Studied by 11 people

0.0(0)

get a hint

hint

1

Domain

The set of possible inputs for a function.

New cards

2

Range

The set of possible outputs of a function.

New cards

3

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.

New cards

4

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.

New cards

5

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.

New cards

6

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.

New cards

7

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.

New cards

8

Gradient of Equation

m = (y₂ - y₁) ÷ (x₂ - x₁)

New cards

9

Equation of a Line

y - y₁ = m(x - x₁)

with coords (x₁, y₁)

New cards

10

Distance Formula

√((x₂ - x₁)² + (y₂ - y₁)²)

from (x₁, y₁) to (x₂, y₂)

New cards

11

Perpendicular Bisector

-1/m

where m is original gradient

New cards

12

Standard Equation of a Circle

(x - a)² + (y - b)² = r²

with centre (a, b) and radius r

New cards

13

Equation of a Circle (fg)

x² + y² + 2fx + 2gy + c = 0

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

New cards

14

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.

New cards

15

Factor Theorem

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

New cards

16

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

New cards

17

Truth by Exhaustion

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

New cards

18

Truth by Counter-Example

Find one example that does not work for the statement.

New cards

19

Pascal's Triangle

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

New cards

20

Factorial Formula

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

New cards

21

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)!)

New cards

22

Binomial Expansion

(a + b)ⁿ = (ⁿCᵣ)(aⁿ⁻¹bʳ)

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

New cards

23

Cosine Rule (a²)

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

New cards

24

Cosine Rule (cos(A))

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

New cards

25

Sine Rule

New cards

26

Sine Rule Solutions

Sometimes produces two possible solutions for a missing angle:

sin(θ) = sin(180 - θ)

New cards

27

Principal Value

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

New cards

28

Sine and Cosine Formulae

sin²(θ) + cos²(θ) = 1

tan(θ) = sin(θ) ÷ cos(θ)

New cards

29

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

New cards

30

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

New cards

31

Vector Magnitude

a = xi + yj → |a| = √(x² + y²)

New cards

32

Unit Vector

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

New cards

33

Position Vector

O→P = pi + qj

A→B = O→B - O→A

New cards

34

First Derivative Formula

x=(x+h)

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

New cards

35

Differentiation Formula

dy/dx = anxⁿ⁻¹

New cards

36

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.

New cards

37

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.

New cards

38

Integration Formula

y = k/(n+1) x xⁿ⁺¹ + c

New cards

39

Area of a Triangle

½ x ab x sin(C)

New cards

40

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

New cards

41

Differentiating eˣ

y = eᵏˣ → dy/dx = keᵏˣ

New cards

42

Logarithm Laws

logb(xy) = logb(x) + logb(y)

logb(x/y) = logb(x) - logb(y)

logb(x^y) = y logb(x)

logb(x) = logc(x) / logc(b)

New cards

43

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).

New cards

44

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).

New cards

45

Multiplying - rules of indices

a³ x a⁵ = a⁸ add the powers

New cards

46

Dividing - rules of indices

a⁶ ÷ a² = a⁴ minus the powers

New cards

47

Brackets - rules of indices

(a⁴)² = a⁸ multiply the powers

New cards

48

Negative and Fractional indices

a^1/m = m√a

a^n/m = m√aⁿ

a^-n = 1/aⁿ

New cards

49

Multiplying surds

√a x √b = √ab

New cards

50

Dividing surds

√a ÷ √b = √a/b

New cards

51

Rationalising denominators with surds

1/√a x √a/√a

1/a+√b x a-√b/a-√b

New cards

52

What is the form of a quadratic equation?

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

New cards

53

What is the quadratic formula?

x = (-b±√b²-4ac)/2a

New cards

54

completing the square

x² + bx = (x+b/2)² - (b/2)²

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

New cards

55

The discriminant

b²−4ac > 0 two real, distinct roots.

b²−4ac = 0 one repeated root.

b²−4ac < 0 no real roots.

New cards

56

Gradient of a straight line

m = y₁ - y / x₁ - x

New cards

57

Equation of a straight line

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

New cards

58

Equation of a line using two known points.

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

New cards

59

Parallel lines

have the SAME gradient.

New cards

60

Perpendicular lines

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

New cards

61

Distance between two points

√(x₁-x₂)² + (y₁-y₂)²

New cards

62

equation for midpoint

(x₁+x₂/2 , y₁+y₂/2)

New cards

63

Equation of a circle

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

New cards

64

What is a tangent?

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

New cards

65

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).

New cards

66

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.

New cards

67

Factor Theorem

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

New cards

68

What is proof by deduction?

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

New cards

69

What is proof by exhaustion?

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

New cards

70

What is disproof by counter-balance?

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

New cards

71

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.

New cards

72

Trigonometry for right-angled triangles

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

New cards

73

What is the cosine rule?

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

New cards

74

What is the sine rule?

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

New cards

75

Area of a triangle with two sides and angle in between

area = ½absinC

New cards

76

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.

New cards

77

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.

New cards

78

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.

New cards

79

What are the two trig identities?

sin²θ + cos²θ ≡ 1

tanθ ≡ sinθ/cosθ

New cards

80

Derivative or gradient function equation

Where h represents a small change.

New cards

81

Differentiating x^n

If f(x) = xⁿ then f'(x) = nxⁿ⁻¹

If f(x) = axⁿ then f'(x) = anxⁿ⁻¹

New cards

82

Differentiating a quadratic

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

New cards

83

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).

New cards

84

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)

New cards

85

Increasing and decreasing functions.

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

New cards

86

Second order derivative

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

New cards

87

Types of stationary points.

Local maximum, local minimum and point of inflection.

New cards

88

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

New cards

89

the discriminant

b^2-4ac=0

>for two roots =for one repeated root <no real roots

New cards

90

turning point of a graph

complete the square

New cards

91

quadratic formula

x=(-b±√(b^2-4ac))/2a

New cards

92

y=f(x)+a

moves (0/a)

New cards

93

y=f(x+a)

moves (-a/0)

New cards

94

y=af(x)

movement of a in y (vertical) direction

New cards

95

y=f(ax)

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

New cards

96

y=-f(x)

reflection in x axis

New cards

97

y=f(-x)

reflection in y axis

New cards

98

gradient equation

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

New cards

99

Distance between two points

d = √(x₁-x₂)² + (y₁-y₂)²

New cards

100

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2

New cards