1. Understanding the Slope Between Two Points
When two points on a line are known, we can find the slope by comparing how much the line moves upward or downward relative to how much it moves horizontally. This helps us understand the direction and steepness of the line.
The slope is always calculated as the change in the y-values divided by the change in the x-values.
1.1. Definition
The slope between two points is the ratio of the difference in y-coordinates to the difference in x-coordinates.
2. Slope Formula Between Two Points
If the two points are \((x_1, y_1)\) and \((x_2, y_2)\), the slope of the line joining them is:
2.1. Formula
\( m = \dfrac{y_2 - y_1}{x_2 - x_1} \)
This formula finds the vertical change compared to the horizontal change.
2.2. Meaning of the Formula
- Numerator (y₂ − y₁): rise (vertical change)
- Denominator (x₂ − x₁): run (horizontal change)
- The slope may be positive, negative, zero, or undefined depending on the points.
3. Interpreting the Sign of the Slope
The sign of the slope gives important information about the direction of the line.
3.1. Slope Interpretation
- Positive slope: line rises to the right
- Negative slope: line falls to the right
- Zero slope: horizontal line
- Undefined slope: vertical line
4. Examples
These examples show how the slope formula is applied in different cases.
4.1. Example 1: Positive Slope
Find the slope between \((2, 3)\) and \((5, 9)\).
\( m = \dfrac{9 - 3}{5 - 2} = \dfrac{6}{3} = 2 \)
4.2. Example 2: Negative Slope
Find the slope between \((4, 7)\) and \((10, 1)\).
\( m = \dfrac{1 - 7}{10 - 4} = \dfrac{-6}{6} = -1 \)
4.3. Example 3: Zero Slope
Points: \((3, 5)\) and \((11, 5)\)
\( m = \dfrac{5 - 5}{11 - 3} = 0 \)
The line is horizontal.
4.4. Example 4: Undefined Slope
Points: \((6, 2)\) and \((6, -4)\)
\( m = \dfrac{-4 - 2}{6 - 6} = \dfrac{-6}{0} \)
The slope is undefined because the denominator is zero.