Distinguish linear from exponential functions using constant difference versus constant ratio, and recognize that a quantity growing by equal factors over equal intervals is exponential (A.FGR, Functional and Graphical Reasoning).

What this topic is asking

This Functional and Graphical Reasoning (A.FGR) standard asks you to distinguish linear from exponential behavior and to choose the right model for a situation. The decisive test in a table is constant difference versus constant ratio: a linear function adds the same amount each step, while an exponential function multiplies by the same factor each step. The Georgia Milestones EOC tests this as classification items (linear or exponential, and the key parameter) and as modeling items that contrast a fixed-amount change with a percent change. The big conceptual idea is that exponential growth eventually overtakes linear growth, no matter how large the linear rate.

The deciding test: difference versus ratio

Given a table with equally spaced inputs, run two checks.

• First differences (subtract consecutive outputs). If they are constant, the function is linear, and that difference is the slope.
• Ratios (divide consecutive outputs). If they are constant, the function is exponential, and that ratio is the base.

For $2, 6, 18, 54$: differences $4, 12, 36$ are not constant, but ratios $3, 3, 3$ are, so the function is exponential with base 3, $g(x) = 2(3)^x$.

Matching a context to a model

Word problems signal the model by how the quantity changes.

• "adds $50 each year," "increases by 3 inches per week," "earns$12 per hour" describe a fixed amount, so linear: $y = mx + b$.
• "grows 25 percent each year," "doubles every hour," "loses 10 percent annually" describe a factor or percent, so exponential: $y = a(1 \pm r)^t$.

The phrase "per" plus a fixed unit is linear; "percent" or "times" per period is exponential.

Why exponential always wins eventually

The most important idea here is that exponential growth overtakes linear growth in the long run, regardless of the constants. A linear function adds a fixed amount each period, so its increases stay the same size forever. An exponential function multiplies by a fixed factor greater than 1, so its increases themselves grow each period: the bigger the value, the bigger the next jump. Early on, a linear function with a large slope can be far ahead of an exponential one with a small base, which is why Account A leads after one year. But the exponential function's compounding jumps eventually become larger than the linear function's fixed jumps, and once that happens the exponential pulls ahead and never looks back. This is why a 25-percent-per-year account beats a fixed-50-per-year account in the long run even though it starts behind, and it is a favorite EOC reasoning point.

How the Milestones examines this topic

• Multiple choice. Classify a table as linear or exponential and identify the slope or ratio.
• Inline choice. Choose "linear" or "exponential" for a described situation.
• Constructed response. Model two contrasting situations and compare their long-run behavior.

Reading graphs of the two families

The two families also look distinct on a graph, and the EOC sometimes asks you to match a graph to a model. A linear graph is a straight line with a constant steepness. An exponential growth graph curves upward, getting steeper as it rises, and has a horizontal asymptote it approaches on one side; an exponential decay graph drops quickly then levels off toward an asymptote. Spotting "straight" versus "curving and steepening" is a quick visual classifier that backs up the difference-versus-ratio test, and recognizing the asymptote (a line the curve approaches but never crosses) is itself a feature the EOC may ask you to identify.

Try this

Q1. A table has $f(0) = 5, f(1) = 8, f(2) = 11, f(3) = 14$. Linear or exponential, and the key value? [1 point]

• Cue. Constant difference 3, so linear with slope 3.

Q2. A bacteria count doubles every hour starting at 50. Linear or exponential? Write the model. [1 point]

• Cue. Doubling is a constant factor, so exponential: $N(t) = 50(2)^t$.