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Line
Definition
The equation of a line consists of a multiple of $x$ plus some constant: \[ f(x) = mx+b. \]
Parameters
- $m$ is the slope.
- $b$ is the $y$-intercept. The value of the function when $x=0$.
- $b/m$ is the $x$-intercept. Another word for $x$-intercept is root, i.e., a value where $f(x)=0$.
Properties
- There is a unique linear function that passes through any two points $(x_1,y_1)$ and $(x_2,y_2)$ if $x_1 \neq x_2$.
- The inverse function of $f(x)=\frac{1}{m}(x-b)$.
Graph
Here is the graph of \[ f(x) = 2x - 3. \]
A line can also be described in a more symmetric form. \[ Ax + By = C, \] This is known as the general equation of a line. We have $b=\frac{C}{B}$ and $m=\frac{-A}{B}$.