which graph represents a function with direct variation?

Dou Vamel yi

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16-01-2025

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Direct variation describes a relationship between two variables where one is a constant multiple of the other. In simpler terms, as one variable increases, the other increases proportionally, and as one decreases, the other decreases proportionally. Understanding how to identify direct variation graphically is crucial in algebra and beyond. This article will show you how to spot a direct variation graph.

Understanding Direct Variation

Before diving into graphs, let's solidify our understanding of direct variation. A direct variation can be represented by the equation:

y = kx

Where:

• y and x are the variables.
• k is the constant of variation (a non-zero constant).

This equation tells us that y is directly proportional to x. The constant k determines the rate of this proportionality. If k is positive, both variables increase or decrease together. If k is negative, one variable increases as the other decreases (inverse relationship, not direct variation).

Identifying Direct Variation on a Graph

A graph representing a direct variation will always have these key characteristics:

• Passes through the origin (0, 0): When x = 0, y will also always equal 0. This is a defining feature.

• Forms a straight line: The relationship between x and y is linear.

What to look for:

• A straight line that goes through (0,0): This is the most important visual cue. If the line doesn't pass through the origin, it's not a direct variation.

• Constant slope: The slope of the line represents the constant of variation (k). A consistent slope indicates a direct proportional relationship.

Example:

Imagine a graph showing the relationship between the number of hours worked (x) and the amount of money earned (y). If the graph is a straight line passing through (0,0), with a constant positive slope (say, $15/hour), it represents a direct variation. Every additional hour worked earns an additional $15, demonstrating direct proportionality.

Graphs That Don't Represent Direct Variation

Several graphs might look similar but fail to show direct variation:

• Any straight line that doesn't pass through (0,0): This indicates a linear relationship, but not a direct variation. The equation would be of the form y = mx + c, where 'c' is the y-intercept. Direct variation requires c = 0.

• Nonlinear graphs (curves): Parabolas, exponentials, etc., clearly don't represent direct variation as the relationship between x and y is not linear.

• Graphs with a discontinuous line: A line with gaps or breaks cannot represent a continuous direct variation.

How to Determine if a Graph Shows Direct Variation: A Step-by-Step Guide

1. Check for a straight line: Is the graph a straight line? If not, it's not a direct variation.

2. Check the origin: Does the line pass through the point (0,0)? If not, it's not a direct variation.

3. Analyze the slope: If the line passes through (0,0) and is straight, examine its slope. A consistent slope indicates a direct variation.

Conclusion

Identifying a direct variation graph is straightforward. Look for a straight line that passes through the origin (0,0). If both conditions are met, you've found a graph depicting direct variation. Remember, the slope of the line represents the constant of variation, k, showing the rate of proportionality between the variables. Understanding these visual clues allows you to quickly and accurately determine if a given graph represents a direct variation.