1.10: Lines
In Calculus I, you study something called a derivative.
In simple terms, the derivative is the slope of particular type of line.
Let's talk about slopes.
The slope of a line that runs through two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is \[m = \text{slope} = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}\]
The slope of a line that runs through two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is \[m = \text{slope} = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}\]
There are four possible slopes, depending on the rise:
Point-Slope Form of a Line
To define a line, you only need one point and a slope.
The point-slope form of a line is an equation of a line that passes through $(x_1, y_1)$ and has slope $m$: \[y - y_1 = m(x - x_1)\]
Find the equation of the line through $(1, -3)$ with slope $-\dfrac{1}{2}$.
Find the equation of the line that passes through $(-1, 2)$ and $(3, -4)$. Sketch a graph of the line.
Vertical and Horizontal Lines
- The vertical line through $(a, b)$ is $x = a$.
- The horizontal line through $(a, b)$ is $y = b$.
Graph the equation $y = 2$ and $x = -3$.