Pure

• Gradient = change in y/change in x, $\frac{y_1-y_2}{x_1-x_2}$
• Midpoint = $(x₁+x₂)/2 +$$(y₁+y₂)/2$
• Equation of line = $y-y₁ =$$m(x-x₁)$
• m₁ x m₂ = -1
• A perpendicular bisector is a line that passes through the midpoint between two points and is 90 degrees to the original line
• The perpendicular gradient is the negative reciprocal of the parallel gradient
• If lines intersect then use simultaneous equations to find the coordinate of intersection
• The linear model is y= ax + b

  b is the value of y when x = 0

  a is the value as x is increased by 1 unit, y changes by m units
• $b²-4ac > 0$ then the quadratic equation will have two distinct solutions
• b²-4ac = 0 then the quadratic will have one solution (repeated or equal root)
• b²-4ac < 0 then the quadratic equation will have no solutions
• The quadratic model if the vertex is known is $a(x-b)²+$$c$
• The quadratic model if the x-intercepts are known is $y =$$a(x-b)(x-c)$
• The quadratic model if the x-intercepts or vertex are unknown is $ax^2+$$bx+$$c$
• Greatest height = vertex
• If you divide or multiply by a negative number the inequality direction must be reversed
• If an unknown variable is on the denominator in an inequality then multiply by the square of that number
• Remainder theorem states that when f(x) is divided by ax-b then the remained will be f(b/a)
• One polynomial is a factor of another polynomial if when dividing the polynomials the remainder is zero
• y = x^3

  
• y = x^4

  
• Circle standard form is $(x-a)^2 +$$(y-b)^2 =$$r^2$
• Two circle theorems:

  • The angle in a semicircle is a right angle
  • Line from the centre of the circle to the midpoint of a chord meets the chord at a right angle
• Normals and tangents are perpendicular to each other
• dy/dx > 0 so positive increasing gradient
• dy/dx < 0 negative decreasing gradient
• dy/dx = 0 when at a turning point or at a maximum or minimum point
• If the second derivative equals zero then the point could be a minimum, maximum or a point of inflection
• If a region is below the x-axis the integral will be negative
• If a region is above the x-axis the integral will be positive
• Integration can be used to find enclosed regions between either a line and curve or between two curves
• The trapezium rule is a numerical technique that can approximate the value of a definite integral
• More strips used in the trapezium rule = more accurate answer
• 0! = 1
• $(1+x)^n$expansion is only valid for |x| < 1
• If a>1 then a^x models exponential growth

  
• If 0<a<1 then a^x models exponential decay
• $logn(n^a) =$$a$
• n^logn(a) = a
• logn(b^a) = alogn(b)