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AS Math Pure Maths Flash Cards

front 1

Complete the square, 2 ways

back 1

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

*if there is a coefficient in front of x2 then factorise the x2 + 6x part, solve and then multiply it all again. Then add the +22

front 2

For a quadratic function how to sketch a graph and find turning point. and cubics

back 2

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.

front 3

Discriminant formula and rules

back 3

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.

front 4

Inequalities equations and on graph rules

back 4

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 the roots of x and if x is smaller shade inside.

front 5

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

back 5

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.

front 6

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

back 6

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.

front 7

Different types of curves based off equations

back 7

front 8

Exponential curve rules

back 8

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.

front 9

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

back 9

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.

front 10

Inverse proportion

back 10

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

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

front 11

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

back 11

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.

front 12

Equation of a circle.

back 12

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.