In This Guide
Quick Comparison
The fastest way to distinguish these functions is to ask what remains constant as x increases by equal amounts.
Linear Function
Equal changes in x produce equal first differences in y. The rate of change is constant.
Quadratic Function
First differences change, but equal x-steps produce constant second differences.
Exponential Function
Equal changes in x multiply y by the same factor. The common ratio is constant.
| Feature | Linear | Quadratic | Exponential |
|---|---|---|---|
| Typical form | y = mx + b | y = ax² + bx + c | y = abˣ |
| Constant pattern | First difference | Second difference | Ratio |
| Graph shape | Straight line | Parabola | Growth or decay curve |
| Rate of change | Constant | Changes linearly | Changes proportionally |
| Common examples | Hourly pay, constant speed | Projectile height, area models | Population growth, depreciation |
The key question
Does the output change by the same amount, does the change itself grow by the same amount, or is the output repeatedly multiplied by the same factor?
What Is a Linear Function?
A linear function has a constant rate of change. When x increases by one unit, y changes by the same amount every time. Its graph is a straight line.
The slope-intercept form is:
Here, m is the slope and b is the y-intercept. The slope measures the constant change in y for each one-unit increase in x:
Example of a linear function
Every time x increases by 1, y increases by 3. The first difference is constant, so the function is linear.
What Is a Quadratic Function?
A quadratic function has degree 2. Its graph is a parabola that opens upward or downward. Unlike a line, its rate of change is not constant. However, when x-values are equally spaced, the second differences of the y-values are constant.
The standard form is:
The coefficient a cannot equal zero. Its sign determines whether the parabola opens upward or downward.
Example of a quadratic function
The x squared term identifies the expression as quadratic. Its graph has a turning point called the vertex and is symmetric about a vertical line.
What Is an Exponential Function?
In an exponential function, the variable appears in the exponent. Equal increases in x multiply the output by the same factor. This creates rapid growth when the base is greater than 1 and decay when the base is between 0 and 1.
A common form is:
The value a is the initial value when x = 0, and b is the growth or decay factor.
Example of exponential growth
Every increase of 1 in x doubles the value of y. The constant ratio is 2.
Example of exponential decay
Each increase of 1 in x multiplies the output by 0.8, which represents a 20% decrease.
How to Identify the Function from an Equation
Linear equation
The variable has exponent 1, and it is not multiplied by another variable or placed in an exponent.
Quadratic equation
The highest power of the variable is 2.
Exponential equation
The variable appears in the exponent.
Common confusion
The function y = x cubed is neither quadratic nor exponential. It is a polynomial function of degree 3. A function is exponential only when the variable is in the exponent.
How to Identify the Function from a Table
Tables are often the best place to compare patterns. First confirm that the x-values change by equal amounts. Then examine differences or ratios in the y-values.
Linear table: constant first differences
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 2 | 5 | 8 | 11 | 14 |
| First difference | - | +3 | +3 | +3 | +3 |
The first difference is always 3, so the function is linear.
Quadratic table: constant second differences
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 1 | 4 | 9 | 16 | 25 |
| First difference | - | 3 | 5 | 7 | 9 |
| Second difference | - | - | 2 | 2 | 2 |
The first differences are not constant, but the second differences are always 2. Therefore, the function is quadratic.
Exponential table: constant ratio
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 3 | 6 | 12 | 24 | 48 |
| Ratio | - | 2 | 2 | 2 | 2 |
Each output is multiplied by 2, so the function is exponential.
Check the x-spacing first
Difference and ratio tests work directly only when the x-values are equally spaced. If the x-steps are unequal, compare rates over equal intervals or determine the model another way.
How to Identify the Function from a Graph
Linear graph
A linear graph is a straight line. It has the same steepness everywhere. A positive slope rises from left to right, a negative slope falls, and a zero slope is horizontal.
Quadratic graph
A quadratic graph is a parabola. It has one turning point, called the vertex, and a vertical line of symmetry. The graph may decrease and then increase, or increase and then decrease.
Exponential graph
A basic exponential graph is curved and changes slowly on one side but rapidly on the other. For a positive initial value, it usually approaches a horizontal asymptote without crossing it.
| Graph clue | Likely function |
|---|---|
| Straight line with constant slope | Linear |
| U-shaped or upside-down U-shaped curve | Quadratic |
| Growth or decay curve with a horizontal asymptote | Exponential |
How to Identify the Function from a Word Problem
The language of the situation often reveals whether change is additive, accelerating, or multiplicative.
Linear language
- increases by the same amount each hour;
- charges a fixed fee plus a constant amount per unit;
- travels at a constant speed;
- earns the same amount for every item sold.
Quadratic language
- area depends on a changing length and width;
- height of a thrown object over time;
- maximum or minimum value occurs at a vertex;
- the rate of change itself changes at a constant rate.
Exponential language
- grows by the same percentage each period;
- doubles, triples, or is multiplied repeatedly;
- depreciates by a fixed percent each year;
- population or investment grows proportionally to its current amount.
Amount versus percentage
"Increases by 50 each year" describes linear change. "Increases by 5% each year" describes exponential change because the amount added depends on the current value.
Worked Examples
Example 1: Identify from an equation
The variable has exponent 1, and the equation matches the form y = mx + b.
Answer
The function is linear.
Example 2: Identify from a table
The first differences are:
The second differences are:
Answer
The constant second difference shows that the function is quadratic.
Example 3: Identify from a real-world description
The balance is multiplied by 1.04 each year. The amount added is not constant because it depends on the current balance.
Answer
The situation is modeled by an exponential growth function.
Example 4: Compare linear and exponential growth
Plan A has constant additive growth:
Plan B has constant percent growth:
Answer
Plan A is linear; Plan B is exponential.
Common Mistakes
Looking only for an x squared term
A quadratic may be written in factored or vertex form, so expand or recognize the structure before deciding.
Calling every curved graph exponential
Quadratic, cubic, logarithmic, rational, and radical functions can also be curved. Use shape and rate patterns together.
Using first differences with uneven x-values
Difference tests require equal x-steps. Uneven intervals can create misleading patterns.
Confusing repeated addition with repeated multiplication
Adding the same amount is linear. Multiplying by the same factor is exponential.
Assuming all percent problems are exponential
A one-time percent calculation is not automatically an exponential function. Repeated percent change over equal intervals is exponential.
Ignoring the domain
A real-world model may use only whole-number times or nonnegative inputs even though the algebraic function is defined more broadly.
A Fast Identification Checklist
- Is the graph a straight line? If yes, the function is linear.
- Does the equation have degree 2? If yes, it is quadratic.
- Is the variable in the exponent? If yes, it may be exponential.
- Are first differences constant for equal x-steps? Linear.
- Are second differences constant? Quadratic.
- Are consecutive y-values multiplied by a constant ratio? Exponential.
- Does the situation use a constant amount, changing rate, or constant percent?
Why the Difference Matters
Choosing the correct function type determines which equation, graph, prediction, and solving method are appropriate. A linear model may work over a short interval but fail when growth is proportional. A quadratic model can describe a maximum or minimum that a basic exponential model does not have. An exponential model may eventually exceed a linear or quadratic model because its growth compounds.
When solving a modeling problem, do not select a function only because it fits the first few values. Use the context, the rate pattern, and the graph together.
Practice Functions and Graphs
Practice identifying functions from equations, tables, graphs, and real-world models. Start with one function type and then continue with mixed comparison questions.