Introduction to Linear Functions
Learn what linear functions are, how to identify them, and how they create straight lines on a graph.
Definition
- is the slope (how steep the line is)
- is the y-intercept (where the line crosses the y-axis)
- is the input (independent variable)
- is the output (dependent variable)
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Worked Examples
Is a linear function?
Check the form
Compare to → It matches the pattern
Identify the slope
The coefficient of is →
Identify the y-intercept
The constant term is →
Verify linearity
No exponents on , no in denominator → It is linear
Answer: Yes, is a linear function with slope and y-intercept .
Common Mistakes
Thinking is linear because it has an
Why it's wrong: The exponent matters! means multiplied by itself, which creates a curve, not a straight line.
Correct: Linear functions have to the first power only. Check that there's no , , , or .
Confusing slope and y-intercept in
Why it's wrong: It's easy to mix them up. The slope () is the coefficient of , while is the constant term.
Correct: In , the slope is (multiplies ) and the y-intercept is (stands alone).
Thinking is not a linear function
Why it's wrong: This is actually , a horizontal line with slope .
Correct: is linear - it's a special case with zero slope. The graph is a horizontal line at .
Interactive Visual
Linear Function Explorer
| x | y |
|---|---|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
Interactive Sandbox
Interactive Grapher
Try these examples:
y = 2x + 1
m=2, b=1
Expression Calculator
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Practice Problems
16 problemsWhich of the following is a linear function?
Why It Matters
- Cell phone plans: A plan that costs 20 dollars per month plus 5 cents per text is linear:
- Distance and time: Driving at a constant speed of 60 mph creates a linear relationship:
- Temperature conversion: Celsius to Fahrenheit is linear:
- Business: A company's costs often include fixed costs plus variable costs per item
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 is extra minutes, then .
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 and substitute:
Distance and Speed
When traveling at a constant speed, distance is a linear function of time.
Example:
Driving at 55 mph: , where is distance in miles and is time in hours.
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 :
Earnings and Hours Worked
Hourly wages create a linear relationship between hours worked and money earned.
Example:
Earning 12 dollars per hour: , where is hours and is earnings in dollars.
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 , where is the slope and is the y-intercept
- 2The graph of a linear function is always a straight line
- 3Linear functions have a constant rate of change - when increases by 1, always changes by
- 4To identify a linear function, check that has no exponent other than 1 and doesn't appear in a denominator or under a radical
Frequently Asked Questions
Glossary
- Linear function
- A function whose graph is a straight line, written as
- Slope
- The rate of change of a line, represented by in . It measures steepness.
- Y-intercept
- The point where the line crosses the y-axis, represented by in
- Rate of change
- How much the output () changes for each unit change in input ()
- Constant rate of change
- When the rate of change is the same between any two points - the defining feature of linear functions