Precalculus | mathXplain

GRAPHING AND TRANSFORMING FUNCTIONS

Domain and range - Those lucky x values for which the function assigns something, are called the Domain, and the part of the y axis that is assigned to the x values is called the Image, or the Range.
Transformations - Internal and external transformations.
Shifting and reflecting - Internal and external transformations.

The unit circle - The unit circle is a circle, centered at the origin and with a radius of 1.
Initial ray - In a unit circle, the ray on the x-axis is called the initial ray.
Terminal ray - In a unit circle, the ray going through terminal point P is called the terminal ray.
Rotation angle - The rotation angle between these two rays can be positive, or negative.
Cosine - The x coordinate of terminal point P is called cosine.
Sine - The y coordinate of terminal point P is called sine.
Trigonometric functions - Let’s talk a bit about trigonometric functions, and the definitions of sine and cosine.
Periodical functions - This means they repeat themselves at certain intervals.
Trigonometric equations - A reminder on how to solve trig equations.
Quadratic trigonometric equations - A very typical case is when a quadratic equation disguises itself as a trigonometric equation.

The inverse function - Calculating the inverse.
The inverses of common functions - Let’s talk about the geometric meaning of the inverse.

Permutation - The number of permutations on a set of n elements is given by n!
k-permutation of n - Number of permutations of k items chosen from n different items.
Combination - How many ways can we choose k items out of n items?

Events - Let's start with a very simple thing. We have a die, roll it once, and see what kind of events could occur. We may roll a 1.It's also possible, that we roll a 2. Then, it is also possible that before the die stops, a meteorite hits Earth and destroys the die, along with the entire human civilization.
Independence - In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability of the other.
Mutually exclusive events - Two events are mutually exclusive if they cannot occur at the same time.
Conditional probability - A given B it answers the question of what chance does A have, if B definitely occurs.
Total probability theorem - This formula is a fundamental rule relating marginal probabilities to conditional probabilities.
Bayes-theorem - We use this theorem if we want to calculate the probability of an earlier event (B_k) in light of a later occurring event (A).

Dot product - The dot product, or sometimes inner product, is an algebraic operation that takes two vectors and turns into a single number.
Cross product - The cross product, is an algebraic operation that takes two vectors and turns into anather vector.
Dyadic product - The dyadic product, is an algebraic operation that takes two vectors and turns into a matrix.
Angle between the two vectors - To calculate the angle between the two vectors, we write their dot product using two formulas.
The equation of the line - We will develop the various forms for the equation of lines in plane and three dimensional space.
The equation of the plane - Here we will develop the equation of a plane.

Matrices - Matrices are really harmless creatures in mathematics. An nXk matrix is simply a rectangular array of numbers, arranged in n rows and k columns.
Matrix operations - Scalar multiplication, addition and multiplication.
Square and diagonal matrices - It is a square-shaped matrix with the same number of rows and columns.The diagonal matrix is a square matrix where all elements outside the main diagonal are zero.
Transpose - The transpose matrix is created by swapping the rows and the columns of the matrix

Real numbers - At every point of the number line there is a real number.
Imaginary numbers - The imaginary numbers live on the imaginary axis perpendicular to the real number line.
Complex numbers - Numbers that consist of real and imaginary parts are called complex numbers.
Operation on complex numbers - Let’s see what kind of operations we can do on complex numbers.
Complex cojugate - Geometrically, conjugation is a reflection about the real axis.
Fundamental theorem of algebra - One significant benefit of complex numbers is that using complex numbers, all polynomials can be factored into linear factors.
Absolute value of complex numbers - Let’s see what kind of operations we can do on complex numbers.
Complex plane - Complex numbers are located on the so called complex plane.
Polar form - The polar form makes it surprisingly simple to multiply and divide complex numbers.
Trigonometric form - The trigonometric form makes it surprisingly simple to multiply and divide complex numbers.