Complete the square, 2 ways
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
For a quadratic function how to sketch a graph and find turning point. and cubics
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.
Discriminant formula and rules
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.
Inequalities equations and on graph rules
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.
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.