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Mom's Basement University

Slope in the Coordinate Plane

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn what slope means, how to calculate it between two points, and how to interpret positive, negative, zero, and undefined slopes.

The Concept

Section titled “The Concept”

Slope (m) measures how steep a line is and whether it rises or falls.

Formula for slope between two points (x₁, y₁) and (x₂, y₂):

m=y2−y1x2−x1=riserunm = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}

What different slopes mean:

  • Positive slope (> 0): Line rises from left to right (uphill).
  • Negative slope (< 0): Line falls from left to right (downhill).
  • Zero slope (m = 0): Horizontal line.
  • Undefined slope: Vertical line (division by zero).

Slope is the same anywhere on a straight line.

Worked Example

Section titled “Worked Example”

1. Find the slope between points (2, 3) and (5, 9).

m=9−35−2=63=2m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2

Positive slope of 2 (rises 2 units for every 1 unit right).

2. Find the slope between points (−1, 4) and (3, −2).

m=−2−43−(−1)=−64=−1.5m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -1.5

Negative slope (falls 1.5 units for every 1 unit right).

3. Points (0, 5) and (0, 8).

m=8−50−0=30m = \frac{8 - 5}{0 - 0} = \frac{3}{0}

Undefined slope - vertical line.

Real-World Application

Section titled “Real-World Application”

Slope helps describe real situations:

  • Road grade or hill steepness (a 10% grade means slope = 0.1).
  • Roof pitch in construction (rise over run).
  • Ramps for accessibility (maximum allowed slope).
  • Budget lines or cost graphs (slope = cost per unit).
  • Navigation: How steeply a trail climbs.

Example: A road rises 300 feet over 1 mile (5,280 feet). Slope = 300/5,280 ≈ 0.057 (about a 5.7% grade).

Quiz

Section titled “Quiz”
What is the slope between points (1, 2) and (4, 8)?
A line with negative slope does what from left to right?
The slope of a horizontal line is:
Points (3, 5) and (3, 9) have what type of slope?
Find the slope between $(-2, 3)$ and $(4, -9)$.