Slope of a Line (Leaving Cert Mathematics): Revision Notes

Step-by-Step Process: Calculating Slope

1. Identify your points: Label them as $(x_1, y_1)$ and $(x_2, y_2)$
2. Find the vertical change: Subtract $y_1$ from $y_2$
3. Find the horizontal change: Subtract $x_1$ from $x_2$
4. Divide: Put the vertical change over the horizontal change

Example 1: Basic Slope Calculation

Problem: Find the slope of line AB where A = (3, -1) and B = (5, 2).

Solution: Using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point A: $(x_1, y_1) = (3, -1)$
• Point B: $(x_2, y_2) = (5, 2)$

$m = \frac{2 - (-1)}{5 - 3} = \frac{2 + 1}{2} = \frac{3}{2}$

Therefore, the slope of line AB is $\frac{3}{2}$.

Example 2: Proving Perpendicular Lines

Problem: Show that line AB is perpendicular to line CD where A(-1, 0), B(3, 2), C(-1, 4), and D(2, -2).

Solution: First, find the slope of AB: $m_1 = \frac{2 - 0}{3 - (-1)} = \frac{2}{4} = \frac{1}{2}$

Next, find the slope of CD: $m_2 = \frac{-2 - 4}{2 - (-1)} = \frac{-6}{3} = -2$

Check if the product equals -1: $m_1 \times m_2 = \frac{1}{2} \times (-2) = -1$

Since the product of the slopes is -1, line AB is perpendicular to line CD.

Example 3: Identifying Slope from a Graph

When looking at a graph, you can estimate slope by:

• Picking two clear points on the line
• Counting the vertical change (up or down)
• Counting the horizontal change (left or right)
• Dividing vertical by horizontal

Remember that going up or right gives positive values, while going down or left gives negative values.