๐ Understanding Slope: A Complete Guide
Slope is a measure of how steep a line is. It tells us how much the line rises or falls for every unit of horizontal change. We can calculate slope using the formula:
\[slope (m) = \frac{rise}{run} = \frac{y_2 - y_1}{x_2 - x_1}\]
๐ Positive Slope
A line with a positive slope goes upwards from left to right. Think of climbing a hill โ you're going up!
- ๐ Visual: Imagine a person walking from left to right. If they are walking uphill, the slope is positive.
- โ Characteristics: As the x-value increases, the y-value also increases.
- ๐ Example: A line passing through (1, 2) and (3, 6). The slope is $\frac{6-2}{3-1} = \frac{4}{2} = 2$.
๐ Negative Slope
A line with a negative slope goes downwards from left to right. Think of skiing down a hill โ you're going down!
- ๐ถ Visual: Imagine a person walking from left to right. If they are walking downhill, the slope is negative.
- โ Characteristics: As the x-value increases, the y-value decreases.
- ๐ Example: A line passing through (1, 6) and (3, 2). The slope is $\frac{2-6}{3-1} = \frac{-4}{2} = -2$.
โ๏ธ Zero Slope
A line with a zero slope is a horizontal line. It's flat โ like a straight road.
- ๐ Visual: A horizontal line is parallel to the x-axis.
- โ๏ธ Characteristics: The y-value remains constant for all x-values.
- ๐ Example: A line passing through (1, 3) and (5, 3). The slope is $\frac{3-3}{5-1} = \frac{0}{4} = 0$.
- ๐ Equation: The equation of a horizontal line is always in the form $y = c$, where $c$ is a constant.
โ ๏ธ Undefined Slope
A line with an undefined slope is a vertical line. It goes straight up and down.
- ๐ฆ Visual: A vertical line is parallel to the y-axis.
- ๐ฅ Characteristics: The x-value remains constant for all y-values.
- ๐ Example: A line passing through (2, 1) and (2, 5). The slope is $\frac{5-1}{2-2} = \frac{4}{0}$, which is undefined.
- ๐ Equation: The equation of a vertical line is always in the form $x = c$, where $c$ is a constant.
โ๏ธ Quick Summary Table
| Slope Type |
Direction |
Value |
Example |
| Positive |
Upward (left to right) |
Greater than 0 |
$\frac{1}{2}$, $3$, $5$ |
| Negative |
Downward (left to right) |
Less than 0 |
$\frac{-1}{2}$, $-3$, $-5$ |
| Zero |
Horizontal |
0 |
$\frac{0}{4} = 0$ |
| Undefined |
Vertical |
Undefined (division by zero) |
$\frac{4}{0}$ |
๐ง Practice Quiz
Determine the slope type (positive, negative, zero, or undefined) for each of the following pairs of points:
- ๐(1, 1) and (2, 2)
- ๐(3, 4) and (1, 2)
- ๐(5, 2) and (5, 7)
- ๐(2, 6) and (4, 3)
- ๐(7, 3) and (9, 3)
- ๐(0, 0) and (1, -1)
- ๐(-1, 5) and (-1, 0)
Answers:
- Positive
- Positive
- Undefined
- Negative
- Zero
- Negative
- Undefined