Midpoint of a line and distance between two points (length of line segment)

the midpoint is found by finding the mean of x and y. add the x coordinates together and divide by 2 to find the new x coordinate and same for y.

To find equation for perpendicular line using x and y points and gradient.

Find the negative reciprocal and sub into formula where y1 and x1 are the x and y coordinates:

y-y1 = gradient (x-x1)

When doing it to find the tangent line equation where centre of circle is zero and given a point. Find gradient from 0 to the point, fill in point as x and y.

Different types of curves based off equations

Exponential curve rules

y = xⁿ

When n is an even integer the curve is more than zero and when odd its less.

if n is bigger then it will go above the curve where n is smaller.

For a recipricoral function how to sketch a graph and find the asymptotes. reciprocal rules.

standard form equation is y = a/x + q

when a is negative the graph will be top left and bottom right and vice versa.

The graph will intercept x if q is not equal to 0. To find the x intercept make y = 0 and find for x in the above equation.

One asymptote is the y axis and the other is equal to y = q

for k/xⁿ form when k is positive constant, if n is odd then curves will be diagonal across each other, when n is even then they will be opposite.

Inverse proportion

Inverse proportion = The statement A is inversely proportional to is written:

A ∝ 1/B then to solve replace 1 with the constant.

Find the 2 points that the diameter goes between when give 3 points on a circle.

Option 1 ; find the lengths between each point and with pythagorus theorem, two lengths squared that = third length squared, the third is the diameter.

Option 2: find gradient of each line. Two lines gradients will multiply to -1, the diameter is the one that doesn't.

Equation of a circle.

equation at centre (0.0) = x² + y² = radius²

general equation anywhere = radius² = (x-a)² + (y-b)² where (a,b) is the centre of the circle and (x,y) is a point on the circumference.

To see if it passes through a point sub x and y of that point in and see if it adds up.

Sequences definitions and types. Periodic, arithmetic, geometric. Fibonacci numbers, triangle numbers, natural numbers.

A periodic sequence repeats itself regularly. The number of terms before the sequence repeats is called the period. An example of a periodic sequence is the sequence 5,7,−1,3,5,7,−1,3,5,7,... This has period 4.

An arithmetic sequence has the same difference between each term and the next term. (common difference) This difference can be positive or negative. An example of an arithmetic sequence is 8,5,2,−1,... This has a difference of −3

A geometric series has the same ratio between each term and the previous term.

A Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones.

Triangle numbers represent dots arranged in rows, where each row has one more dot than the previous one, forming a triangle.

Natural numbers are the counting numbers 1,2,3,4,5...

Sigma notation

Represents the sum of a sequence within a range. equation is to the right, At the bottom is the starting integer and the end is at the top, add up values in between.

Inductive vs deductive

Deductive is the direct formula to find a value of the sequence, the term to term role.

Inductive is the formula to get the next value in the sequence. a is the start number and put k as the value of the previous term to the one we are finding.

formula to find sum of sequences, infinity and value of term for geometric. plus logarithms

for arithmetic

S_n = n/2( 2a + (n + 1 ) d) or if we know the last term of the sequence L

S_n = n/2 ( a + L)

The S_n for natural numbers = n(n+1)/2

for geometric - ratio is r, first term a. kth term = ar^k-1

when r is over 1 = a(rⁿ - 1) / r-1

when r below 1 = a(1-rⁿ ) / 1-r

can find sum of infinity when r is a fraction -

a / 1-r

When solving for ⁿ use logarithm to bring it down.

say have 0.9ⁿ < 0.37 become n x log(0.9) > log (0.37) then solve with calculator. (this < sign was reversed as 0.9 is less than 1)

Binomial expansions

eg. (x+2)³ the x terms will be going down gradually. x³ x² x and 1. To decode the coefficients the second term in bracket in this case 2, which has the power of whatever is needed to make the two powers add up to 3 or whatever the exponent is. This figure is multiplied by the corresponding numbers from the row pascals triangle (as exponent is 3, will be third row).

Always starts and ends in 1. other numbers are the two numbers above added together.

To find the pascal triangle to fit in without drawing triangle for each use equation - n! / r! x (n/r)!

n is the power of the equation and r is power of second figure in bracket.

transformation equations from y=f(x) curve

For f(x+a), f(x-a), f(x)+a, and f(x)-a moves either y or x up or down. When the negative is outside the bracket it gets taken from y. and when it's inside it gets added to x, and vise versa. (normal for y outside, opposite for x always inside)

for y=f(ax) it stretches x by 1/a and for and y = af(x) it stretches y by a. so sub x/a or ay into new equation.

-f(x) is a refection on the x axis and f(-x) is a reflection across your axis.

To differentiate with fractions and roots also

multiply coefficient by the power snd the way 1 from the power. nx^n-1

for roots first rewrite with coefficient and if the root is the denominator just write it as a negative or vise versa if already negative.

how to find stationary points on a cubic graph and if max or min

at stationary point dy / dx = 0 so differentiate the equation to find its new version and sub y for zero to find the values for x. then sub x into original equation to find y. 42 second differential when finding max and min points on a cubic graph card image first differentiate once, then do it a second time to find the new equation. Into the new equation sub the stationary point value for x and if it is positive it is the minimum point.

approximate average curve gradient by adding h. to find the limit

f stands for the curve equation

f(x+h) - f(x) / h

to get add h to the point for x you are given and put it into the equation. and then take away just the point x in the equation. and then divide h on bottom.

the chain rule

for differentiating function in a function . differentiate the outside then multiply it by derivative of inside.

(x² + 1)³ becomes 3(x² + 1)² x by 2x

formula like dV/dt= dV/Dx x Dx/dt the Dx's cancel out to give dV/dt. in this case V is for volume and t for time so it is to get volume change but can do for any with same cancel out principal.