Compare linear, quadratic, and exponential functions across representations and observe that exponential growth eventually exceeds the others (NC.M1.F-LE.3, F-IF.9).

What this topic is asking

NC.M1.F-LE.3 asks you to observe (from graphs and tables) that an increasing exponential function eventually grows faster than any linear or quadratic function. NC.M1.F-IF.9 asks you to compare functions given in different representations. Together they are about telling the three families apart and knowing how they behave.

Telling the families apart from a table

The differences and ratios reveal the family.

So for inputs $0, 1, 2, 3$: outputs $2, 5, 8, 11$ are linear ($+3$ each); $1, 4, 9, 16$ are quadratic (second differences all $2$); $2, 6, 18, 54$ are exponential (times $3$ each).

A worked classification

Why exponential growth eventually wins

A line adds a fixed amount each step; a parabola adds an increasing (but bounded-by-a-formula) amount; an exponential multiplies. Multiplication compounds, so even a slow-looking exponential like $1.1^x$ eventually overtakes $1000x$ or $x^2$. The EOC tests this as a concept: for large enough inputs, exponential growth is the steepest.

Comparing across representations

F-IF.9 may give one function as an equation and another as a graph or table, and ask which has the greater rate or value. The technique is to read the same feature from each, evaluate both at the same input, or compare both y-intercepts, or compare both average rates over the same interval. This connects to average rate of change.

How the NC Math 1 EOC examines this topic

• Multiple choice. Classify a function from a table or graph, or choose which grows fastest.
• Multiple select. Select all true statements comparing two functions.
• Technology-enhanced. Match tables or graphs to function families.

This topic ties together exponential functions (constant ratio), sequences (arithmetic is linear, geometric is exponential), and quadratic behavior from interpreting key features.

Why the pattern of change is the fingerprint

Each family has a signature you can detect without graphing: add the same amount (linear), add a steadily increasing amount whose own change is constant (quadratic), or multiply by the same factor (exponential). This is powerful because a short table is often enough to identify the model, no curve-fitting required. It also explains the long-run behavior: addition keeps a steady or steadily widening lead, but multiplication snowballs, which is why exponential growth always overtakes the others eventually. Carrying this single diagnostic, first difference, second difference, or ratio, lets you classify any function the EOC presents and predict how it will behave for large inputs.

Try this

Q1. Outputs $5, 8, 11, 14$ for inputs $0, 1, 2, 3$. Which family? [1 point]

• Cue. First differences all $+3$: linear.

Q2. Outputs $2, 6, 18, 54$ for inputs $0, 1, 2, 3$. Which family? [2 points]

• Cue. Constant ratio $\times 3$: exponential, $y = 2\cdot 3^x$.