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
