Evaluate every representation below for proportionality. Report the complete set that shows proportional relationships.
A. Table: \(x~=~2,~4,~6,~8\) and \(y~=~5,~10,~15,~20\)
B. Equation: \(y~=~4x~+~1\)
C. Graph: a straight line that passes through the origin and the point \((3,~9)\)
D. Context: The total cost of joining a gym is \(\$12~\text{for}~a~\text{sign}\text{-}\text{up}~\text{fee}\) plus \(\$8~\text{for}~\text{each}~\text{visit}\).
E. Table: \(x~=~1,~2,~3,~4\) and \(y~=~3,~6,~10,~12\)
What this problem is really about
We need the complete set, so each representation must be tested with the shortcut suited to its form. We’ll compare output-to-input ratios in both tables, inspect the equation and context for added starting amounts, and require the graph to be a straight line through the origin. One failed row or a nonzero start rules a representation out.