It represents all solutions to the linear equation.
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A straight line.
The curved line on a time vs. distance graph represents that the object is accelerating.
The slope of a line on a position vs. time graph would represent the a velocity of the object being described.
Yes, x = -3 would represent a vertical line at abscissa -3, parallel to the y-axis.
The "vertical line test" will tell you if it is a function or not. The graph is not a function if it is possible to draw a vertical line through two points.