Difficulty: Foundational to Intermediate | Prerequisites: Part 1 of these notes (Coordinate Plane and Parent Functions).
Tags: identifying functions, classify functions, linear table, quadratic table, exponential table, first difference, second difference, common ratio, growth, decay, function equations, function graphs
Once you can recognise the three parent functions in their simplest forms, the next step is classification: given any representation (a table of values, an equation, or a graph), determine which family the function belongs to. This is a skill you will use constantly. Modelling questions on exams hand you raw data or an equation and ask you to name the type, pick the right approach, or predict behaviour. The classification techniques here, first differences, second differences, and common ratios, are the mechanical tools that make those calls reliable rather than guesswork.
To classify a function from a table, make sure x-values change by a constant amount, then check the y-values: constant first differences mean linear, constant second differences mean quadratic, and a constant ratio between consecutive y-values means exponential. From an equation, look at what happens to x: multiplied by a coefficient gives linear, squared gives quadratic, used as an exponent gives exponential. From a graph, straight lines are linear, U-shapes (parabolas) are quadratic, and steep curves that approach but never touch the x-axis are exponential.
First difference
The change in y between consecutive rows of a table (each y-value minus the one before it), provided the x-values increase by a constant step. If all first differences are equal, the function is linear. Think of it as asking: "Am I adding the same number every time?"
Second difference
The change between consecutive first differences. You take the first differences, then find the differences of those differences. If the second differences are all equal, the function is quadratic. Think of it as: "The amount I'm adding is itself changing by a constant amount."
Common ratio
The factor you multiply each y-value by to get the next one (again, assuming constant steps in x). If every consecutive pair of y-values shares the same ratio, the function is exponential. Think of it as: "Am I multiplying by the same number every time?"
Exponential growth
An exponential function where the graph rises as x increases. The base b in y = a · bˣ is greater than 1. In simple terms, the values get bigger and bigger, faster and faster.
Exponential decay
An exponential function where the graph falls toward zero as x increases. The base b is between 0 and 1 (a proper fraction). In simple terms, the values shrink toward zero but never quite reach it.
The method only works when x-values are evenly spaced. Always check that first.
Step-by-step process:
Write out the y-values in order.
Calculate the first differences (subtract each y from the next).
If the first differences are all the same, the function is linear. Stop here.
If the first differences are not constant, calculate the second differences (differences of the first differences).
If the second differences are all the same, the function is quadratic. Stop here.
If neither differences pattern works, go back to the original y-values and divide each y by the previous y.
If that ratio is constant, the function is exponential.
If no pattern appears, classify as "none of the above."
Worked example, linear:
x | -10 | -6 | -2 | 2 |
|---|---|---|---|---|
y | -4 | -6 | -8 | -10 |
x changes by +4 each time. First differences in y: -2, -2, -2. Constant first difference, so the function is linear.
Worked example, quadratic:
x | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
y | -5 | 25 | 59 | 97 |
x changes by +2 each time. First differences in y: 30, 34, 38. Not constant. Second differences: 4, 4. Constant second difference, so the function is quadratic.
Worked example, exponential decay:
x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
y | 135 | 45 | 15 | 5 |
x changes by +1 each time. Ratios: 45/135 = 1/3, 15/45 = 1/3, 5/15 = 1/3. Common ratio of 1/3 (a fraction between 0 and 1), so this is exponential decay.
Worked example, exponential growth:
x | 1 | 4 | 7 | 10 |
|---|---|---|---|---|
y | 3 | 12 | 48 | 192 |
x changes by +3 each time. Ratios: 12/3 = 4, 48/12 = 4, 192/48 = 4. Common ratio of 4 (greater than 1), so this is exponential growth.
The key is to look at what role x plays in the equation.
Linear indicators:
x appears only to the first power (no exponent written, or exponent of 1).
The equation can be rearranged into the form y = mx + b.
Examples: y = 4x - 7, y = 8 - (1/3)x, 2x + 5y = 0, y = 6 (a horizontal line is a special linear case with slope 0).
Quadratic indicators:
x is squared (x² appears somewhere in the equation).
The equation fits or can be expanded into y = ax² + bx + c.
Examples: y = 2x² + 9x - 1, y = (x - 2)(x + 7) (this multiplies out to give an x² term), y = -(x + 3)², y/x = x + 1 (rearranges to y = x² + x).
Exponential indicators:
x appears as an exponent (in the power position), not as the base.
The equation fits the form y = a · bˣ.
Examples: y = 5ˣ, y = 2 · (1/4)ˣ (decay, because the base 1/4 is between 0 and 1), y = (1/4) · (3/2)ˣ (growth, because the base 3/2 is greater than 1), y = (1/2) · 6ˣ.
Quick decision rule: Where is x? Multiplied in front (linear), squared in the expression (quadratic), or up in the exponent (exponential).
Linear: A perfectly straight line, going in any direction. Can be steep, flat, rising, or falling, but it never curves.
Quadratic: A parabola, a smooth U-shape (opening up) or an inverted U (opening down). It has exactly one turning point (the vertex) and is symmetric about a vertical line through that vertex.
Exponential growth: A curve that starts close to the x-axis on the left and sweeps sharply upward to the right. It never touches or crosses the x-axis.
Exponential decay: A curve that starts high on the left and drops toward the x-axis on the right, getting closer but never reaching zero. Mirror image of growth, in a sense.
None of the above: If the graph fails the vertical line test (a vertical line crosses it in more than one place), it is not a function at all. Circles, sideways parabolas, and similar shapes fall here.
Table classification summary:
Pattern in y-values (with constant x-step) | Function type |
|---|---|
Constant first difference (add the same value) | Linear |
Constant second difference (the increase itself increases at a steady rate) | Quadratic |
Constant ratio (multiply by the same value) | Exponential |
Exponential sub-classification:
Common ratio | Sub-type |
|---|---|
Greater than 1 | Exponential growth |
Between 0 and 1 | Exponential decay |
When scientists collect data points in a lab (temperature over time, population counts per year, distance vs. speed), they use exactly these techniques to decide which model fits before writing a prediction equation. A financial analyst checking whether revenue is growing linearly or exponentially will plot the data and look for the same visual cues. The classification step always comes before the modelling step.
Students try to identify the function type from a table without first checking that the x-values have a constant step. If x jumps unevenly, the first-difference and second-difference methods do not apply directly.
Confusing the common ratio with the first difference. A common ratio involves dividing consecutive y-values; a first difference involves subtracting them. Mixing the two operations up leads to a wrong classification.
Assuming any curve is exponential. A parabola is also a curve, but it has a turning point and is symmetric. Exponential curves do not turn around; they keep going in one direction.
Forgetting to specify growth or decay when identifying an exponential function. The lesson notes explicitly say to always include this distinction. On an exam, writing just "exponential" may lose marks.
⚠️ Tables are the most common format for classification questions on assessments. Practise the first-difference, second-difference, and ratio checks until they are automatic.
⚠️ When classifying equations, watch for disguised forms: y = (x - 2)(x + 7) is quadratic even though it does not look like y = ax² + bx + c at first glance. Expand if unsure.
⚠️ For exponential functions, always state whether it is growth or decay. This is a frequent mark lost on exams.
⚠️ The lesson stresses: "When looking at tables, make sure there is a constant change in x!" Expect at least one exam question with uneven x-steps designed to trip you up.
True or false: If the first differences in a table are 3, 3, 3, 3, the function is quadratic.
Fill in the blank: To identify an exponential function from a table, you look for a constant ______ between consecutive y-values.
True or false: The equation y = (1/2) · 6ˣ represents exponential decay.
Fill in the blank: In the equation y = a · bˣ, exponential decay occurs when b is between ______ and ______.
True or false: A graph that curves steeply upward and never crosses the x-axis could be exponential growth.
Answers: 1. False (constant first difference means linear). 2. Ratio. 3. False (the base is 6, which is greater than 1, so it is growth). 4. 0 and 1. 5. True.
Q: Given the table below, classify the function. x: 3, 4, 5, 6 and y: 14, 26, 44, 68.
A: First differences: 12, 18, 24. Not constant. Second differences: 6, 6. Constant second difference, so the function is quadratic.
Q: Classify y = 2 · (1/4)ˣ. Is it growth or decay?
A: It is exponential decay. The variable x is in the exponent (making it exponential), and the base 1/4 is between 0 and 1 (making it decay).
Q: A graph shows a smooth curve that starts high on the left, falls steeply, then levels off approaching but never touching the x-axis. What type of function is it?
A: Exponential decay. The curve decreases toward zero without reaching it, which is the signature shape.
Q: Why can you not use the first-difference method on this table? x: 1, 2, 5, 10 and y: 3, 6, 15, 30.
A: The x-values do not increase by a constant step (the gaps are 1, 3, 5). The first-difference and second-difference methods require evenly spaced x-values to be valid.
Q: The equation y/x = x + 1 does not look quadratic at first glance. Explain why it is.
A: Multiply both sides by x to get y = x² + x. The x² term makes this a quadratic function in the form y = ax² + bx + c, with a = 1, b = 1, c = 0.
The classification skills from this lesson feed directly into curve fitting and regression analysis later in the course, where you choose a model type before running calculations. They also connect to sequences and series: a linear function corresponds to an arithmetic sequence (common difference), and an exponential function corresponds to a geometric sequence (common ratio). If your course covers logarithms, those are the inverse of exponential functions, so solid familiarity with exponential behaviour here makes the logarithm unit considerably easier.
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