The slope of a line measures how steeply it rises or falls as you move from left to right. When a line makes an angle θ with the positive x-axis, that steepness is directly connected to the tangent of the angle. This connection matters because it links algebra, geometry, and trigonometry in one simple idea.
It also helps students interpret graphs, ramps, motion diagrams, and linear models.
Key Facts
- Slope is the ratio of vertical change to horizontal change: m = rise/run.
- For a line making angle θ with the positive x-axis, tan θ = rise/run.
- Because both equal rise/run, the slope of a line is m = tan θ.
- If the slope is known, the angle can be found with θ = arctan(m).
- A horizontal line has θ = 0° and slope m = 0 because tan 0° = 0.
- A vertical line has undefined slope because run = 0, and tan 90° is undefined.
Vocabulary
- Slope
- Slope is the ratio of vertical change to horizontal change along a line.
- Rise
- Rise is the vertical change between two points on a graph.
- Run
- Run is the horizontal change between two points on a graph.
- Tangent
- Tangent is a trigonometric ratio equal to opposite side divided by adjacent side in a right triangle.
- Arctangent
- Arctangent is the inverse tangent function used to find an angle from a tangent value or slope.
Common Mistakes to Avoid
- Using run/rise instead of rise/run is wrong because slope and tangent both compare vertical change to horizontal change.
- Forgetting the sign of the slope is wrong because a line rising left to right has positive slope, while a line falling left to right has negative slope.
- Using degrees when the calculator is in radians is wrong because arctan gives different-looking angle values depending on the angle mode.
- Treating a vertical line as having slope 0 is wrong because its run is 0, so rise/run is division by zero and the slope is undefined.
Practice Questions
- 1 A line rises 6 units while running 8 units to the right. Find its slope, then find the angle θ it makes with the positive x-axis to the nearest degree.
- 2 A line has slope m = 1.5. Use θ = arctan(m) to find the angle it makes with the positive x-axis to the nearest degree.
- 3 Two lines have slopes 0.5 and 3. Explain which line is steeper and how the tangent function supports your answer.