How do you tell linear, quadratic, and exponential functions apart from a table, an equation, or a graph, and why does exponential growth eventually overtake the others?
Distinguish among linear, quadratic, and exponential functions using their rates of change from tables, equations, and graphs, and recognize that a quantity growing exponentially eventually exceeds one growing linearly or quadratically (MA.912.F.1.6, MA.912.AR.5.6).
A B.E.S.T. Algebra 1 EOC answer on distinguishing function families (MA.912.F.1), constant differences for linear, constant second differences for quadratic, constant ratios for exponential, and why exponential growth eventually dominates.
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What this topic is asking
MA.912.F.1 asks you to tell apart the three function families, linear, quadratic, and exponential, from a table, an equation, or a graph, and to know that exponential growth eventually dominates. The B.E.S.T. Algebra 1 EOC tests recognizing the family from its pattern of change and predicting long-run behavior.
The fingerprint of each family
The way the outputs change for equal steps in the input identifies the family:
| Family | Pattern in a table (equal input steps) | Equation |
|---|---|---|
| Linear | constant first difference (same amount added) | |
| Quadratic | constant second difference | |
| Exponential | constant ratio (same factor multiplied) |
So for outputs the first differences are all (linear). For the first differences are and the second differences are all (quadratic). For each output is times the last (exponential).
How the B.E.S.T. EOC examines this topic
- Multiple choice and editing task. Classify a function from a table, equation, or graph.
- Matching. Pair tables or graphs with linear, quadratic, or exponential.
- Context items. Compare two growing quantities and decide which is larger long-term.
A clarifying idea: linear is "add the same each step," exponential is "multiply by the same each step," and quadratic is in between (its rate of change itself changes at a constant rate). Naming the operation, add, multiply, or accelerate, is the quickest classifier.
Why exponential growth eventually wins
The long-run dominance of exponential growth is one of the most tested ideas, and it follows from the difference between adding and multiplying. A linear function adds a fixed amount each step, so its increases are all the same size. An exponential function multiplies by a fixed factor each step, so each increase is a percentage of an ever-larger amount, meaning the increases themselves keep growing. Early on a linear (or quadratic) function can be far ahead, because exponential growth starts slow. But because the exponential's step-by-step increase compounds, it accelerates without bound and will, given enough steps, overtake any linear or quadratic function and stay ahead forever. This is why \100 growing 20 percent a year eventually buries \100 plus $50 a year, even though the linear plan leads at the start. The takeaway for the EOC: when one quantity multiplies and another adds, the multiplier wins in the long run.
Reading the family from a graph
From a graph, the shapes are distinct. A line is straight, with constant steepness. A parabola is a symmetric U that turns around at a vertex. An exponential curve never turns around; it hugs a horizontal asymptote on one end and shoots up (or, for decay, falls toward the asymptote) on the other, with no axis of symmetry. If a graph keeps curving upward and steepening without a turning point, it is exponential, not quadratic, a distinction the EOC tests with look-alike curves.
Try this
Q1. Outputs for inputs . Which family? [1 point]
- Cue. Constant ratio , so exponential.
Q2. Outputs . Which family? [1 point]
- Cue. Constant second difference , so quadratic.
Exam-style practice questions
Practice questions written in the style of FLDOE exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
B.E.S.T. (style)1 marksMultiple choice. A table shows outputs for inputs . What type of function is this? (A) exponential (B) linear (C) quadratic (D) constantShow worked answer →
The correct answer is (A).
Check how the outputs change. They are not adding a constant ( but ), so not linear. But each output is the previous one times (), a constant ratio, which is exponential. A constant difference signals linear; a constant ratio signals exponential.
B.E.S.T. (style)2 marksTwo savings plans start at \100. Plan A adds \50 each year (linear). Plan B grows 20 percent each year (exponential). Explain which plan is larger after many years and why.Show worked answer →
Plan B (exponential) is eventually larger, even though Plan A may lead at first.
Plan A is , adding a fixed amount. Plan B is , multiplying by each year. For small , the linear plan can be ahead, but exponential growth compounds, each year's increase is larger than the last, so Plan B's growth accelerates and eventually surpasses any linear amount. Markers reward the key idea: a quantity increasing by a constant factor eventually exceeds one increasing by a constant amount.
Related dot points
- Write, evaluate, and interpret exponential functions that model growth and decay, identifying the initial value and the growth or decay factor and rate (MA.912.AR.5.4, MA.912.F.1.6).
A B.E.S.T. Algebra 1 EOC answer on exponential models (MA.912.AR.5), the growth and decay forms y = a(1 + r)^t and y = a(1 - r)^t, the initial value, the growth or decay factor, and interpreting in context.
- Graph exponential functions and identify key features including the y-intercept, the horizontal asymptote, domain, range, and whether the function is increasing or decreasing (MA.912.F.1.3, MA.912.AR.5.6).
A B.E.S.T. Algebra 1 EOC answer on graphing exponentials (MA.912.F.1, AR.5), the y-intercept at the initial value, the horizontal asymptote at y = 0, the domain and range, and growth versus decay shape.
- Calculate and interpret the average rate of change of a function over a specified interval from a graph, a table, or an equation (MA.912.F.1.4).
A B.E.S.T. Algebra 1 EOC answer on average rate of change (MA.912.F.1.4), the change-in-output over change-in-input formula, reading it from tables and graphs, and interpreting it as a slope in context.
- Write and evaluate explicit and recursive formulas for arithmetic and geometric sequences, and relate arithmetic sequences to linear functions and geometric sequences to exponential functions (MA.912.AR.5 and MA.912.F).
A B.E.S.T. Algebra 1 EOC answer on sequences, the explicit and recursive formulas on the reference sheet, finding the common difference or ratio, and linking arithmetic to linear and geometric to exponential growth.
- Compare key features (intercepts, rate of change, maximums, and minimums) of two functions each represented differently, such as one as an equation and one as a table or graph (MA.912.F.1.5).
A B.E.S.T. Algebra 1 EOC answer on comparing functions (MA.912.F.1.5), extracting slopes, intercepts, and maximums from equations, tables, and graphs, and comparing them when the two functions are shown in different forms.
Sources & how we know this
- B.E.S.T. Mathematics Standards — Florida Department of Education (2020)
- B.E.S.T. Algebra 1 EOC Computer-Based Practice Test — Florida Department of Education (2024)