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Teacher Guide: Introduction to Linear Functions

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

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Class quiz

10 questions on Linear Functions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define a linear function and identify its key components (slope and y-intercept)
  • Determine whether a function is linear by examining its equation
  • Create tables of values for linear functions and recognize the constant rate of change
  • Graph linear functions using the slope and y-intercept
  • Apply linear functions to real-world situations
Prerequisites
  • • Understanding of coordinate planes and ordered pairs
  • • Ability to substitute values into algebraic expressions
  • • Basic understanding of functions as input-output machines
  • • Familiarity with positive and negative numbers
Discussion Starters
  • 1. Why do you think these functions are called 'linear'? What does that word suggest?
  • 2. Can you think of a real-life situation where a relationship is NOT linear?
  • 3. What happens to the graph when the slope is negative? When it's zero?
  • 4. If two lines have the same slope but different y-intercepts, what do their graphs look like?
Common Misconceptions

Any equation with x is linear

Remediation: Show counterexamples like y=x2y = x^2 or y=1xy = \frac{1}{x}. Graph them to show they are curves, not lines.

The y-intercept is where the line crosses the x-axis

Remediation: Stress that the y-intercept lies on the y-axis, where x=0x = 0. Have students substitute x=0x = 0 into y=2x+3y = 2x + 3 to get the point (0,3)(0, 3) and mark it on the y-axis. The point where y=0y = 0 is the x-intercept, a different point.

Slope is always a whole number

Remediation: Provide examples with fractional and decimal slopes like y=0.5x+3y = 0.5x + 3 or y=23x−1y = \frac{2}{3}x - 1.

Differentiation Ideas

For Struggling Students:

  • • Start with positive integer slopes only
  • • Use concrete examples (cost per item, distance per hour) before abstract equations
  • • Provide graphing grids with scales already marked

For On-Level Students:

  • • Practice with negative slopes and fractional values
  • • Convert between different representations (table, graph, equation)
  • • Solve real-world problems involving linear functions

For Advanced Students:

  • • Explore systems of linear equations (where two lines intersect)
  • • Investigate parallel and perpendicular lines
  • • Analyze piecewise linear functions
Standards Alignment
  • 8.F.A.3 (CCSS.MATH.CONTENT.8.F.A.3)

    Interpret the equation y = mx + b as defining a linear function whose graph is a straight line

  • 8.F.B.4 (CCSS.MATH.CONTENT.8.F.B.4)

    Construct a function to model a linear relationship between two quantities

  • F-LE.A.1 (CCSS.MATH.CONTENT.HSF.LE.A.1)

    Distinguish between situations that can be modeled with linear functions and with exponential functions

Lesson Resources
  • visualInteractive Line Explorer

    Adjust slope and y-intercept to see how the line changes

  • activityLinear or Not? Card Sort

    Sort equations into linear and non-linear categories

  • worksheetReal-World Linear Functions

    Practice identifying and writing linear functions from scenarios

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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).

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.

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

What makes a function linear vs non-linear?

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.

Can the slope be zero or negative?

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).

What's the difference between a linear function and a linear equation?

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

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