Mathematics Model Descriptions


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Linear equation are the simplest form of equation you may deal with; it restricts the problem to a unique unknown variable (usually called x) and no exponent (e.g. f(x) = 2x + 15).

The general form of such equation is: $$\textbf{f(x) = ax + b}$$

Any linear expression can be drawn as a line in a 2D graphic. The graph drawing is very simple: we have y, or f(x), on the vertical axis (the result) and for a range of x values (horizontal abscisse) we compute and plot the points (x, y).

In brief:
Changing a is changing the slope of the line.
Changing b is changing the offset of the line.


Quadratic equation are just a step further from linear equation, we still restrict the problem to a unique unknown variable (usually called ‘x’) and add one term: x² (or x*x). The general form of such equation is: $$\textbf{f(x) = ax² + bx + c } ,\: \: where\: \: a \neq 0$$

They will frequently turn up in many areas and very often make an appearance as part of the overall solution within most of the real world problems in the fields of physics, astronomy, engineering, computing, architecture...

Let's play with it online and we will quickly see that:
Changing a is changing the opening of the parabola.
Changing b is changing the slope of the parabola at x = 0.
Changing c is changing the offset of the parabola.


Just like transformations in geometry, we can translate / shift a mathematical object by changing its function.

Already seen with the previous models (linear, quadratic), we can move it up or down by adding a constant to the function:
$$f(x) → f(x) + b$$

To move it left or right, we add a constant to the function variable (x-value):
$$f(x) → f(x + a)$$ We can think of this as moving the "abscissa origin" to be more in advance or a bit late.

Putting it together:
$$\textbf{f(x) → f(x + a) + b}$$

Why adding a positive number a moves the function to the left (the negative direction)?

Well imagine you want to start recording a movie at 8:00 o’clock with the function:
Start(t) = 8:00 o’clock.

If you change your mind to say that we want to record 5 minutes before, we will have the function:
Start(t + 5 mins) = 8:00 o’clock

Adding 5 minutes to the current time will make the record starting 5 minutes earlier (i.e. to the left direction).

Scaling (Stretch / Compress)

If we understand the Linear and Translate functions above, we may imagine what could be the scaling function : instead of playing with additions, we will play with multiplication factors to stretch or compress our mathematical object. Please note that a scale is a non-rigid transformation : it alters the shape and size of the graph function.

We can stretch or compress it in the y-direction by multiplying the whole function by a constant:
$$f(x) → a * f(x)$$

We can stretch or compress it in the x-direction by multiplying the function variable x by a constan:
$$f(x) → f(b * x)$$

Putting it together (both are the same expression):
$$\textbf{f(x) → a * f(b * x)}$$

I’m sure you can now imagine why bigger b value cause more compression on the x-scale.
We could say : we put more information within the same base unit.

Inverse - Anti-function - Reverse - Reflect

Commonly written f-1(x), they are extremely useful to solve equations and are therefore a very interesting tool to express several concepts. They allow mathematical operations to be reversed (e.g.minus inverses sum, multiplication reverses division, logarithms inverses exponential etc.). Whenever a mathematical procedure is introduced, one of the most important questions is how to invert it.

Two functions are inverses of one another if they "undo each other" in the following sense: if the output of one is used as input to the other, it results to the original input.

We all naturally know a tons of basic functions and their inverse, here is some you may have been across:

The graph of f-1(x) can be obtained from the graph of f by switching the positions of the x and y axes : this is equivalent to reflecting the graph across the line y = x.

The cool thing about the inverse is that we are able to get back to the original value only with the result. When the function f(door) turns for instance an open door into a close one, then the inverse function f-1(door) turns it back to open. Thoses equations express the same idea:

$$f(x) = y \iff g(y) = x$$ $$f(g(x))=x \quad and \quad g(f(x))=x$$ $$f^{-1}( f(x) ) = x \quad and \quad f( f^{-1}(x) ) = x$$




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