- Given two points \(P_1(x_1,y_1)\) and \(P_2(x_2, y_2)\) in the \(xy\)-plane,
\begin{align*}
\Delta x =\amp x_2-x_1\qquad \text{is called the change in x, or the ``run'',}\\
\Delta y =\amp y_2-y_1\qquad \text{is called the change in y, or the ``rise''.}
\end{align*}
If \(x_1\neq x_2\text{,}\) the line passing through the points \(P_1\) and \(P_2\) is non-vertical, and its slope is
\begin{equation*}
m=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}=\frac{\text{rise}}{\text{run}}
\end{equation*}
- If \(P_1 \neq P_2\) but \(\Delta y = 0\) then \(m=0\) and the line is horizontal. If \(P_1 \neq P_2\) but \(\Delta x = 0\) then the line is vertical, and its slope is undefined. A vertical line is not the graph of a function, since it does not pass the vertical line test.
- The equation of a line has the form \(y = m x + b\text{,}\) where \(m\) is the slope and \(b\) is the \(y\)-intercept.
- Two lines \(y = m_1 x + b_1\) and \(y = m_2 x + b_2\) are parallel if \(m_1 = m_2\text{.}\) They are perpendicular if \(m_1 = - 1/m_2\text{.}\)
- To find the equation of the line with a given slope \(m=a\) passing through a point \(P(x_1,y_1)\text{,}\) we can use the point-slope formula:
\begin{equation*}
y-y_1 = m(x-x_1).
\end{equation*}
We expand and gather terms to get an equation of the form \(y=mx + b\text{.}\)
- To find the equation of the line passing through two points \(P_1(x_1,y_1)\) and \(P_2(x_2,y_2)\text{,}\) we first determine that the slope of the line is
\begin{equation*}
m = \frac{y_2-y_1}{x_2-x_1},
\end{equation*}
and then substitute into the point-slope formula to get
\begin{equation*}
y-y_1 = \frac{y_2-y_1}{x_2-x_1} (x-x_1).
\end{equation*}
We expand and gather terms to get an equation of the form \(y=mx+b\text{.}\)
- The distance between two points \(P_1(x_1,y_1)\) and \(P_2(x_2, y_2)\) is
\begin{equation*}
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.
\end{equation*}