Pre-Calculus

Stuff to know when delving into Calculus I.

Lines

Lines connects at least two points and they don’t curve.
They have a slope.

Any points drawn into space have two coordinates $(x,y)$, see $fig.1$
The slope is calculated as:

$$slope=\frac{rise}{run}$$

Where the $rise$ is the difference in height between two points in the line ($y_2-y_1$) and the $run$ is instead the difference of the values of $x_2-x_1$.
So we have:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

we fix only $x_1$ and $y_1$ and let the other point free:

$$m=\frac{y-y_1}{x-x_1}$$

we can see that this is equivalent too:

$$m(x-x_1)=y-y_1$$

By re-ordering it we obtain the Point-Slope equation:

$$y-y_1=m(x-x_1)$$

This can be used to obtain the equation of a line in the classical form of:

$$y=mx+q$$

where $m$ is the slope and $q$ is the intercept.

Note: a equation of a line in the form of $y=c$ where $c$ is constant equals to an horizontal line that crosses the $y$-axis with a intercept of $c$.

Parallel Lines have the same exact slope as one another.