Lines and Quadratics
3.1 Lines
Slope as Rate
A line is defined by constant rate of change.
Definition: Slope
For points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_1\ne x_2$,
$$m=\frac{y_2-y_1}{x_2-x_1}. $$
Example: Find slope from two points
Find the slope through $(1,3)$ and $(5,-1)$.
Show Solution
$$m=\frac{-1-3}{5-1}=\frac{-4}{4}=-1.$$
Definition: Common line forms
- Slope-intercept form: $y=mx+b$
- Point-slope form: $y-y_1=m(x-x_1)$
Example: Write an equation of a line
Find the equation of the line with slope $3$ through $(2,-4)$.
Show Solution
Use point-slope:
$$y-(-4)=3(x-2)\Rightarrow y+4=3x-6\Rightarrow y=3x-10.$$
MyOpenMath: Practice: slope and line equations (placeholder)
Parallel and Perpendicular Lines
Slope relationships let us classify line orientation quickly.
Fact: Slope relationships
- Parallel lines have equal slopes.
- Perpendicular lines have slopes that are negative reciprocals.
Example: Parallel and perpendicular equations
Find a line parallel to $2x-y=7$ through $(0,3)$.
Show Solution
Rewrite given line:
$$2x-y=7\Rightarrow y=2x-7,$$
so slope is $2$.
Parallel line through $(0,3)$ has slope $2$:
$$y-3=2(x-0)\Rightarrow y=2x+3.$$