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Higher Formula Revision Flashcards

A few things to remember - not exhaustive !

🔄 Click or tap the card to reveal notes, worked examples and revision help videos.

Algebra

Functions

function

Domain: set of input values
Codomain: set of possible outputs
Range: set of actual outputs

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Functions

function
Reflection in the x-axis gives \(y = -f(x)\)

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Functions

function
Reflection in the y-axis gives \(y = f(-x)\)

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Functions

function
Translate in the x direction to get \(y = f(x \pm c)\)

Remember to go the opposite way to the sign:
For \((x - c)\), go right \(c\) spaces.

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Exponentials

\[ y=a^x \] \[ x=\log_a y \]

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Exponential and Logarithmic Questions

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Logarithms

Converting Between Bases
\[ \log_{a} x = \frac{\log_{b} x}{\log_{b} a} \]

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Exponential and Logarithmic Questions

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Exponentials

exponential

An ordinary exponential function always passes through:

\[ (0,1),\quad (1,a),\quad (-1,\tfrac{1}{a}) \]

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Logarithms

exponential

An ordinary logarithmic function always passes through:

\[ (1,0),\quad (a,1) \]

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Log linear graph

log linear a straight line with a logarithmic y-axis but an ordinary x-axis
\[\log y = (\log b)x + \log a\] \[y = ab^x\]

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Log log graph

log linear
\[\log y = b\log x + \log a\] \[y = ax^b\]

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Recurrence Relation

A linear recurrence relation is of the form:

\[ u_{n+1} = au_n + b \quad , a \ne 0\]

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Sequences Questions

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Recurrence Relation

If the sequence converges to a limit \(L\), then:

\[ L = aL + b \] \[ L(1 - a) = b \] \[ L = \frac{b}{1 - a} \]

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Section Formula

section formula
\[ \mathbf{p} = \frac{n}{m+n}\,\mathbf{a} + \frac{m}{m+n}\,\mathbf{b} \]

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Vector Questions

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Vectors

vector scalar dot product

Dot Product

\[ \mathbf a\cdot\mathbf b = |\mathbf a| |\mathbf b| \cos\theta \]

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Vector Questions

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Vectors

Component Form of Dot Product

\[ \text{If }\mathbf{a} = \begin{pmatrix} a_1 \\[4pt] a_2 \\[4pt] a_3 \end{pmatrix} \quad\text{and}\quad \mathbf{b} = \begin{pmatrix} b_1 \\[4pt] b_2 \\[4pt] b_3 \end{pmatrix} \] \[ \text{then }\mathbf{a}\cdot\mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \] \[ \text{(From the Cosine Rule)} \]

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Vectors

Angle Between Vectors

\[ \cos\theta = \frac{\mathbf a\cdot\mathbf b} {|\mathbf a||\mathbf b|} \]

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Calculus

Basic Differentiation

\[ \text{If } f(x)=x^{n}, \] \[ \text{then } f'(x)=n\,x^{\,n-1}, \] \[ \text{where } n \text{ is a rational number.} \]

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Basic Differentiation

\[ \text{If } f(x)=a\,x^{n}, \] \[ \text{then } f'(x)=a\,n\,x^{\,n-1}, \] \[ \text{where } a \text{ is a constant,} \] \[ \text{and } n \text{ is a rational number.} \]

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Differentiation

\[ f(x)=\sin(ax) \] \[ f'(x)=a\cos(ax) \]

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Differentiation

\[ f(x)=\cos(ax) \] \[ f'(x)=-a\sin(ax) \]

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Chain Rule

\[ h'(x)=g'\!\big(f(x)\big)\,\cdot\,f'(x) \]

or in Leibniz notation:

\[ \frac{dy}{dx} = \frac{dy}{du} \;\times\; \frac{du}{dx} \]

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Differentiation

\[ \text{If } f(x) = (x+a)^{\,n} \] \[ \text{then } f'(x) = n(x+a)^{\,n-1} \] \[ \text{where \(n\) is a rational number and \(a\) is real.} \]

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Differentiation

\[ \text{If } f(x) = (ax + b)^{\,n} \] \[ \text{then } f'(x) = a n (ax + b)^{\,n-1} \] \[ \text{where \(n\) is a rational number and \(a\) is real.} \]

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Basic Integration

\[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \qquad (n \ne -1) \]

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Integration

\[ \int (ax + b)^{n}\,dx = \frac{(ax + b)^{\,n+1}}{a(n+1)} + c \qquad (n \neq -1) \]

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Integration

\[ \int \sin x\,dx = -\cos x + C \]
\[ \int \sin(ax + b)\,dx = -\frac{1}{a}\cos(ax + b) + c \]

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Integration

\[ \int \cos x\,dx = \sin x + C \]
\[ \int \cos(ax + b)\,dx = \frac{1}{a}\sin(ax + b) + c \]

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Integration Questions

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Integration

area \[ \int_{a}^{b} f(x)\,dx \quad\text{is called a definite integral.} \]
\[ \int_{a}^{b} f(x)\,dx = F(b) - F(a) \quad (a \le x \le b) \]

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Integration

area
When calculating the area enclosed by a graph and the x-axis:
  • Always draw a sketch.
  • Calculate areas above and below the x-axis separately.
  • Ignore negative signs and add.

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Integration Questions

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Integration

area
\[ \text{Area between curves} = \int_a^b (\text{upper}-\text{lower})\,dx \]

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Geometry

Circle Equation

\[ x^2 + y^2 = r^2 \]

Radius: \(r\)

Centre: \((0,0)\)

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Circle Equation

\[ (x-a)^2 + (y-b)^2 = r^2 \]

Radius: \(r\)

Centre: \((a,b)\)

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General Equation of a Circle

\( x^2+y^2+2gx+2fy+c=0 \)
where \(g\), \(f\) and \(c\) are constants.

Radius: \[ \sqrt{g^2+f^2-c} \] provided that \( g^2+f^2-c>0 \)

Centre:\( (-g,-f) \)

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Touching Circles

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Touching circles externally

\[ AB=r_1+r_2 \]

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Touching Circles

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Touching circles internally

\[ AB=r_1-r_2 \]

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Distance Formula

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\[ d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]

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Straight Line Questions

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Midpoint Formula

Trig gradient
\[ \text{The midpoint of } AB \text{ is} \] \[ M\!\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

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Perpendicular Lines

perpendicular lines \[ m_1m_2=-1 \]

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Gradient

Trig gradient
\[ \text{Gradient} = \tan \theta \]

where \( \theta \) is measured anticlockwise from the x‑axis.

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Tangency

If \(b^2 - 4ac = 0\), the line is a tangent.
If \(b^2 - 4ac > 0\), the line cuts the curve at two distinct points.
If \(b^2 - 4ac < 0\), the line does not touch the curve.

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Circle Questions

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Trigonometry

Radians and Degrees

\[ 180^\circ = \pi \text{ rads} \qquad \text{Divide degrees by } 180^\circ \text{, then multiply by } \pi \text{ rads.} \]
\[ \pi \text{ rads} = 180^\circ \qquad \text{Multiply rads by } 180^\circ \text{, then divide by } \pi. \]

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Exact Values : Radians

trig exact values

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Wave Theory

trig exact values

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Compound Formulae

\[ \sin(A-B) = \sin A\cos B-\cos A\sin B \]
\[ \sin(A+B) = \sin A\cos B+\cos A\sin B \]

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Compound Formulae

\[ \cos(A+B) = \cos A\cos B-\sin A\sin B \]
\[ \cos(A-B) = \cos A\cos B+\sin A\sin B \]

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Compound Formulae

\[ \sin 2A = 2\sin A\cos A \]
\[\cos 2x = \cos^2 x - \sin^2 x\] \[ = 1 - 2\sin^2 x\] \[ = 2\cos^2 x - 1 \]

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