11. Slope on the Coordinate Plane

Learning goals

By the end of this lesson, you should be able to:

• calculate slope from two points using a consistent subtraction order;
• interpret positive, negative, zero, and undefined slopes; and
• recognize parallel and perpendicular line relationships.

Slope measures how steep a line is -- how much it rises for each step to the right. It is "rise over run":
\[ \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}. \]

Like distance, slope uses the differences of the coordinates -- but it divides them instead of squaring and adding.

Worked example: the slope through \((1, 2)\) and \((4, 8)\) is
\[ \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2. \]
The line rises 2 for every 1 it moves right.

Reading slope:

• Positive slope rises left to right; negative slope falls.
• A horizontal line has slope \(0\); a vertical line has an undefined slope.

Negative slope and units

For points \((-1,5)\) and \((3,-3)\):
\[ m=\frac{-3-5}{3-(-1)}=\frac{-8}{4}=-2. \]
The line falls 2 vertical units for every 1 unit moved to the right.

In context, slope is a rate with units. If \(y\) is dollars and \(x\) is hours, a slope of 18 means $18 per hour. If a tank's water level has slope -3 centimeters per minute, it decreases 3 centimeters each minute.

Special lines

A vertical line's slope is undefined because its run is zero, and division by zero is not defined.

Parallel and perpendicular lines

• Parallel nonvertical lines have the same slope.
• Perpendicular nonvertical lines have negative reciprocal slopes. For example, a slope of \(\frac{2}{3}\) is perpendicular to \(-\frac{3}{2}\).

These relationships help classify figures and check whether streets, ramps, or structural lines meet at right angles.

Common mistake: flipping the formula to "run over rise," or subtracting the x's and y's in a different order. Keep the same point first in both the top and bottom.