Test every representation for multiplicative change: for each increase of \(1\) in the input, the output must be multiplied by the same factor. Report the complete set of labels that meet this condition.
A. A table has \(x\text{-}\text{values}~1,~2,~3,~4\) and \(y\text{-}\text{values}~5,~10,~20,~40\).
B. \(y~=~7x~+~3\)
C. A gym plan has total cost \(c~=~25~+~8v\) for \(v\) visits.
D. As \(x\) increases by \(1\) each time, the \(y\text{-}\text{values}\) are multiplied by \(2\), producing a graph that is not a straight line.
E. As the input increases by \(1\), the output increases by \(6\) each time.
What this problem is really about
Different forms can hide the same kind of change, so apply one invariant test to each representation. For equal one-unit input steps, look for a constant ratio between consecutive outputs; fixed increases in a rule, story, or table indicate a different structure even when multiplication appears in the notation. Test every representation before collecting the labels that truly share the required structure.