Slope (Leaving Cert Mathematics): Revision Notes

Slope

What is Slope?

The slope of a line measures its steepness and direction in the Cartesian plane. It is a numerical representation of how much the line rises or falls for a given horizontal distance. Slope is commonly denoted by the letter $m$.

Slope Formula

For a straight line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is calculated as:

$$m = \frac{y_2 - y_1}{x_2 - x_1}, \quad x_1 \neq x_2$$

Interpreting Slope

• Positive Slope: The line rises as it moves from left to right.
• Negative Slope: The line falls as it moves from left to right.
• Zero Slope: The line is horizontal.
• Undefined Slope: The line is vertical (division by zero).

Slope as a Rate of Change

In real-life contexts, the slope represents the rate of change. For example:

• In a graph of distance vs. time, the slope represents speed.
• In a cost vs. quantity graph, the slope shows the cost per unit.

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Worked Examples

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Summary

• Slope describes the steepness and direction of a line:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• A positive slope means the line rises; a negative slope means it falls.
• Zero slope indicates a horizontal line; undefined slope indicates a vertical line.
• Slope represents rate of change in real-world problems.
• Practice calculating and interpreting slope for deeper understanding.