inflexion?
By differentiating the differential by d²y/dx² and subbing in the values of x of the stationary points into this.
When is the point a minimum point in terms of d²y/dx²?
When d²y/dx² equals a positive number
When is the point a maximum point in terms of d²y/dx²?
When d²y/dx² equals a negative number
When is the point a point of inflexion in terms of d²y/dx²?
When d²y/dx² equals 0
How do you tell if the point of inflexion is positive or negative?
By subbing in numbers to the dy/dx that are around the same value to see how the gradient changes around the stationary point
When is the gradient positive?
When dy/dx is greater than 0
When is the gradient negative?
When dy/dx is less than 0
How do you work out the points of which the equation has a negative gradient?
Work out dy/dx of the equation and put it to zero using < if you're trying to find out when it's negative in gradient
How do you work out the points of which the equation has a positive gradient?
Work out dy/dx of the equation and put it to zero using > if you're trying to find out when it's negative in gradient
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