To represent the slope of a line, we can use the formula:
\[ \text{slope (m)} = \frac{\text{rise}}{\text{run}} \]
Step 1: Understand the Terms
- Rise: The vertical change between two points on the line (how much you go up or down).
- Run: The horizontal change between those same two points (how far you go left or right).
Step 2: Create a Slope Triangle
Imagine you have a coordinate plane and a straight line that passes through two points, (x₁, y₁) and (x₂, y₂). You can visualize a right triangle (called a slope triangle) to help find the rise and run.
- Start at the first point (x₁, y₁).
- Move vertically to the second point (x₂, y₂) to find the rise (this is the vertical side of the triangle).
- Then move horizontally from that point to find the run (this is the horizontal side of the triangle).
Step 3: Calculate Rise and Run
- Rise = \( y₂ - y₁ \) (the change in y)
- Run = \( x₂ - x₁ \) (the change in x)
Step 4: Write the Expression
So, you can write the expression for the slope of the line as:
\[ m = \frac{y₂ - y₁}{x₂ - x₁} \]
Why This Expression Represents the Slope
This expression gives you the rate of change of y with respect to x, which is what the slope tells you. It shows how much the y-value increases (or decreases) when the x-value increases by 1 unit. A larger absolute value of the slope means the line is steeper, while a slope of 0 means the line is flat (horizontal).
Conclusion
By using the rise over run formula, you're calculating how steep the line is, and this helps to understand how two variables are related in a linear relationship.