All pure formulas Flashcards
As of current, up till term 3 holidays
Rationalising surds (2)
x / √a = { x / √a } x { √a / √a }
a ± √b has a surd conjugate of a ∓ √b
a + √b has a conjugate of a - √b
a - √b has a conjugate of a + √b
Difference of 2 squares
a² – b² = (a + b)(a - b)
Perfect squares (2)
a² - 2ab + b² = (a - b)²
a² + 2ab + b² = (a + b)²
Completing the square
ax² + bx + c = a(x ± h)² ± k
Testing for number of solutions
b² - 4ac
> 0 , 2 real solutions
= 0 , 1 repeated solution
< 0 , no real solutions
Sum of roots
-b / a
Product of roots
c / a
Horizontal asymptote •
y = a/c
Vertical asymptote •
x = -d / c
Vertex for graph of absolute value function
(in form y = mx + c)
x = -c / m
Graphical translations by a translation vector (2)
f(x) ± b translates vertically
f(x ∓ a) translates horizontally
Graphical reflections (2)
-f(x) reflects in x-axis
f(-x) reflects in y-axis
Graphical stretches (2)
pf(x) stretches vertically by scale factor p
f(qx) stretches horizontally by scale factor 1/q
Combined transformation rule
vertical transformations are normal
horizontal transformations (transformations within brackets) are either ∓, reciprocals, or follow reverse order of operations
Distance between 2 points
d = √(x₂ - x₁)² + (y₂ - y₁)²
Equation of line making angle θ with x-axis
tan(θ) = m
Circle equation
(x - p)² + (y - q)² = r²
Chain rule
dy/dx = dy/du • du/dx
Second derivative
d²y/dx²
Vertex in completed square form
a(x - h) + k
where (h, k) is the vertex
Hyperbolic curve equation
(ax + b) / (cx + d)
Square root function graph equation
y = √ax + b
Properties of composite and inverse functions (4)
ff ⁻¹(x) = f ⁻¹ f(x) = x
(f ⁻¹) ⁻¹(x) = f(x)
(fg) ⁻¹(x) = g ⁻¹ f ⁻¹ (x)
f ⁻¹(x) ≠ 1 / f(x)
Differentiations rules (2 - 2)
f(x) = xⁿ
f’(x) = n • xⁿ⁻¹
f(x) = axⁿ
f’(x) = n • axⁿ⁻¹
Standard form of linear equation
y - y₁ = m(x - x₁)
Stationary points
f’(x) = 0
Increasing function
f’(x) > 0
Decreasing function
f’(x) < 0
Points of inflection
f’(x) = 0
f’‘(x) = 0
Maximum stationary point
f’‘(x) < 0
Minimum stationary point
f’‘(x) > 0
Applications of rates of change, calculus (2)
v = s’ = ds/dt
a = v’ = dv/dt
Sine rule (2)
sinA/a = sinB/b = sinC/c
a/sinA = b/sinB = c/sinC
Cosine rule (2)
a² = b² + c² - 2bc x cosA
cosA = [a² - b² - c²] / -2bc
Area of a non-right angle triangle
[bc sin(A)] / 2
Radian/Degree conversions
Radian to degrees:
x radian • 180/π
Degrees to radian:
xº • π/180
Arc length
S = rθ
Area of a sector
A = 1/2r²θ
Area of a segment
A = 1/2r²(θ - sinθ)
General transformation notation (5)
y = A ___ (Bx - C) + D
where ___ is either; sin, cos, or tan
(4) A enlargement in y direction
(2) B enlargement in x direction
(1) C changes position along x-axis
(3) D changes position along y-axis
Trigonometric quadrants
CAST:
Q1: All positive in A (0º - 90º) , [principle values]
Q2: Sin positive in S (90º - 180º) , [180 - principle value]
Q3: Tan positive in T (180º - 270º) , [180 + principle value]
Q4: Cos positive in C (270º - 360º) , [360 - principle value]
Trigonometric identity rules (2)
sin²(x) + cos²(x) = 1
tanθ = sinθ/cosθ
Integration (2 - 2)
∫ f(x) dx
∫ axⁿ dx = [axⁿ⁺¹ / n + 1] + c where n ≠ -1
∫ k(ax + b)ⁿ dx
{ [ k(ax + b)ⁿ⁺¹ ] / a(n + 1) } + c
Properties of indefinite integrals (3)
∫[ f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
∫[ f(x) - g(x)] dx = ∫ f(x) dx - ∫ g(x) dx
∫ kf(x) dx = k x ∫ f(x) dx , where k is a constant
Definite integral
∫ ᵇₐ f’ (x) dx = [f(x)ᵇₐ] = f(b) - f(a) where b > a
Properties of definite integral
∫ᵇₐ f(x) dx = - ∫ᵃᵦ f(x) dx where b > a
∫ ᵃₐ f(x) dx = 0
∫βₐ f(x) dx + ∫ᶜᵦ f(x) dx = ∫ᶜₐ f(x) dx
Area enclosed by a curve and the x-axis
∫ᵇₐ f(x) dx where b > a
Area above and below the x-axis
∫ᵇₐ f(x) dx + | ∫ᶜᵦ f(x) dx | where c > b > a
Area enclosed by a curve and the y-axis
∫ᵇₐ f(y) dy where b > a
Area bounded by a curve and a line or by two curves
A = ∫ᵇₐ [top function] dx - ∫ᵇₐ [bottom function] dx
where b > a
Volumes of solids around x-axis, y-axis
π ∫ᵇₐ f(x)² dx ,
π ∫ᵇₐ f(y)² dy
Binomial expansion; term bʳ, term aʳ
(nCr)aⁿ⁻ʳbʳ
[nC(n-r)]aʳbⁿ⁻ʳ
Geometric sequence
t₁ = a
t₂ = ar
t₃ = ar²
t₄ = ar³
r = tₙ₊₁ / tₙ
