The simplest of functions is the straight line.

As you define a function to be a constant you get a horizontal straight line
$y = c$

As well, line equations are those that follow the formula

y = c + m·x

this means that when a factor $x$ changes, $y$ changes also proportionally, by a slope factor of $m$. It also begins at $c$ when $x=0$

We can calculate the slope, or inclination, as:

m = \frac{rise}{run} = \frac{y_1-y_0}{x_1-x_0} 

Point-slope equation of a line

Knowing a point $P(x_0, y_0)$ and the slope $m$. A point $P(x,y)$ is in the line if

m = \frac{y-y_0}{x-x_0} 

And therefore the equation can be expressed as:

y-y_0 = m(x-x_0) 

Which rearranges into

y = m·x - m·x_0 + y_0 = c + m·x

with $c$ being the point where the $x=0$.

Vertical Lines

As well as horizontal lines at a point $(a,b)$ is $x=a$ then vertical lines at that point is $y=b$

General Equation of a Lineal Equation

L: ⠀Ax + By + C = 0

Where $A$, $B$, and $C$ are constant.

Parallel Lines

Two Lines are parallel if they have the same slope

L_1: m_1 · x + c_1 

L_2: m_2 · x + c_2 

L_1 ∥ L_2 ⟺  m_1 = m_2 

Perpendicular Lines

Two Lines are perpendicular if they have inverse slopes

L_1 ⟂ L_2 ⟺  m_1 = - \frac{1}{m_2} = - m̈_2  ⟷  m_1·m_2 = -1

Linear Lines

It is worth mentioning that "Linear" and "Line" are not always synonyms. There is a general understanding by common sense that they are, but Linear means something specific in the formalities of Mathematics.

• That two points in the line can be added together and fall into the line.
  • $P_1 + P_2 = P_c$
• That a point can be scaled and fall into this same line.
  • $k·P_0 = P_k$
  • This implies that the Origin is always a point in the line. As $k=0$

By this measure, then, the formula for linear line is:

y = k·x