Lesson 6 — Slope
Big idea: Slope measures steepness as rise over run, and it tells you immediately whether two lines are parallel or perpendicular.
Words to know
| Word | Meaning |
|---|---|
| Rise | The vertical change between two points on a line. |
| Run | The horizontal change between two points on a line. |
| Slope | The ratio of rise to run; usually written m. |
| Angle of inclination | The angle a line forms with the horizontal. |
| Undefined slope | The slope of a vertical line, because the run is 0. |
| Opposite reciprocals | Two numbers whose product is −1, such as 2/3 and −3/2. |
Concepts to know
- Slope = rise ÷ run.
- On a coordinate grid, slope = (y₂ − y₁) ÷ (x₂ − x₁); either point may be chosen first, as long as you subtract in the same order.
- A horizontal line has slope 0; a vertical line has undefined slope.
- Theorem 52: Two nonvertical lines are parallel iff their slopes are equal.
- Theorem 53: Two nonvertical lines are perpendicular iff the product of their slopes is −1.
- The angle of inclination is related to slope by the tangent ratio: tan θ = slope.
Formulas / rules
- m = rise ÷ run
- m = (y₂ − y₁) ÷ (x₂ − x₁)
- Parallel lines: m₁ = m₂
- Perpendicular lines: m₁ · m₂ = −1, or m₂ = −1/m₁
- tan θ = m (for a line rising from left to right), so θ = tan⁻¹(m)
- Point-slope equation: y − y₁ = m(x − x₁)
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find slope from a graph | Count rise and run between two clear points; divide. |
| Find slope from two points | Substitute into (y₂ − y₁)/(x₂ − x₁) and simplify. |
| Decide whether lines are parallel | Compare slopes; equal slopes mean parallel. |
| Decide whether lines are perpendicular | Multiply the slopes; a product of −1 means perpendicular. |
| Find a missing coordinate | Put the known slope and points into the slope formula and solve. |
| Find the angle of inclination | Take the inverse tangent of the slope. |
| Write a line’s equation | Use point-slope form with a point and the slope. |
Watch out
- Subtract coordinates in the same order on top and bottom.
- A vertical line’s slope is undefined, not 0; a horizontal line’s slope is 0.
- Perpendicular slopes are negative reciprocals, not just reciprocals: 2/3 and −3/2.
- A product of −1 is required; a product of 1 is not perpendicular.
- The angle of inclination is measured from the horizontal, not from the vertical.