ACT® · Science: Interpretation of Data (13-17 items)
Complex Nonlinear Relationships in Data (IOD 602)
A nonlinear relationship is one where equal steps in the input don't produce equal steps in the output. You identify its shape by watching how the gaps between consecutive values themselves grow or shrink.
A straight line means the output changes by the same amount every time the input ticks up by one. A curve that bends upward breaks that rule, and the way it breaks it tells you about the process making the data. A process that doubles every step, like a cell culture splitting each hour, produces gaps that themselves double. A process tied to squared time, like falling distance, produces gaps that grow by a fixed amount added each step, not a fixed factor multiplied each step. Both curves bend upward on a graph, but they get there by different arithmetic, and the test wants you to tell them apart by checking the pattern in the differences, not just by eyeballing the bend.
Work it from a table. Values of 2, 8, 18, 32 at inputs 1, 2, 3, 4 have gaps of 6, 10, 14: each gap is 4 more than the last, a constant amount added. That constant-amount-added pattern in the gaps is the signature of a squared relationship, so the rule is 2 times the input squared. If instead the gaps were 6, 12, 24, each one double the last, that constant-factor pattern would signal exponential growth instead.
A curve that levels off, an asymptote, is still climbing, just by smaller and smaller amounts; it has not stopped increasing, it has slowed. Confusing leveling off with stopping leads to bad extrapolation. Once you know the shape, pushing a trend past the last measured point means continuing that shape's own pattern, a ratio for exponential data, a fixed second difference for squared data, never a straight line.
Key terms
- Nonlinear relationship
- A pattern where equal increases in the input do not produce equal increases in the output.
- Exponential growth
- A pattern where each successive increase is a fixed multiple of the increase before it.
- Squared relationship
- A pattern where the output grows with the square of the input, giving gaps that increase by a constant added amount.
- Asymptote
- A level a curve approaches but keeps slowly climbing or falling toward, never fully flat.
- Extrapolate
- To estimate a value beyond the range of the measured data by continuing its pattern.
Watch it explained
- How it worksHow do you tell an exponential rise from a squared rise just by watching how fast the gaps grow?
- Worked exampleFind the relationship behind a table reading 1 at 2, 2 at 8, 3 at 18 and 4 at 32 and name its shape.
- Common mistakeWhy isn't a curve that levels off the same thing as a curve that has stopped increasing?
- Ties togetherHow does recognizing a nonlinear shape change the way you push a trend past the last measured point?
Check yourself
End of topic.