Distinguish linear, quadratic, and exponential functions by their rate of change and recognize that a quantity growing by a constant factor eventually exceeds one growing linearly (LA A1: F-LE.A.1, F-LE.A.3).

What this topic is asking

Standards A1: F-LE.A.1 and F-LE.A.3 ask you to distinguish linear, quadratic, and exponential functions and to recognize that a quantity growing by a constant factor (exponential) eventually exceeds one growing by a constant amount (linear). On LEAP 2025 these are Type I and Type II items, identifying the family from a table, graph, or context, and reasoning about long-run behavior.

The three families and their signatures

Identifying from a table

With equally spaced $x$-values, examine the outputs:

If instead the outputs had a constant ratio, the function would be exponential.

Identifying from a graph or context

A line is linear. A parabola (one turning point, symmetric) is quadratic. A curve that keeps bending upward faster and faster (or decaying toward zero) is exponential. In words: "adds the same amount" is linear, "doubles or grows by a percent" is exponential, "area or projectile height" is often quadratic.

Exponential eventually beats linear

F-LE.A.3 makes a specific claim: exponential growth overtakes linear growth eventually. A linear quantity adds a fixed amount each step; an exponential quantity multiplies, so the amount it adds keeps growing. However large the linear rate, the exponential one passes it and never looks back. This is why compound interest, populations, and viral spread are modeled exponentially.

How LEAP examines this topic

• Multiple choice. Classify a function from a table, graph, or description.
• Type II reasoning. Explain why exponential growth exceeds linear, or compare two models.
• Drag and drop. Sort functions or contexts into linear, quadratic, and exponential.

A clarifying idea: the test of what is constant unifies these families with sequences: constant difference = linear (arithmetic), constant ratio = exponential (geometric), constant second difference = quadratic.

Why constant second differences mean quadratic

The reason a quadratic shows a constant second difference is worth understanding because it is the fingerprint that separates parabolas from lines and exponentials. For a linear function the first differences are constant, each equal step in $x$ changes $y$ by the same amount, so the rate of change is fixed. A quadratic's rate of change is not fixed; it speeds up steadily, which is why a parabola gets steeper as you move away from the vertex. But the rate at which the rate changes is itself constant: the "difference of the differences" settles to a fixed number (equal to $2a$ for $f(x) = ax^2 + bx + c$ over unit steps). So the first differences grow by a constant amount, and that constant is the second difference. An exponential, by contrast, never has constant differences at any level, because each step multiplies, the differences themselves grow by a constant ratio, not a constant amount. This layered view, constant differences (linear), constant second differences (quadratic), constant ratios (exponential), gives you a single reliable procedure for classifying any function from a table, and it explains why exponential growth, with its compounding multiplier, ultimately outruns the steady acceleration of a quadratic and the fixed pace of a line.

Try this

Q1. Equally spaced outputs are $4, 8, 16, 32$. Which family? [2 points]

• Cue. Constant ratio $2$: exponential.

Q2. Equally spaced outputs are $1, 4, 9, 16$. Which family? [2 points]

• Cue. First differences $3, 5, 7$; second differences constant ($2$): quadratic.