Question

Which explains whether or not the graph represents a direct variation?

On a coordinate plane, a line goes through points (0, 0) and (1, 3).
The graph has a constant of variation of 3, so it represents a direct variation.
The graph has a slope of 3, so it represents a direct variation.
The graph has a positive slope, so it does not represent a direct variation.
The graph does not begin at the origin, so it does not represent a direct variation.

Answer 1

To determine whether the graph represents a direct variation, we need to consider the definition of direct variation. A relationship between two variables, \(y\) and \(x\), is considered a direct variation if it can be described by the equation \(y = kx\), where \(k\) is a constant and the line passes through the origin (0, 0).

Let's analyze the given information:

- The points (0, 0) and (1, 3) indicate that the line passes through the origin.
- The slope of the line can be calculated using the two points: the change in \(y\) (3) over the change in \(x\) (1) gives us a slope of 3.
- The equation of the line can be expressed as \(y = 3x\), which fits the form \(y = kx\) where \(k = 3\).

Since the line passes through the origin and can be represented in the form \(y = kx\), we conclude that this relationship is a direct variation.

From the given options, the statement "The graph has a constant of variation of 3, so it represents a direct variation." and "The graph has a slope of 3, so it represents a direct variation." are both correct in expressing that it represents a direct variation.

In summary, the appropriate explanations confirming that the graph represents a direct variation are:

1. The graph has a constant of variation of 3, so it represents a direct variation.
2. The graph has a slope of 3, so it represents a direct variation.