Preliminary Notes:
Types of numbers: real numbers - whole, rational, irrational and unreal numbers and unreal numbers - imaginary numbers, etc.
Standard form + scientific notation - 0.0015 → 1.5 x 10^-3
Symbol R - Set of real numbers
Direct Proportion:
Things can be in direct proportion.
y=kx
"For two quantities that are in direct proportion, as one quantity is multiplied by a certain number then the other quantity is also multiplied
Inverse Proportion:
if x and y are inversely proportional the relationship will have an equation of the form
$y=k/x$
Graphing of Inverse Proportion (also known as hyperbola)
Function notation:
"Given the rule y = 3x - 1 and a particular value of x, say 5, we can determine the corresponding value of y, in this case 14."
"In mathematics any rule that takes any given input value and assigns to it a particular output value is called a function."
Notation: $f(x)$
Chapter 1: Trigonometry
Subtopics:
Unit Circle:
The sine of the angle that AO makes with the positive x-axis is given by the y-coordinates of A.
The cosine of the angle AO makes with the positive x-axis is given by the x-coordinate of A.
Here, the point represents cosine value and sine value, with cosine being the x and sine being the y.
IMPORTANT:
Given A is obtuse angle,
$sinA = sin(180°- A)$
Given A is obtuse angle,
$cosA = -cos(180°-A)$
Given A is obtuse angle,
$tanA = -tan(180 - A)$
Area of triangle given 2 sides + angle between them:
$Area = absinC/2$
The Sine Rule:
The Cosine Rule:
$c^2 =a^2+b^2-2abcos(C)$
Exact Values:
Memorise later, what you need to know is that they can be derived from 1 right triangles:
1 where the hypotenuse is 2 and one side is 1 unit. The angle where the hypotenuse and the side of 1 unite meet is 60 degrees. The other angle that is not 60 is 30 degrees.
2 where both sides that aren't the hypotenuse are 1. This creates 45 degrees.
Chapter 2: Radian measure:
Note: Radians refer to a cooler way to measure angles. It derived from the idea that a radius, given the circumference of a circle $2\pi r$ can be lined across the entirety of the circumference $2 \pi$ times, this full encompassing of the circle, which would be a revolution in degrees (360$\degree$), is $2 \pi$ in radians.
In general,
$$ 360\degree = 2 \pi \ Radians $$
$\therefore$
Degrees → Radians:
$x\degree = \frac{x\pi}{180}$
Radians → Degrees:
$x \ Radians= \frac{180x}{\pi}$
Arcs, sectors and segments formulas:
Given radius of circle (r) and angle at centre (MEASURED IN RADIANS)
$Arc length = rθ$
$Sector area = 1/2 r^2θ$
$Segmentarea=1/2r^2(θ-sinθ)$
Chapter 3: Functions:
Rules of a function:
3 Potential Types of Functions:
Many to One function: numerous elements from the domain can result in a single element of the range. e.g. f(x) = x^2, this will create numerous many-to-win scenarios, e.g. if range = 4, then the domain could be -2, or 2 (sqrt(4) = +-2)
One to One function: self explanatory, e.g. f(x) = x+1
One to Many function: ONLY WORKS when all elements = ONLY 1 element of the range, btw this isnt a function
If a vertical line is moved from the left of the domain to the right it must never cut the graph in more than one place
Similar horizontal line test to determine whether a function is a one to one function or not.
GENERAL RULE OF FUNCTIONS:
e.g: $f(x) = \sqrt{x}$ : Natural domain is ≥ 0 as if it were negative then you would create a complex number (i.e i (sqrt(-1)), which is NOT a real number)
Otherwise, natural domain + range SHOULD be: {x ∈ R}
FORMAT OF NATURAL DOMAIN AND RANGE:
({x ∈ R}) + : + limitations, e.g. ≤,≥,≠, etc.
e.g. Given $f(x)=sqrt(x)$
{x ∈ R: x ≥ 0}
Chapter 5: Quadratic Functions:
Information about a parabola/quadratic function on a graph:
The graph is symmetrical with the line of symmetry (x value of the turning point)
It can either be:
for larger and larger positive x values, y takes even larger positive values. We can say that:
"as x approaches infinity then y approaches infinity"
In other words, $As \ x \rightarrow \infin \ then \ y \rightarrow \infin$
Similarly, $As \ x \rightarrow -\infin \ then \ y \rightarrow \infin$
All quadratics that are considered "functions" can be written in the form:
$y=ax^2+bx+c$ for $a \neq 0$
