Distinguish linear, quadratic, and exponential functions from tables, graphs, and contexts using constant differences and ratios, and compare their long-run growth (Ohio F-LE.1, F-LE.3, F-IF.4).

What this topic is asking

Ohio standards F-LE.1 and F-LE.3 ask you to distinguish linear, quadratic, and exponential behavior from a table, graph, or description, and to compare how fast they grow. The test rewards spotting the pattern: a constant difference (linear), a constant second difference (quadratic), or a constant ratio (exponential). This is a high-value Functions skill across both parts.

The three signatures in a table

For a table with equal input steps, each family leaves a distinct fingerprint.

So test in order: subtract for a constant first difference; if not, subtract again for a constant second difference; also divide for a constant ratio.

Working an example

A short routine settles most table items.

Reading the description and the graph

In words: a constant per-unit change is linear ("\5$per month"); a constant **percent** change is **exponential** ("grows$3%$ a year"); a quantity tied to area or a squared variable, or a path that rises then falls, is quadratic. In a graph: a straight line is linear, a parabola is quadratic, and a curve that rises ever more steeply (or falls toward but never reaches an axis) is exponential.

Long-run growth

For large inputs, the family beats the coefficients: exponential eventually overtakes quadratic, which overtakes linear, no matter how large the linear slope or how small the exponential base (as long as the base exceeds $1$). This is why "for large $x$, which is greatest?" is answered by the family, not the leading number.

How Ohio examines this topic

• Multiple choice and multiple-select. Classify a table, graph, or context as linear, quadratic, or exponential.
• Tables. Compute first or second differences, or ratios, to justify the family.
• Reasoning items. Compare two functions' long-run behavior, or match a scenario to a model.

Why each family has its own fingerprint

The three fingerprints come straight from the rules. A linear function adds $m$ for each unit of input, so consecutive outputs differ by a constant, the first difference is $m$. A quadratic function's rate of change itself changes at a constant rate, so the first differences form an arithmetic pattern and the second differences are constant. An exponential function multiplies by $b$ each step, so consecutive outputs have a constant ratio $b$, not a constant difference. Because each family is built on a different operation, add, add-then-add-again, multiply, the diagnostic is reliable: find the level at which the pattern becomes constant and you have named the family.

Why the family beats the coefficients in the long run

It is tempting to think a linear function with a huge slope, like $1000x$, will always stay ahead of an exponential with a small base, like $1.01^x$. For small inputs it does, but the operations decide the long run. Adding a fixed amount (linear) or even squaring (quadratic) cannot keep pace with repeated multiplication (exponential), because each exponential step scales the whole accumulated amount, so the gap it opens keeps widening. Given enough input, $1.01^x$ overtakes $1000x$ and every polynomial. This is why the ordering exponential > quadratic > linear holds eventually for any positive coefficients, and why the test phrases these comparisons as "for large values of $x$."

Try this

Q1. A table's outputs go $4, 12, 36, 108$. Which family? [2 points]

• Cue. Ratios all $3$, so exponential.

Q2. A car depreciates by a fixed \1500$ each year. Linear or exponential? [1 point]

• Cue. Constant dollar amount, so linear (a fixed percent would be exponential).