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.

Worked Examples

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Example 1: Find the Slope of a Line

Problem: Find the slope of the line passing through the points $A(2, 3)$ and $B(5, 7)$

Solution:

Using the slope formula:

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

Substitute $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (5, 7)$

m = \frac{7 - 3}{5 - 2} = \frac{4}{3}

Answer: The slope is $\frac{4}{3}$

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Example 2: Interpret the Slope of a Line

Problem: A car travels 100 km in 2 hours.

Represent the car's journey as a graph of distance vs. time and find the slope.

Solution:

Step 1: Consider the points $(0, 0)$ (start point) and $(2, 100)$ (end point).

Step 2: Calculate the slope:

m = \frac{100 - 0}{2 - 0} = \frac{100}{2} = 50

Interpretation: The slope of $50$ represents the car's speed, $50$ $km$ per hour.

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.

Slope Simplified Revision Notes for Leaving Cert Mathematics