Let's analyze the given line passing through the points [tex]\((-4, 3)\)[/tex] and [tex]\((4, 3)\)[/tex] to determine if it represents a direct variation.
1. Understanding Direct Variation:
- A direct variation implies a linear relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that can be written as [tex]\(y = kx\)[/tex], where [tex]\(k\)[/tex] is a constant.
- The graph of a direct variation is a straight line that passes through the origin [tex]\((0, 0)\)[/tex].
2. Nature of the Given Line:
- The points given are [tex]\((-4, 3)\)[/tex] and [tex]\((4, 3)\)[/tex].
- Both points share the same [tex]\(y\)[/tex]-coordinate, 3.
3. Classification of the Line:
- Since the [tex]\(y\)[/tex]-coordinates are the same for both points, the line is horizontal.
- A horizontal line has the form [tex]\(y = c\)[/tex], where [tex]\(c\)[/tex] is a constant. In this case, [tex]\(y = 3\)[/tex].
4. Checking for Direct Variation:
- For the line to represent a direct variation, it must pass through the origin [tex]\((0, 0)\)[/tex].
- The equation [tex]\(y=3\)[/tex] does not pass through the origin because when [tex]\(x = 0\)[/tex], [tex]\(y\)[/tex] is not 0; instead, it remains 3.
Therefore, the correct description is:
The line does not represent a direct variation because it does not go through the origin.
