Maths Year 1 Notes

P → Q

P implies Q or if P is true then Q is true or P is sufficient for Q

P ← Q

P is implied by Q or if Q is true then P is true or P is necessary for Q

P ↔ Q

P is equivalent to Q or Q is true if and only if P is true

x ∈ ( a , b )

a < x < b

x ∈ \[ a , b \]

a ≤ x ≤ b

x ∈ \[ a , b )

a ≤ x < b

x ∈ ( a , b \]

a < x ≤ b

x ∈ A ∪ B

Union of A and B where x can be in either A or B or both

x ∈ A ∩ B

Intersection of A and B where x is in both A and B

x ∈ ∅

No solutions to the inequality

∅

Symbol for an empty set

a^n × a^m

a^(m+n)

a^(1/n)

n√a

a^m ÷ a^n

a^(m-n)

a^(m/n)

n√a^m

(a^m)^n

a^(m × n)

a^n × b^n

(ab)^n

a^0

1

a^n ÷ b^n

(a/b)^n

a^(-n)

1/a^n

Rationalize the denominator

Multiply the top and bottom of the fraction by an appropriate expression to create a difference of two squares

Positive parabola

a > 0

Negative parabola

a < 0

y-intercept

Value of c in quadratic equation, (0,c)

Quadratic equation

ax² + bx + c

Complete the square formula

a(x+p)²+q

Turning point of graph

(-p,q) from completing the square

Line of symmetry

x = -p = (x₁+x₂)/2

Quadratic formula

(-b±√b²-4ac)/2a

Discriminant

∆ = b² - 4ac

Graph has two real roots

∆ > 0

Graph has one real root

∆ = 0

Graph has no real roots

∆ < 0

Factor theorem

(ax+b) is a factor of a polynomial f(x) if and only if f(-b/a) = 0

Polynomial

ax^k

y = f(x) + c

Translation c up

y = f(x+d)

Translation d to the left

y = pf(x)

Vertical stretch, scale factor p relative to the x-axis

y = f(qx)

Horizontal stretch, scale factor 1/q relative to the y-axis

y = -f(x)

Reflection in the x-axis

y = f(-x)

Reflection in the y-axis

Direct proportion

y = kx

Inverse proportion

y = k/x

Distance between two points

√(x₂ - x₁) + (y₂ - y₁)

Midpoint of two points

( (x₁+x₂)/2 , (y₁+y₂) )

Gradient of a line

m = (y₂-y₁) / (x₂-x₁)

Method to find equation of straight line

y - y₁ = m (x - x₁)

Equation of a straight line

y = mx + c / ax + by + c = 0

Parallel lines

Have the same gradient

Perpendicular lines

m₁ m₂ = -1

Equation of a circle

(x - a)² + (y - b)² = r²

Tangent

Perpendicular to the radius

Normal

Same direction as radius

b^c = a

logb(a)

log10(x)

log(x)

loga(x^a)

x=a^(loga(x))

loge(x) / natural logarithm

In(x)

loga(x) + loga(y)

loga(xy)

loga(x) - loga(y)

loga(x/y)

loga(1/x)

\-loga(x)

loga(x^k)

kloga(x)

loga(1)

0

loga(a)

1

Graph of y=a^x

y-intercept is (0,1) and x-axis is an asymptote

Gradient of e^(kx)

ke^(kx)

Ae^(kt) model

Initial value is A and the rate of change equals ky

Graph of y=In(k)

Passes through (1,0) and y-axis is an asymptote

Convert y=ax^n

log(y) = log(a) + nlog(x)

log(y) = log(a) + nlog(x)

y-intercept is log(a) and gradient is n

Binomial theorem

(a+b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁aⁿ⁻¹b¹ + … + ⁿCₙa⁰bⁿ

ⁿCr

n! / r!(n-r)!

n!

n × (n-1) × … × 3 × 2 × 1

0!

1

sin(x)

sin(180-x) = sin(x+360)

sin(180+x)

sin(-x) = -sin(x)

cos(x)

cos(-x) = cos (x+360)

cos(180-x)

cos(180+x) = -cos(x)

tan(x)

tan(x+180) = tan(x+360) = …

Tan graph asymptotes

x = 90° , x = 270° etc

Period of sine and cosine graph

360°

Period of tan graph

180°

cos(90-x)

sin(x)

sin(90-x)

cos(x)

tan(x) identity

tan(x) ≡ sin(x) / cos(x)

sin(x) and cos(x) identity

sin²(x) + cos²(x) ≡ 1

Sine rule

a / sin(A) = b / sin(B) = c / sin(C)

Two answers by sine rule

A and 180-A

cosine rule

c² = a² + b² - 2abcos(C)

Area of a triangle

Area = 1/2 absin(C)

Magnitude of vector a = pi + qj

√p² + q²

Unit vector

Magnitude is one

Parallelogram

AB = DC

Rhombus

|AB| = |BC|

If vectors a and b are parallel

b = ta

Displacement from A to B

AB = b-a

Midpoint of line AB

1/2 (a+b)

If graph is increasing

Positive gradient

If graph is decreasing

Negative gradient

When gradient = 0

Stationary point

Differentiation from first principles

f’(x) = lim h→0 (f(x+h) - f(x)) / h