17.1 - Lines (analytic geometry)

Links and useful resources

Refresher: Completing the Square
Refresher: Graphing functions
The graph of an equation is the shape made by drawing in all points that satisfy the equations - mark all places where the coordinates result in a true equation.

A line equation always:

Has powers of 1 for all variables
Can be written in the form (except for the vertical line $x = c$ , which is not a function)
- $(y - a) = m (x - b)$ (point slope, where the point $(a, b)$ is on the line and the slope is $m$
- $y = m x + b$ (slope-intercept, the slope is $m$ , and the line crosses the $y$ -axis at $b$ )

Starting with two points $a = (x_{a}, y_{a})$ and $b = (x_{b}, y_{b})$ on the line, you can find the equation of the line with these steps:

Find the slope: remember, $slope = \frac{rise}{run}$ :
1. $m = \frac{y_{b} - y_{a}}{x_{b} - x_{a}}$
Write the point-slope equation: $(y - y_{b}) = m (x - x_{b})$

Find an equation whose graph is the line passing through the points $(- 1, 7)$ and $(- 5, - 3)$
Find an equation whose graph is the line with slope 4 and $y$ -intercept $(0, - 3)$ .
- (a) Find the length of the segment with endpoints $(- 4, - 3)$ and $(6, 1)$ .
- (b) Find a formula for the distance between the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .
Let point $A$ be $(x_{1}, y_{1})$ and point $B$ be $(x_{2}, y_{2})$ . Let $M$ be the point with coordinates $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) .$ . In this problem we show that $M$ is the midpoint of $\overset{―}{A B}$ .
- (a) Show that $A M = M B$ .
- (b) Show that $M$ is on $\overset{\leftrightarrow}{A B}$ .
Explain why two different lines that have the same slope are parallel.