A level maths: pure

Domain

The set of possible inputs for a function.

Range

The set of possible outputs of a function.

Discriminant

b² - 4ac \> 0 then two distinct real roots.

b² - 4ac \= 0 then one repeated real root.

b² - 4ac < 0 then a quadratic function has no real roots.

Types of Lines for Regions

If y < f(x) or y \> f(x) then the curve y \= f(x) is not included in the region, and is represented by a dotted line.

If y ≤ f(x) or y ≥ f(x) then the curve y \= f(x) is included in the region, and is represented by a solid line.

Graph Translations

y \= f(x) + a is a translation of the graph y \= f(x) by a upwards.

y \= f(x + a) is a translation of the graph y \= f(x) by a to the left.

Graph Stretches

y \= af(x) is a stretch of the graph y \= f(x) by a scale factor of a in the vertical direction.

y \= f(ax) is a stretch of the graph y \= f(x) by a scale factor of 1/a in the horizontal direction.

Graph Reflections

y \= -f(x) is a reflection of the graph of y \= f(x) in the x-axis.

y \= f(-x) is a reflection of the graph of y \= f(x) in the y-axis.

Gradient of Equation

m \= (y₂ - y₁) ÷ (x₂ - x₁)

Equation of a Line

y - y₁ \= m(x - x₁)

with coords (x₁, y₁)

Distance Formula

√((x₂ - x₁)² + (y₂ - y₁)²)

from (x₁, y₁) to (x₂, y₂)

Perpendicular Bisector

-1/m

where m is original gradient

Standard Equation of a Circle

(x - a)² + (y - b)² \= r²

with centre (a, b) and radius r

Equation of a Circle (fg)

x² + y² + 2fx + 2gy + c \= 0

with centre (-f, -g) and radius √(f² + g² - c)

Circle Theorems

• Tangent to a circle is perpendicular to the radius of the circle at the point of intersection.
• Perpendicular bisector of a chord will go through the circle centre.
• If triangle forms across the circle, its diameter is the hypotenuse of the right-angled triangle.
• Equations of the perpendicular bisectors of two different chords will intersect at the circle centre.

Factor Theorem

If f(p) \= 0, (x - p) is a factor of f(x)

Mathematical Proofs

• State any info/assumptions
• Show every step clearly
• Make sure every step follows logically from the previous step
• Cover all possible cases
• Write a statement of proof at the end of your working

Truth by Exhaustion

Break the statement into smaller cases and prove each case separately.

Truth by Counter-Example

Find one example that does not work for the statement.

Pascal's Triangle

The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a + b)ⁿ

Factorial Formula

n! \= n x (n - 1) x (n - 2) x ... x 2 x 1

Factorials in Pascal's Triangle

The number of ways of choosing r from a group of n items is:
ⁿCᵣ \= n! ÷ (r! x (n - r)!)

Binomial Expansion

(a + b)ⁿ \= (ⁿCᵣ)(aⁿ⁻¹bʳ)

(a + b)ⁿ \= aⁿ + (ⁿC₁)(aⁿ⁻¹b) + (ⁿC₂)(aⁿ⁻²b²) + ... + (ⁿCᵣ)(aⁿ⁻ʳbʳ) + ... + bⁿ

Cosine Rule (a²)

a^2=c^2+b^2-2bcCosA

Cosine Rule (cos(A))

CosA=a^2+b^2-c^2/ 2ab

Sine Rule

Sine Rule Solutions

Sometimes produces two possible solutions for a missing angle:

sin(θ) \= sin(180 - θ)

Principal Value

When you use the inverse trig function on your calculator, the angle you get is the principal value.

Sine and Cosine Formulae

sin²(θ) + cos²(θ) \= 1

tan(θ) \= sin(θ) ÷ cos(θ)

Triangle Law for Vector Addition

A→B + B→C \= A→C

If A→B \= a, B→C \= b and A→C \= c, then a + b \= c

Vector Rules

• P→Q \= R→S, then line segments PQ and RS are equal in length and are parallel.
• A→B \= -(B→A)
• Any vector parallel to the vector a may be written as λa

Vector Magnitude

a \= xi + yj → |a| \= √(x² + y²)

Unit Vector

In the direction of a, unit vector is
a ÷ |a|

Position Vector

O→P \= pi + qj

A→B \= O→B - O→A

First Derivative Formula

* x=(x+h)
* Any workings on x must be applied to (x+h)

Differentiation Formula

dy/dx \= anxⁿ⁻¹

Function's Gradient

Increasing on the interval [a, b] if f'(x) ≥ 0 for all values of a < x < b.

Decreasing on the interval [a, b] if
f'(x) ≤ 0 for all values of a < x < b.

Stationary Point

Any point on the curve y \= f(x) where f'(x) \= 0.
• if f''(a) \> 0, the point is a local minimum.
• if f''(a) < 0, the point is a local maximum.

Integration Formula

y \= k/(n+1) x xⁿ⁺¹ + c

Area of a Triangle

½ x ab x sin(C)

Natural Log (y \= ln x)

The graph of y \= ln(x) is a reflection of the graph y \= eˣ in the line y \= x.

eˡⁿ⁽ˣ⁾ \= ln(eˣ) \= x

Differentiating eˣ

y \= eᵏˣ → dy/dx \= keᵏˣ

Logarithm Laws

\

1. logb(xy) = logb(x) + logb(y)
2. logb(x/y) = logb(x) - logb(y)
3. logb(x^y) = y logb(x)
4. logb(x) = logc(x) / logc(b)

y \= axⁿ

If y \= axⁿ then the graph of log(y) against log(x) will be a straight line with gradient n and vertical intercept log(a).

y \= abˣ

If y \= abˣ then the graph of log(y) against x will be a straight line with gradient log(b) and vertical intercept log(a).

Multiplying - rules of indices

a³ x a⁵ \= a⁸
add the powers

Dividing - rules of indices

a⁶ ÷ a² \= a⁴
minus the powers

Brackets - rules of indices

(a⁴)² \= a⁸
multiply the powers

Negative and Fractional indices

a^1/m \= m√a

a^n/m \= m√aⁿ

a^-n \= 1/aⁿ

Multiplying surds

√a x √b \= √ab

Dividing surds

√a ÷ √b \= √a/b

Rationalising denominators with surds

1/√a x √a/√a

1/a+√b x a-√b/a-√b

What is the form of a quadratic equation?

ax² + bx + c \= 0
where a, b and c are real constants and a≠0

What is the quadratic formula?

x \= (-b±√b²-4ac)/2a

completing the square

x² + bx \= (x+b/2)² - (b/2)²

ax² + bx + c \= a(x + b/2a)² + (c - b²/4a²)

The discriminant

b²−4ac \> 0 two real, distinct roots.

b²−4ac \= 0 one repeated root.

b²−4ac < 0 no real roots.

Gradient of a straight line

m \= y₁ - y / x₁ - x

Equation of a straight line

y \= mx + c
OR
ax + by + c \= 0

Equation of a line using two known points.

y - y₁ \= m(x - x₁)
where m \= gradient and (x₁,y₁) are the coordinates used.

Parallel lines

have the SAME gradient.

Perpendicular lines

They meet at a right angle. If line 1 has a gradient of m, the gradient of line 2 is -1/m

Distance between two points

√(x₁-x₂)² + (y₁-y₂)²

equation for midpoint

(x₁+x₂/2 , y₁+y₂/2)

Equation of a circle

(x-a)² + (y-b)² \= r²
with centre (a,b) and radius r.

What is a tangent?

A tangent is a line that only touches one point on the circumference of a circle.

What is a chord?

A line joining two points on the circumference of a circle, but not passing through the centre (that is the diameter).

Tangent and chord properties on circles.

A tangent to the circle is perpendicular to the radius at the point of intersection.
The perpendicular bisector of a chord will go through the centre of a circle.

Factor Theorem

if f(p)\=0 then (x-p) is a factor of f(x).

What is proof by deduction?

Starting from known facts or definitions, then using logical steps to reach the desired conclusion.

What is proof by exhaustion?

Breaking the statement into smaller cases and proving each case separately.

What is disproof by counter-balance?

Giving an example that doesn't work for the statement.

Pascal's Triangle

It is formed by adding adjacent pairs of numbers to find the numbers of the next row.
The (n+1)th row of Pascal's Triangle gives the coefficients in the expansion of (a+b)ⁿ.

Trigonometry for right-angled triangles

sinθ \= opp/hyp
cosθ \= adj/hyp
tanθ \= opp/adj

What is the cosine rule?

Length:
a² \= b² + c² - 2bc cosA
Angle:
cosA \= (b² + c² - a²)/2bc

What is the sine rule?

Length:
a/sinA \= b/sinB \= c/sinC
Angle:
sinA/a \= sinB/b \= sinC/c

Area of a triangle with two sides and angle in between

area \= ½absinC

Graph of y \= sin(x)

Repeats every 360°.
Crosses x-axis at -180, 0, 180, 360 etc.
Has a max value of 1 and a min of -1.

Graph of y \= cos(x)

Repeats every 360°.
Crosses x-axis at -90, 90, 270, 450 etc.
Has a max value of 1 and a min of -1.

Graph of y \= tan(x)

Repeats every 180°.
Crosses x-axis at -180, 0, 180, 360 etc.
Has no max or min value
Has vertical asymptotes at x\=-90, 90, 270 etc.

What are the two trig identities?

sin²θ + cos²θ ≡ 1

tanθ ≡ sinθ/cosθ

Derivative or gradient function equation

Where h represents a small change.

Differentiating x^n

If f(x) \= xⁿ then f'(x) \= nxⁿ⁻¹

If f(x) \= axⁿ then f'(x) \= anxⁿ⁻¹

Differentiating a quadratic

If y \= ax² + bx + c
then dy/dx \= 2ax + b

Tangents to curves.

The tangent to the curve y\=f(x) at (a, f(a)) has the equation:
y-f(a) \= f'(a)(x-a).

Normals to curves.

The normal to the curve y\=f(x) at (a, f(a)) has the equation:
y-f(a) \= -1/f'(a) x (x-a)

Increasing and decreasing functions.

Increasing: f'(x) \> 0
Decreasing: f'(x) < 0

Second order derivative

f''(x) or d²y/dx² a.k.a differentiate twice.

Types of stationary points.

Local maximum, local minimum and point of inflection.

Sketching gradient functions

y\=f(x) y\=f'(x)
max or min cuts x-axis
point of inflection touches x-axis
+ve grad above x-axis
-ve grad below x-axis
vertical asymp. vertical asymp.
horizontal asymp. horiz asymp at x-axis

the discriminant

b^2-4ac\=0

\>for two roots
\=for one repeated root

turning point of a graph

complete the square

quadratic formula

x\=(-b±√(b^2-4ac))/2a

y\=f(x)+a

moves (0/a)

y\=f(x+a)

moves (-a/0)

y\=af(x)

movement of a in y (vertical) direction

y\=f(ax)

movement by scale factor of 1/a in x (horizontal) direction

y\=-f(x)

reflection in x axis

y\=f(-x)

reflection in y axis

gradient equation

m \= (y2-y1) / (x2-x1)

Distance between two points

d \= √(x₁-x₂)² + (y₁-y₂)²

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2