AS Math Pure Maths Flash Cards Flashcards

x²+ 6x + 22

Like normal half bx and then take away or add whats necessary, where (x+3)squared and then + or - n to match equation.

Shortcut for this is the formula. (x+b/2)squared - (b/2)squared + c

*if there is a coefficient in front of x² then factorise the x² + 6x part, solve that inside the bracket and then multiply it back out. Then add the c.

If coefficient of x² is positive, the graph will be a positive U-shaped curve and vice versa.

Find y and x intercepts by making x and y 0 in the equation.

Find the turning point by either: using the equation x = -b/2a from the equation. This gives x and to find y sub value of x in original equation. If a is positive it is a min point, and if negative then max point. The original equation is y = ax² + bx + c.

or complete the square to form a(x+p)² + q where the coordinate are (-p, q)

For a cubic its the same thing however 3 set of brackets to find 3 values of x. and then make x zero to find y.

*When x is large and positive, y is large and negative and vice versa when negative leading coefficient. When positive. Large positive x → large positive y and vise versa. This is basically the slope direction. it tells you whether its going up or down and how much at certain point for exponential curve.

b² - 4ac

If greater than zero quadratic graph will cross the x - axis twice.

If zero only once, if less then 0 times resulting in no solutions for x.

If the discriminant is a square number x will be 2 rational numbers.

Write equations like -2<x<5. If you divide or multiply both sides by a negative reverse the sign.

For a line graph or a quadratic with an equation in form y= - when y is greater shade above the parabola and when y is less shade below.

For quadratic when you in form x= - if x is greater than shade outside/right the roots of x and if x is smaller shade inside/left. this is not always rule though so use inequalities to get actual answer or test intervals.

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.

Find the negative reciprocal and sub into formula where y1 and x1 coordinates for the point we are given, x and y just get left as x and y to represent any point on the line:

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 x1 and y1.

y = e^x

• For when include constants for e and x. eg y = ae^bx. If x is 0 curve will cross y axis at (0,a) if b is positive means growth, if negative decay.
• The x exponent differentiates itself so just multiply b constant by a.
• When n is an even integer the curve positive and when odd it has both negative and positive values.
• if x is bigger then it will go above the curve where n is smaller, except the opposite when x is between 0 and 1.
• ln means natural logarithm which makes it the inverse so ln(e⁷) = 7
• For integrating the exponent the ^x exponent stays the same don't add one, and if it has a constant as in e^ax then multiple it by 1/a.

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

• Vertical asymptote: x=0
• Horizontal asymptote: 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 = The statement A is inversely proportional to is written:

A ∝ 1/B then to solve replace 1 with the constant. Figure out what the constant is from info given.

Option 1 ; find the lengths between each point and with pythagorus theorem, two lengths squared = 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 at centre (0.0) = x² + y² = radius²

general equation when not at centre = 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.

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...

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.

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.

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

all above formulas are in sheet

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)

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.

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 y axis.

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.

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.

Find the second differential when finding max and min points on a cubic graph 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.

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.

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. These given in formula sheet

reverse of differentiation

formulas (1 / n + 1) xⁿ⁺¹ + c. this formula is simpler put increase the power by 1, divide be new power.

Put in x and n is the number over xⁿ when given that figure. Solve for c from the rest of equation ie. 3 + c = 2

The symbol f with an and b shows range. put in b and an into equation and take a figure from b figure. the x is the expression being integrated. Use this to find the area of a curve and to get an and b factorise the curve equation to get the values of x. when for two lines that intersect find x points where the lines intercept.

For shaded area between curve and line. Find point they intercept and use this for range. Calculate the range of each line to get area of each, and then subtract one value form area to get shaded area. usually take away from whichever is higher, can figure this out by subbing in random number for x and seeing which value is higher.

This gets you volume in graph curve . You can find area by doing this but just with tubing in your or x with out it being a squared power and no pie.

for rotating around x axis make y the subject and vice versa. the above equation includes for x axis but for y axis just swap out the y squared for x squared and make x the subject.

substitute y or x in, after that then integrate it with the integration rule

• Add 1 to the power
• Divide by the new power

Then do it for the two limits taking one away from the other.

360 degrees is 2pie radians, half a semi circle is equal to 1 pie radian and 180 degrees.

arc length = radius x angle in radians

area - 1/2 radius² x angle in radians

1. When multiplying indices with the same base, add the powers.
2. When dividing indices with the same base, subtract the powers.
3. When there is a power outside the bracket multiply the powers.
4. x to the power of 0 = 1
5. When the index is negative, put it over 1 and flip (write its reciprocal) to make it positive x^-m = 1/x^m
6. When the index is a fraction, the denominator is the root of the number or letter, then raise the answer to the power of the numerator. x^a/b =( ^b√x)^a
7. A value raised to the power of ½ means take the square root and a value raised to the power of ⅓ means take the cube root and so on.

You remove any square roots (surds) from the denominator by multiplying the entire fraction (numerator and denominator) by the surd in the denominator.

For denominators with a sum or difference of surds (e.g., a + b√c), you multiply by the same expression with the opposite sign (the conjugate) to use the difference of squares formula (a² - b²) and eliminate the surd.

Example with a rational number and a surd e.g., 1 / (√5 + √2)

1. Identify the conjugate: The conjugate of √5 + √2 is √5 - √2.
2. Multiply by the conjugate: (1 * (√5 - √2)) / (√5 + √2) * (√5 - √2)).
3. Expand and simplify:
   • Numerator: √5 - √2.
   • Denominator: (√5)² - (√2)² = 5 - 2 = 3.
4. Final answer: (√5 - √2) / 3.

to get other answers as can be multiple. for sin 180-x, for cos 360-x, for tan 180 + x

a is side length

For not equilateral A= 1/2​ ab sinC

The area of a segment of a circle is found by subtracting the area of the triangle from the area of the sector.

also here as point is the centre of circle to the tangent d is also the radius.

when with triangle only up to 180 degrees so only 1st and 2nd quadrant