1. What Are the Types of Slopes?
The slope of a line tells us how the line is slanted on the Cartesian plane. Depending on the direction in which the line moves, the slope falls into one of four types: positive, negative, zero, or undefined.
Each type describes how the line behaves when moving from left to right.
1.1. Definition of Slope
Slope \((m)\) is the ratio of the change in y to the change in x between any two points on a line:
\( m = \dfrac{y_2 - y_1}{x_2 - x_1} \)
2. Positive Slope
A line has a positive slope if it rises upward as we move from left to right. In this case, the numerator \((y_2 - y_1)\) and the denominator \((x_2 - x_1)\) have the same sign.
2.1. Characteristics
- Line goes upward
- Slope value: \(m > 0\)
- Rise and run are either both positive or both negative
2.2. Example
Points: \((1, 2)\) and \((3, 6)\)
\( m = \dfrac{6 - 2}{3 - 1} = \dfrac{4}{2} = 2 \)
3. Negative Slope
A line has a negative slope if it falls downward as we move from left to right. Here, the numerator and denominator have opposite signs.
3.1. Characteristics
- Line goes downward
- Slope value: \(m < 0\)
- Rise is negative if run is positive
3.2. Example
Points: \((4, 5)\) and \((7, 1)\)
\( m = \dfrac{1 - 5}{7 - 4} = \dfrac{-4}{3} \)
4. Zero Slope
A line has a zero slope if it is perfectly horizontal. There is no vertical change even though there is horizontal movement.
4.1. Characteristics
- Slope value: \(m = 0\)
- No rise (y-values are equal)
- Line is horizontal
4.2. Example
Points: \((2, 5)\) and \((10, 5)\)
\( m = \dfrac{5 - 5}{10 - 2} = 0 \)
5. Undefined Slope
A line has an undefined slope if it is vertical. The x-coordinates of the two points are the same, so the denominator becomes zero.
5.1. Characteristics
- Slope is undefined because division by zero is not possible
- Line is vertical
- x-values are equal
5.2. Example
Points: \((6, 3)\) and \((6, -1)\)
\( m = \dfrac{-1 - 3}{6 - 6} = \dfrac{-4}{0} \)