To represent the slope of a line mathematically, we typically use the formula for slope, which is defined as the change in the vertical direction (rise) divided by the change in the horizontal direction (run). The formula can be expressed as:
\[ m = \frac{\Delta y}{\Delta x} \]
where:
- \( m \) is the slope of the line,
- \( \Delta y \) is the difference in the y-coordinates of two points on the line (the rise),
- \( \Delta x \) is the difference in the x-coordinates of the same two points (the run).
Drawing a Slope Triangle
To visualize the slope, you can think of two points on the line: Point A \((x_1, y_1)\) and Point B \((x_2, y_2)\). To create a slope triangle:
- Draw a horizontal line from Point A to Point B. This represents the run, \( \Delta x \).
- Draw a vertical line from Point A up or down to meet the level of Point B. This represents the rise, \( \Delta y \).
- The slope of the line is then displayed as the ratio of the vertical leg (rise) to the horizontal leg (run) of this triangle.
Expression of Slope
If you plug the points into the slope formula, the expression for the slope between Point A and Point B can specifically be written as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Explanation of the Expression
This expression captures the essence of slope:
- If both \( y_2 \) and \( y_1 \) increase (i.e., the line goes up), the slope is positive, indicating the line rises as you move from left to right.
- If \( y_2 \) is less than \( y_1 \) (i.e., the line goes down), the slope is negative, indicating the line falls as you move from left to right.
- If either of the coordinates \( x_2 \) or \( x_1 \) is the same while \( y \) values differ, the slope is undefined (the line is vertical).
In conclusion, the expression \( m = \frac{y_2 - y_1}{x_2 - x_1} \) effectively represents the slope of the line as it quantifies the steepness and direction of the line based on two points on it.