Answer
The slope shows the direction of a line: a positive slope means the line goes up from left to right, indicating a positive relationship, while a negative slope means the line goes down, indicating a negative relationship. A zero slope means the line is flat, and an undefined slope means the line is vertical.
Solution
The slope of a line on a graph is a measure of its steepness and direction. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope can be represented mathematically as:
\[
m = \frac{\Delta y}{\Delta x}
\]
where:
- \( m \) is the slope,
- \( \Delta y \) is the change in the y-coordinates (vertical change),
- \( \Delta x \) is the change in the x-coordinates (horizontal change).
### Direction Indicated by the Slope:
1. **Positive Slope (\( m > 0 \))**:
- If the slope is positive, the line rises as it moves from left to right. This indicates a positive relationship between the x and y variables. For example, if you were plotting the relationship between time and distance traveled, a positive slope would indicate that as time increases, distance also increases.
2. **Negative Slope (\( m < 0 \))**:
- If the slope is negative, the line falls as it moves from left to right. This indicates a negative relationship between the x and y variables. For instance, if you were plotting the relationship between temperature and the amount of ice remaining, a negative slope would suggest that as temperature increases, the amount of ice decreases.
3. **Zero Slope (\( m = 0 \))**:
- A slope of zero indicates a horizontal line. This means that there is no change in the y-value as the x-value changes. In practical terms, this could represent a situation where a variable remains constant regardless of changes in another variable.
4. **Undefined Slope**:
- If a line is vertical, the slope is considered undefined because the change in x (\( \Delta x \)) is zero, leading to division by zero in the slope formula. A vertical line indicates that the y-value can change while the x-value remains constant.
### Summary:
The slope provides critical information about the relationship between two variables represented on a graph. By analyzing the slope, one can determine whether the relationship is positive, negative, or non-existent, which is essential for understanding trends and making predictions based on the data.
Reviewed and approved by the UpStudy tutoring team
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