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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Worksheet Answer key

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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 $y = x^2$ or $y = \frac{1}{x}$ . Graph them to show they are curves, not lines.

The y-intercept is where x = 0 on the line

Remediation: This is actually correct but often misunderstood. Clarify that the y-intercept is the y-value when x = 0, which is the point (0, b).

Slope is always a whole number

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

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 + b

Where:

$m$ is the slope (how steep the line is)
$b$ is the y-intercept (where the line crosses the y-axis)
$x$ is the input (independent variable)
$y$ 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

x

, the value of

y

changes by the same amount (

m

Worked Examples

Is $y = 3x - 2$ a linear function?

Check the form

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

Identify the slope

The coefficient of $x$ is $3$ → $m = 3$

Identify the y-intercept

The constant term is $-2$ → $b = -2$

Verify linearity

No exponents on $x$ , no $x$ in denominator → It is linear

Answer: Yes, $y = 3x - 2$ is a linear function with slope $3$ and y-intercept $-2$ .

Common Mistakes

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

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

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

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

Why it's wrong: It's easy to mix them up. The slope ( $m$ ) is the coefficient of $x$ , while $b$ is the constant term.

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

Thinking $y = 5$ is not a linear function

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

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

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: $y = 0.05x + 20$
Distance and time: Driving at a constant speed of 60 mph creates a linear relationship: $d = 60t$
Temperature conversion: Celsius to Fahrenheit is linear: $F = 1.8C + 32$
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 $x$ is extra minutes, then $y = 0.10x + 25$ .

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 $y = mx + b$ and substitute:

Distance and Speed

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

Example:

Driving at 55 mph: $d = 55t$ , where $d$ is distance in miles and $t$ 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 $d = 15t$ :

Earnings and Hours Worked

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

Example:

Earning 12 dollars per hour: $y = 12x$ , where $x$ is hours and $y$ 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 $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept
2The graph of a linear function is always a straight line
3Linear functions have a constant rate of change - when $x$ increases by 1, $y$ always changes by $m$
4To identify a linear function, check that $x$ 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 + 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 = 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 is any equation whose graph is a line. A linear function is specifically written with

y

alone on one side:

y = mx + b

. Every linear function is a linear equation, but not every linear equation is written as a function.

Glossary

Linear function: A function whose graph is a straight line, written as $y = mx + b$
Slope: The rate of change of a line, represented by $m$ in $y = mx + b$ . It measures steepness.
Y-intercept: The point where the line crosses the y-axis, represented by $b$ in $y = mx + b$
Rate of change: How much the output ( $y$ ) changes for each unit change in input ( $x$ )
Constant rate of change: When the rate of change is the same between any two points - the defining feature of linear functions

Teacher Guide: Introduction to Linear Functions

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Cell Phone Plans

Distance and Speed

Earnings and Hours Worked

Key Takeaways

Frequently Asked Questions

Glossary

More in This Topic

Slope-Intercept Form (y = mx + b)

Understanding Slope

Point-Slope Form

Standard Form of Linear Equations