Compare and contrast linear, quadratic, and exponential functions using tables, graphs, and equations, and determine which family best models a situation (A.F.2).

What this topic is asking

This part of A.F.2 asks you to compare linear, quadratic, and exponential functions across tables, graphs, and equations, and to decide which family models a situation. On the Virginia Algebra I SOL these are Functions items: identify the type of function from a table or graph, or choose the best model for a context. They appear as multiple choice, table items, and matching.

The table test: differences versus ratios

The fastest way to classify a function from a table (with equal $x$-steps) is to look at how $y$ changes:

• Linear: constant first difference. Each $y$ is the previous plus the same number. For $y = 2, 5, 8, 11$ the difference is $+3$ each time.
• Quadratic: constant second difference. The first differences are not constant, but the differences of those differences are. For $y = 1, 4, 9, 16$ the first differences are $3, 5, 7$ and the second differences are $2, 2$ (constant).
• Exponential: constant ratio. Each $y$ is the previous times the same number. For $y = 3, 6, 12, 24$ the ratio is $\times 2$ each time.

The graph test

Each family has a recognizable shape:

• Linear: a straight line (constant slope).
• Quadratic: a parabola (a U-shape with a vertex and an axis of symmetry).
• Exponential: a curve that rises or falls ever more steeply and approaches a horizontal asymptote without crossing it.

Choosing a model in context

Decide which kind of change the situation describes:

• Linear when a quantity changes by a fixed amount per step (\$5 per ticket,$2$ cm per week).
• Quadratic when there is a squared relationship or a maximum/minimum (area, projectile height with a peak).
• Exponential when a quantity changes by a fixed percent per step (population growth, depreciation, compound interest).

The phrase "per year by $3\%$" signals exponential; "\3$per item" signals linear; "height after$t$ seconds" with a peak signals quadratic.

Why exponential growth eventually wins

A defining comparison the SOL tests is that an increasing exponential function eventually exceeds any linear or quadratic function, no matter how large their coefficients. The reason is the operation behind each: a linear function adds a fixed amount each step, a quadratic adds an amount that grows linearly, but an exponential multiplies by a fixed factor each step, so its increments grow in proportion to its current size. Early on, a steep line or a wide parabola can be far ahead, which is why a graph may show the line above the exponential at first. But because the exponential's growth compounds, after enough steps it overtakes and then races away from any polynomial. This is the practical reason exponential models describe runaway processes (viral spread, compound interest) while linear models describe steady ones. Recognizing additive versus multiplicative change is the single idea that both classifies a table and predicts this long-run behavior.

How the SOL examines this topic

• Multiple choice. Identify the family from a table or graph, or choose the best model for a context.
• Table items. Compare two functions' values, or extend a table by its pattern.
• Matching. Pair tables, graphs, and equations of the same function.

Try this

Q1. A table has differences $+5, +5, +5$. Which family? [1 point]

• Cue. Constant first difference, so linear.

Q2. A savings account earns $2\%$ interest per year. Which model? [1 point]

• Cue. Constant percent change, so exponential.