Introduction to Linear Functions

Learn what linear functions are, how to identify them, and how they create straight lines on a graph.

Intermediate25 minLesson
Printable worksheet

Definition

A linear function is a function whose graph is a straight line. It can be written in the form:
y=mx+by = mx + b
Where:
  • mm is the slope (how steep the line is)
  • bb is the y-intercept (where the line crosses the y-axis)
  • xx is the input (independent variable)
  • yy is the output (dependent variable)
The key characteristic of a linear function is that it has a constant rate of change - for every unit increase in xx, the value of yy changes by the same amount (mm).

Try it now

Which of the following is a linear function?

Worked Examples

Is y=3x−2y = 3x - 2 a linear function?
1

Check the form

Compare to y=mx+by = mx + b → It matches the pattern
2

Identify the slope

The coefficient of xx is 33 → m=3m = 3
3

Identify the y-intercept

The constant term is −2-2 → b=−2b = -2
4

Verify linearity

No exponents on xx, no xx in denominator → It is linear

Common Mistakes

Thinking y=x2y = x^2 is linear because it has an xx

Why it's wrong: The exponent matters! x2x^2 means xx multiplied by itself, which creates a curve, not a straight line.

Correct: Linear functions have xx to the first power only. Check that there's no x2x^2, x3x^3, x\sqrt{x}, or 1x\frac{1}{x}.

Confusing slope and y-intercept in y=mx+by = mx + b

Why it's wrong: It's easy to mix them up. The slope (mm) is the coefficient of xx, while bb is the constant term.

Correct: In y=3x+5y = 3x + 5, the slope is 33 (multiplies xx) and the y-intercept is 55 (stands alone).

Thinking y=5y = 5 is not a linear function

Why it's wrong: This is actually y=0x+5y = 0x + 5, a horizontal line with slope 00.

Correct: y=5y = 5 is linear - it's a special case with zero slope. The graph is a horizontal line at y=5y = 5.

Interactive Visual

Linear Function Explorer

y = 2x + 1
Slope (m)2
Y-Intercept (b)1
b
run
rise
xy
-2-3
-1-1
01
13
25

Function graph

Plot these points:

  • (-1, -1)
  • (0, 1)
  • (1, 3)
  • (2, 5)
-3-2-11234-3-2-112345670xy

Tap the grid to place a point. Tap a point again to remove it.Place at least 2 points to see the line through them.

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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History

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Practice Problems

16 problems
Problem 1 of 16
Easy
Which of the following is a linear function?

Why It Matters

Linear functions appear everywhere in real life:
  • Cell phone plans: A plan that costs 20 dollars per month plus 5 cents per text is linear: y=0.05x+20y = 0.05x + 20
  • Distance and time: Driving at a constant speed of 60 mph creates a linear relationship: d=60td = 60t
  • Temperature conversion: Celsius to Fahrenheit is linear: F=1.8C+32F = 1.8C + 32
  • Business: A company's costs often include fixed costs plus variable costs per item
Understanding linear functions helps you predict outcomes, compare options, and make better decisions!

Real World Applications

Cell Phone Plans

Many phone plans have a base fee plus a per-usage charge, creating a linear function.

Example:

A plan costs 25 dollars monthly plus 10 cents per minute over the limit. If xx is extra minutes, then y=0.10x+25y = 0.10x + 25.

1Try It Yourself

A phone plan charges 30 dollars per month plus 5 cents per text message.

How much would 100 text messages cost in a month?

Step 1: Write the mathematical expression

Write the function as y=mx+by = mx + b and substitute:

Distance and Speed

When traveling at a constant speed, distance is a linear function of time.

Example:

Driving at 50 mph: d=50td = 50t, where dd is distance in miles and tt is time in hours.

2Try It Yourself

A cyclist rides at a constant speed of 15 mph.

How far will they travel in 3 hours?

Step 1: Write the mathematical expression

Use d=15td = 15t:

Earnings and Hours Worked

Hourly wages create a linear relationship between hours worked and money earned.

Example:

Earning 12 dollars per hour: y=12xy = 12x, where xx is hours and yy is earnings in dollars.

3Try It Yourself

A student earns 15 dollars per hour tutoring, plus a 20 dollar bonus for each new client.

If they tutored for 4 hours for a new client, how much did they earn?

Step 1: Write the mathematical expression

Calculate earnings:

Key Takeaways

  • 1A linear function has the form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept
  • 2The graph of a linear function is always a straight line
  • 3Linear functions have a constant rate of change - when xx increases by 1, yy always changes by mm
  • 4To identify a linear function, check that xx has no exponent other than 1 and doesn't appear in a denominator or under a radical

Frequently Asked Questions

A function is linear if its graph is a straight line. Mathematically, this means the equation can be written as y=mx+by = mx + b with no exponents on variables, no variables in denominators, and no variables under radicals.
A function is linear if its graph is a straight line. Mathematically, this means the equation can be written as y=mx+by = mx + b with no exponents on variables, no variables in denominators, and no variables under radicals.
Yes! A slope of 0 gives a horizontal line (y=by = b). A negative slope means the line goes down from left to right (like a hill you ski down).
A linear equation in two variables, like 2x+y=72x + y = 7, has a straight line as its graph. If you solve it for yy, you get a linear function: here y=−2x+7y = -2x + 7. A vertical line such as x=3x = 3 is a linear equation but not a function, because the single input x=3x = 3 has many outputs.

Glossary

Linear function
A function whose graph is a straight line, written as y=mx+by = mx + b
Slope
The rate of change of a line, represented by mm in y=mx+by = mx + b. It measures steepness.
Y-intercept
The point where the line crosses the y-axis, represented by bb in y=mx+by = mx + b
Rate of change
How much the output (yy) changes for each unit change in input (xx)
Constant rate of change
When the rate of change is the same between any two points - the defining feature of linear functions
Today's challengeOne problem, three tries

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