All AS Maths Equations/Calculations and Questions

Description

All of the equations needed for formulae in the AS maths exams and questions on some of the theorems.
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Question Answer
√a x √b =√ab
√a/b =√a/√b
√ab/√b =√a
ax^2+bx+c=f(x) what happens when you: increase b? decrease b? increase a? decrease a? increase b = translates up the y axis decrease b = translates down the y axis increase a = translates left decrease a = translates right
ax^2+bx+c=f(x) what shape is the graph is the quadratic is: positive? negative? positive: the graph will make a 'u' shape negative: the graph will make an 'n' shape
If you use (x1+x2)/2 to find the x value of the root, how do you find the y value? Substitute the x value into the 'y=' equation of the quadratic
To complete the square, what does ax^2+bx+c turn into? (x-d)^2-e
Where does (x+2)^2+2 translate to on a graph? (x+2) = translates left (along x axis) by 2 +2 = translates up the y axis by 2
What is the quadratic formula?
What part of the quadratic equation is the discriminant and what does the discriminant mean in terms of roots? b^2-4ac b^2-4ac<0 = no relevant roots b^2-4ac=0 = two repeated roots b^2-4ac>0 = two distinct roots
What are the three ways of solving a quadratic? 1. factorising (always try first) 2. quadratic equation (if doesn't factorise or 'b' isn't even) 3. complete the square (only use if 'b' is even)
What is the factor theorem? f(7)=0 ∴ (x-7) is a factor of f(x) f(a)=0 ∴ (x-a) is a factor of f(x)
What is the equation of a circle? (x-a)^2+(x-b)^2=r^2
How do you work out the distance between two points? Pythagoras' theorem
What equation do you use to find the midpoint of a line?
What do you use to find the equation of a line? y - y1 = m (x - x1)
What is the difference between a perpendicular and parallel line? Perpendicular: have gradients that are opposite signed (+/-) and reciprocals Parallel: have the same gradients but different y intercepts
What circle theorem is this? Angles in a semi circle: angle on the arch is always a right angle (perpendicular lines)
What circle theorem is this? Any chord, perpendicular line is drawn through center of circle, will bisect the chord
What circle theorem is this? From any point outside the circle, there is exactly two tangents. The radius to the tangent is perpendicular.
What is this line? Tangent (Has one repeated root)
What is this line? Secant (two distinct roots)
What is a scew line? A line that has no real roots/intersections with a circle
What does dy/dx equal? The gradient
What is: f(x) f'(x) f''(x) f(x) = before differentiation f'(x) = first derivative f''(x) = second derivative
What is d^2y/dx^2 > 0 Minimum
What is d^2y/dx^2 < 0 maximum
If f(x) represents displacement, what does f'(x) and f''(x) represent? f'(x) represents velocity f''(x) represents acceleration
What is an 'increasing function'? An increasing function is a function f(x) such that f'(x) > 0 for all values of x> (a function that is always increasing)
What is a 'decreasing function'? A decreasing function is a function f(x) such that f'(x) < 0 for all values of x (A function that is always decreasing)
Describe the graph transformations taking place from: f(x) = x^2 =2x^2 =2(x)^2 x^2 = original formulae 2x^2 = stretched in y direction by scale factor of 2 2(x)^2 = stretched in x axis by scale factor of 0.5
What is the difference in transformations if the number is inside or outside the bracket? inside the bracket : x direction outside the bracket : y direction
What does xn = a + d(n-1) calculate? Any number in a sequence (arithmetic)
What is a Sigma? The big E symbol. It means 'sum up'
What does this mean? S = (n/2) × (2a + (n-1)d) It calculates the sum of the first 'n' terms of an arithmetic series
What does xn = ar(n-1) calculate? Any term in a geometric progression
What is the sine rule?
What is the cosine rule?
How do you find the area of a triangle? 0.5(base x height) OR 0.5absinC
How do you work out the area of a sector? DEGREES: (θ/360) x πr^2 RADIANS: θr^2/2
How do you work arc length? DEGREES: (θ/360) x 2πr RADIANS: rθ
What does a tan graph look like?
What does a cos graph look like?
What does a sin graph look like?
What does a CAST diagram look like? (DEGREES)
What are the trig identities?
What are the log rules?
How do you find the area of a trapezium? A = 0.5(a+b)h
What is the trapezium rule?
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