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Facts currently selected by your teacher · 74 facts · questions and answers

Algebra

01

Discriminant and how it determines the number of roots of a quadratic

Δ=b2−4ac\Delta=b^2-4ac

Δ > 0: two distinct real roots. Δ = 0: one repeated real root. Δ < 0: no real roots.

02

Quadratic formula

x=−b±b2−4ac2ax=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}
03

The factor theorem

(x−a) is a factor of f(x)  ⟺  f(a)=0(x-a)\text{ is a factor of }f(x)\iff f(a)=0
04

The remainder theorem

Remainder when f(x) is divided by (x−a)=f(a)\text{Remainder when }f(x)\text{ is divided by }(x-a)=f(a)
23

Laws of logarithms

log⁡A+log⁡B=log⁡(AB)\log A+\log B=\log(AB)log⁡A−log⁡B=log⁡ ⁣(AB)\log A-\log B=\log\!\left(\dfrac AB\right)log⁡(An)=nlog⁡A\log(A^n)=n\log A

Trigonometry

05

The sine rule

asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}
06

The cosine rule

a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A
07

Area of a triangle using two sides and the included angle

A=12absin⁡CA=\dfrac12 ab\sin C
10

Sketch the graph of y = sin x in degrees and radians

y = sin x
−1100°π/290°π180°3π/2270°2π360°xy

Period 2π (360°); range [−1, 1].

11

Sketch the graph of y = cos x in degrees and radians

y = cos x
−1100°π/290°π180°3π/2270°2π360°xy

Period 2π (360°); range [−1, 1].

12

Sketch the graph of y = tan x in degrees and radians

y = tan x
00°π/290°π180°3π/2270°2π360°xy

Period π (180°); asymptotes at x = π/2 + kπ, for any integer k.

13

Special values of sin, cos and tan (in both degrees and radians)

θ0°30°45°60°90°
Radians0π/6π/4π/3π/2
sin θ01/2√2/2√3/21
cos θ1√3/2√2/21/20
tan θ01/√31√3undefined

Angles: 0°, 30°, 45°, 60°, 90° (or 0, π/6, π/4, π/3, π/2 radians). tan 90° is undefined.

14

Arc length when the angle is in radians

s=rθs=r\theta
15

Area of a sector when the angle is in radians

A=12r2θA=\dfrac12r^2\theta
16

Identity linking tan θ, sin θ and cos θ

tan⁡θ=sin⁡θcos⁡θ\tan\theta=\dfrac{\sin\theta}{\cos\theta}
17

The Pythagorean trigonometric identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1
18

Identity linking tan² θ and sec² θ

1+tan⁡2θ=sec⁡2θ1+\tan^2\theta=\sec^2\theta
19

Identity linking cot² θ and cosec² θ

1+cot⁡2θ=cosec⁡2θ1+\cot^2\theta=\cosec^2\theta
20

Double-angle formula for sin 2θ

sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta=2\sin\theta\cos\theta
21

Three forms of the double-angle formula for cos 2θ

cos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos 2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta
22

Double-angle formula for tan 2θ

tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}
56

Small-angle approximations for sin θ, cos θ and tan θ

sin⁡θ≈θ,cos⁡θ≈1−θ22,tan⁡θ≈θ\sin\theta\approx\theta,\qquad\cos\theta\approx1-\dfrac{\theta^2}{2},\qquad\tan\theta\approx\theta

θ is in radians and close to zero.

Coordinate geometry

08

Equation of a circle

Centre (a, b), radius r:

(x−a)2+(y−b)2=r2(x-a)^2+(y-b)^2=r^2
09

Equation of a straight line with gradient m passing through (x₁, y₁)

y−y1=m(x−x1)y-y_1=m(x-x_1)

Graphs

24

Sketch the graph of y = x²

y = x²
-2-1012(0, 0)xy
25

Sketch the graph of y = x³

y = x³
-2-1012(0, 0)xy
26

Sketch the graph of y = 1/x

y = 1/x
-2-10123xy

Neither axis is an intercept; both axes are asymptotes.

27

Sketch the graph of y = √x

y = √x
-10123(0, 0)xy

Domain x ≥ 0.

28

Sketch the graph of y = aˣ

y = 2ˣ
-2-10123(0, 1)xy

The plot uses a = 2 (> 1), with the x-axis as a horizontal asymptote. If 0 < a < 1, the curve decreases through (0, 1).

29

Sketch the graph of y = logₐ x

y = log₂ x
-10123(1, 0)xy

The plot uses a = 2 (> 1), with the y-axis as a vertical asymptote. If 0 < a < 1, the curve decreases through (1, 0).

30

Transformation y = f(x) + a

Translate by (0a)\text{Translate by }\begin{pmatrix}0\\a\end{pmatrix}
31

Transformation y = f(x + a)

Translate by (−a0)\text{Translate by }\begin{pmatrix}-a\\0\end{pmatrix}
32

Transformation y = af(x)

Stretch parallel to the y-axis by scale factor a.

33

Transformation y = f(ax)

Stretch parallel to the x-axis by scale factor 1/a (for a > 0).

34

Transformation y = −f(x)

Reflect in the x-axis.

35

Transformation y = f(−x)

Reflect in the y-axis.

Calculus

36

Nature of stationary points from the second derivative

d2ydx2>0:local minimum\dfrac{d^2y}{dx^2}>0:\quad\text{local minimum}d2ydx2<0:local maximum\dfrac{d^2y}{dx^2}<0:\quad\text{local maximum}d2ydx2=0:check the gradient on either side\dfrac{d^2y}{dx^2}=0:\quad\text{check the gradient on either side}
37

Definitions of concave and convex

Concave means d2ydx2<0\text{Concave means }\dfrac{d^2y}{dx^2}<0Convex means d2ydx2>0\text{Convex means }\dfrac{d^2y}{dx^2}>0
43

Chain rule for y = (stuff)ⁿ

dydx=stuff′×n(stuff)n−1\dfrac{dy}{dx}=\mathrm{stuff}'\times n(\mathrm{stuff})^{n-1}
44

Chain rule for e to the power of stuff

y=estuffy=e^{\mathrm{stuff}}
dydx=stuff′×estuff\dfrac{dy}{dx}=\mathrm{stuff}'\times e^{\mathrm{stuff}}
45

Chain rule for a to the power of stuff

y=astuffy=a^{\mathrm{stuff}}
dydx=stuff′×(ln⁡a)astuff\dfrac{dy}{dx}=\mathrm{stuff}'\times(\ln a)a^{\mathrm{stuff}}
46

Chain rule for y = ln(stuff)

dydx=stuff′×1stuff\dfrac{dy}{dx}=\mathrm{stuff}'\times\dfrac1{\mathrm{stuff}}
47

Chain rule for y = sin(stuff)

dydx=stuff′×cos⁡(stuff)\dfrac{dy}{dx}=\mathrm{stuff}'\times\cos(\mathrm{stuff})
48

Chain rule for y = cos(stuff)

dydx=stuff′×[−sin⁡(stuff)]\dfrac{dy}{dx}=\mathrm{stuff}'\times[-\sin(\mathrm{stuff})]
49

Chain rule for y = tan(stuff)

dydx=stuff′×sec⁡2(stuff)\dfrac{dy}{dx}=\mathrm{stuff}'\times\sec^2(\mathrm{stuff})
50

Product rule for differentiation

(uv)′=u′v+uv′(uv)'=u'v+uv'
51

Quotient rule for differentiation

(uv)′=u′v−uv′v2\left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}
52

What partial-fraction form would you use for two distinct linear factors?

mx+n(x+p)(x+q)\dfrac{mx+n}{(x+p)(x+q)}
mx+n(x+p)(x+q)=Ax+p+Bx+q\dfrac{mx+n}{(x+p)(x+q)}=\dfrac A{x+p}+\dfrac B{x+q}
53

What partial-fraction form would you use for a repeated linear factor?

mx+n(x+p)(x+q)2\dfrac{mx+n}{(x+p)(x+q)^2}
mx+n(x+p)(x+q)2=Ax+p+Bx+q+C(x+q)2\dfrac{mx+n}{(x+p)(x+q)^2}=\dfrac A{x+p}+\dfrac B{x+q}+\dfrac C{(x+q)^2}
54

What partial-fraction form would you use for an irreducible quadratic factor?

mx+n(x+p)(x2+q2)\dfrac{mx+n}{(x+p)(x^2+q^2)}
mx+n(x+p)(x2+q2)=Ax+p+Bx+Cx2+q2\dfrac{mx+n}{(x+p)(x^2+q^2)}=\dfrac A{x+p}+\dfrac{Bx+C}{x^2+q^2}

Sequences

38

nth term of an arithmetic sequence

un=a+(n−1)du_n=a+(n-1)d
39

Sum of the first n terms of an arithmetic sequence

Sn=n2[2a+(n−1)d]S_n=\dfrac n2[2a+(n-1)d]
40

nth term of a geometric sequence

un=arn−1u_n=ar^{n-1}
41

Sum of the first n terms of a geometric sequence

Sn=a(rn−1)r−1(r≠1)S_n=\dfrac{a(r^n-1)}{r-1}\quad(r\ne1)
42

Sum to infinity of a geometric sequence

S∞=a1−r(∣r∣<1)S_\infty=\dfrac a{1-r}\quad(|r|<1)
59

Sum of the first n positive integers

∑r=1nr=n(n+1)2\sum_{r=1}^n r=\dfrac{n(n+1)}2
60

Sum of the first n squares

∑r=1nr2=n(n+1)(2n+1)6\sum_{r=1}^n r^2=\dfrac{n(n+1)(2n+1)}6
61

Sum of the first n cubes

∑r=1nr3=n2(n+1)24\sum_{r=1}^n r^3=\dfrac{n^2(n+1)^2}{4}

Matrices

62

Matrix for reflection in the x-axis

(100−1)\begin{pmatrix}1&0\\0&-1\end{pmatrix}
63

Matrix for reflection in the y-axis

(−1001)\begin{pmatrix}-1&0\\0&1\end{pmatrix}
64

Matrix for reflection in y = x

(0110)\begin{pmatrix}0&1\\1&0\end{pmatrix}
65

Matrix for reflection in y = −x

(0−1−10)\begin{pmatrix}0&-1\\-1&0\end{pmatrix}
66

Matrix for anticlockwise rotation through θ about O

(cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}
67

Matrix for stretches parallel to the axes

(c00d)\begin{pmatrix}c&0\\0&d\end{pmatrix}

Scale factor c parallel to the x-axis, d parallel to the y-axis.

68

Matrix for a shear with the x-axis invariant

(1k01)\begin{pmatrix}1&k\\0&1\end{pmatrix}

(0, 1) maps to (k, 1).

69

Matrix for a shear with the y-axis invariant

(10k1)\begin{pmatrix}1&0\\k&1\end{pmatrix}

(1, 0) maps to (1, k).

70

Determinant of a 2×2 matrix

det⁡(abcd)=ad−bc\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc
71

Inverse of a 2×2 matrix

M−1=1ad−bc(d−b−ca)M^{-1}=\dfrac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}

For M = [[a, b], [c, d]] and ad − bc ≠ 0.

72

Geometric significance of the determinant of a matrix

The area scale factor is |det M|; the sign of det M indicates orientation.

Polynomials

73

Vieta’s relations for ax² + bx + c = 0

α+β=−ba,αβ=ca\alpha+\beta=-\dfrac ba,\qquad\alpha\beta=\dfrac ca
74

Vieta’s relations for ax³ + bx² + cx + d = 0

α+β+γ=−ba\alpha+\beta+\gamma=-\dfrac baαβ+βγ+γα=ca\alpha\beta+\beta\gamma+\gamma\alpha=\dfrac caαβγ=−da\alpha\beta\gamma=-\dfrac da
75

Vieta’s relations for ax⁴ + bx³ + cx² + dx + e = 0

α+β+γ+δ=−ba\alpha+\beta+\gamma+\delta=-\dfrac baαβ+αγ+αδ+βγ+βδ+γδ=ca\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta=\dfrac caαβγ+αβδ+αγδ+βγδ=−da\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta=-\dfrac daαβγδ=ea\alpha\beta\gamma\delta=\dfrac ea

Complex numbers

76

Modulus of a product of two complex numbers

∣wz∣=∣w∣ ∣z∣|wz|=|w|\,|z|
77

Argument of a product of two complex numbers

arg⁡(wz)≡arg⁡(w)+arg⁡(z)(mod2π)\arg(wz)\equiv\arg(w)+\arg(z)\pmod{2\pi}