Slope as rate of change, slope-intercept form, point-slope form, standard form, intercepts, and horizontal and vertical lines.
Slope
The slope of a line measures how steep it is: how much \(y\) changes for each unit \(x\) increases.
\(m > 0\): the line rises from left to right.
\(m < 0\): the line falls.
\(m = 0\): the line is horizontal.
A vertical line has undefined slope (the run is \(0\)).
Three forms of a linear equation
Horizontal and vertical lines
A horizontal line through \((a, b)\) has equation \(y = b\) (slope \(0\)).
A vertical line through \((a, b)\) has equation \(x = a\) (undefined slope).
The \(x\)-axis is \(y = 0\) and the \(y\)-axis is \(x = 0\).
Slope as a rate of change
In real situations, slope is a rate. A plumber who charges 60 dollars for the visit plus 45 dollars per hour earns \(y = 45x + 60\) dollars for \(x\) hours: the slope \(45\) is dollars per hour, and the intercept \(60\) is the fixed fee.
Practice
Quick check
What is the slope of a vertical line?
The run is \(0\), and you can't divide by zero.
Quick check
What is the slope of the line through \((1, 2)\) and \((3, 2)\)?
The rise is \(0\), so the line is horizontal.
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