1. What is Slope?
The slope of a line shows how steep the line is and the direction in which it moves. It tells us how much the line rises or falls when we move horizontally.
In simple terms, slope measures the tilt of a line.
1.1. Definition
The slope of a line is defined as the ratio of the vertical change to the horizontal change between any two points on the line.
2. Understanding Rise and Run
To find the slope between any two points, we look at two changes:
2.1. Rise (Vertical Change)
This is the change in the y-coordinate. It shows how much the line moves upward or downward.
2.2. Run (Horizontal Change)
This is the change in the x-coordinate. It shows how much we move to the left or right.
2.3. Slope as Rise/Run
Slope is written as:
\( m = \dfrac{\text{rise}}{\text{run}} \)
2.3.1. Important Note
If we move to the right, run is positive. If we move to the left, run is negative. Rise is positive when going up and negative when going down.
3. Slope Formula
When two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line are known, the slope can be found directly.
3.1. Slope Between Two Points
\( m = \dfrac{y_2 - y_1}{x_2 - x_1} \)
This formula shows how the y-values change compared to the x-values.
3.2. Interpretation
- If the value of m is positive → the line rises upward to the right.
- If m is negative → the line goes downward to the right.
- If m = 0 → the line is horizontal.
- If the denominator becomes 0 → slope is undefined (vertical line).
4. Examples
These examples help understand how slope works in different cases.
4.1. Example 1: Positive Slope
Find the slope between \((1, 2)\) and \((3, 6)\).
\( m = \dfrac{6 - 2}{3 - 1} = \dfrac{4}{2} = 2 \)
The line rises upward to the right.
4.2. Example 2: Negative Slope
Find the slope between \((4, 5)\) and \((8, 1)\).
\( m = \dfrac{1 - 5}{8 - 4} = \dfrac{-4}{4} = -1 \)
The line goes downward to the right.
4.3. Example 3: Zero Slope
Points: \((2, 7)\) and \((9, 7)\)
\( m = \dfrac{7 - 7}{9 - 2} = 0 \)
A horizontal line has zero slope.
4.4. Example 4: Undefined Slope
Points: \((4, 3)\) and \((4, -2)\)
\( m = \dfrac{-2 - 3}{4 - 4} = \dfrac{-5}{0} \)
A vertical line has undefined slope because division by zero is not possible.