Linear Functions: Definition, Graph, Value Table and Linear Growth
A linear function is a function of the form f(x) = mx + b whose graph is a straight line, where m is the slope and b is the y-intercept. In brief: in f(x) = 2x + 1, each increase of x by 1 raises the output by 2, and the line crosses the y-axis at (0, 1). Linear growth means adding the same amount in every equal step.
Linear functions are the simplest models of change: a steady speed, a fixed price per item, a constant saving per week. They are also the basis for understanding curves, because any smooth curve looks like a line when viewed closely enough. This guide explains the definition, the value table, the graph and the idea of linear growth with worked examples.
What is a linear function?
A linear function has the form f(x) = mx + b, where m and b are constants. The number m is the slope, which tells how much f(x) changes when x increases by 1. The number b is the y-intercept, the value f(0). The domain is all real numbers, and the range is all real numbers unless m = 0.
For a linear function the rate of change is the same everywhere, and this is its defining property.
How do you make a value table for a linear function?
Pick equally spaced x-values and compute f(x). The outputs will change by the same amount, m, in every row. For f(x) = 3x − 2:
| x | 3x − 2 | f(x) | Change from previous |
|---|---|---|---|
| −2 | −6 − 2 | −8 | – |
| −1 | −3 − 2 | −5 | +3 |
| 0 | 0 − 2 | −2 | +3 |
| 1 | 3 − 2 | 1 | +3 |
| 2 | 6 − 2 | 4 | +3 |
| 3 | 9 − 2 | 7 | +3 |
| 4 | 12 − 2 | 10 | +3 |
The constant change +3 confirms the slope m = 3. The value at x = 0 is −2, which is the y-intercept b.
How do you graph a linear function?
- Plot the y-intercept (0, b).
- Use the slope m = rise/run to find a second point: from (0, b) go 1 right and m up (or down if m is negative).
- Draw a straight line through the two points and extend it in both directions.
- Check with a third point, such as x = 2.
In the sketch the line goes through (0, −2) and (1, 1): from the intercept, one step right and three steps up.
What is linear growth?
A quantity grows linearly if it increases by the same amount in each equal time interval. If a plant is 12 cm tall and grows 2 cm per week, then after t weeks its height is h(t) = 12 + 2t. After 5 weeks, h(5) = 22 cm. A negative slope gives linear decay: a tank with 80 litres losing 4 litres per minute holds V(t) = 80 − 4t litres, which is empty at t = 20 minutes.
Linear growth is different from exponential growth, where a quantity is multiplied by the same factor per step. Over time, the exponential model grows much faster.
Worked example: a phone plan
A plan costs 15 dollars per month plus 0.10 dollars per minute. The cost is C(x) = 0.1x + 15. For 120 minutes: C(120) = 12 + 15 = 27 dollars. The slope 0.1 is the price per minute, and the intercept 15 is the fixed fee. If the budget is 30 dollars, solve 0.1x + 15 = 30 to get x = 150 minutes.
What are special cases of linear functions?
- Proportional function: b = 0, so f(x) = mx. The line passes through the origin.
- Constant function: m = 0, so f(x) = b. The graph is a horizontal line, and the function is neither increasing nor decreasing.
- Identity function: m = 1 and b = 0, so f(x) = x.
A vertical line x = c is not the graph of a function, because it assigns infinitely many outputs to one input.
Where do linear functions come from?
Proportional reasoning was used in ancient Egypt and Babylon for trade, land measurement and astronomy. The graph of a straight line as an equation became possible with the analytic geometry of Descartes and Fermat in the seventeenth century. The term “linear” comes from the Latin word for line, linea.
Where are linear functions used today?
Taxi fares, hourly wages, utility bills with a fixed fee and unit rates are all linear. Converting units is linear: Fahrenheit = 1.8 · Celsius + 32. Engineers approximate complex behaviour with linear models over a small range, and statisticians fit a line to data with a least squares method.
Which mistakes are common?
Mixing up slope and intercept
In y = 5 + 2x the slope is 2 and the intercept is 5. Rewrite the formula as mx + b before reading values.
Counting the run wrongly
Slope is rise over run: up or down divided by right. A slope of 3 is 3/1, not 1/3.
Assuming a table is linear after two points
Check that the change is constant for all rows.
Forgetting units
In applications the slope has units, such as dollars per minute. State them in the answer.