How to Explain Linear Functions to Eighth Graders

"For every hour I work, I earn $15."

That's a linear relationship. Linear functions are everywhere—describing constant rates of change in the world around us. When students master y = mx + b, they've gained a powerful tool for understanding and predicting.

Why Linear Functions Matter

Linear functions are essential for:

• Understanding constant rates of change
• Graphing and interpreting lines
• Making predictions and extrapolations
• Modeling real-world situations
• Foundation for systems of equations
• Preparation for more complex functions

What Makes a Function Linear?

Definition

A linear function graphs as a straight line and has a constant rate of change.

Standard Forms

Slope-Intercept Form:

y = mx + b
  m = slope
  b = y-intercept

Standard Form:

Ax + By = C

Examples of Linear Functions

y = 2x + 3       (slope = 2, y-intercept = 3)
y = -x + 5       (slope = -1, y-intercept = 5)
y = (1/2)x       (slope = 1/2, y-intercept = 0)
f(x) = 4x - 1    (same thing, function notation)

Not Linear

y = x²           (curved - quadratic)
y = 1/x          (curved - rational)
y = 2ˣ           (curved - exponential)

Understanding Slope

What Slope Means

Slope measures the steepness of a line.

It answers: "How much does y change when x increases by 1?"

The Formula

Slope = m = rise/run = (change in y)/(change in x) = (y₂ - y₁)/(x₂ - x₁)

Calculating Slope from Two Points

Example: Find the slope between (2, 3) and (5, 9)

m = (y₂ - y₁)/(x₂ - x₁)
m = (9 - 3)/(5 - 2)
m = 6/3
m = 2

The line goes UP 2 for every 1 it goes RIGHT.

Visual Understanding

Positive Slope     Negative Slope    Zero Slope      Undefined Slope
   (m > 0)            (m < 0)          (m = 0)         (vertical)

     /                  \             ________            |
    /                    \                                |
   /                      \                               |

 "Uphill"             "Downhill"       "Flat"          "Cliff"

Slope Values

Slope	Meaning	Example
m = 2	Go up 2, right 1	y = 2x
m = 1/2	Go up 1, right 2	y = (1/2)x
m = -3	Go down 3, right 1	y = -3x
m = 0	Horizontal line	y = 5
undefined	Vertical line	x = 3

Comparing Slopes

        y
        |   /  /y = 3x (steepest)
        |  /  /
        | /  /  y = x
        |/  /
        +--/------- x
          /
         y = (1/2)x (gentlest)

Larger absolute value = steeper line.

Understanding Y-Intercept

What Y-Intercept Means

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

It's the output when x = 0.

Finding Y-Intercept

In y = mx + b, the y-intercept is b.

y = 3x + 5    → y-intercept is 5, point (0, 5)
y = -2x - 1   → y-intercept is -1, point (0, -1)
y = 4x        → y-intercept is 0, point (0, 0)

Visualizing Y-Intercepts

        y
        |
      5 +  *  ← y = 2x + 5 crosses at (0, 5)
        |   \
      2 + *  \  ← y = x + 2 crosses at (0, 2)
        |  \  \
      0 +-*-+--+---- x
        | \
     -2 +  *  ← y = x - 2 crosses at (0, -2)

Graphing Linear Functions

Method 1: Using Slope and Y-Intercept

Graph y = 2x - 3

Step 1: Plot the y-intercept (0, -3)

Step 2: Use slope (m = 2 = 2/1) to find another point

• From (0, -3): go up 2, right 1 → (1, -1)

Step 3: Draw the line through both points

        y
      4 +
        |        *
      2 +     *
        |  *
      0 +--+--+--+--+-- x
        |
     -2 +  *
        |*
     -4 +

Method 2: Using Two Points

Graph y = -x + 4

Find two points by choosing x-values:

• x = 0: y = -(0) + 4 = 4 → (0, 4)
• x = 4: y = -(4) + 4 = 0 → (4, 0)

Plot both points and connect.

Method 3: Using X and Y Intercepts

Graph 2x + 3y = 6

X-intercept (set y = 0):

2x + 0 = 6
x = 3 → (3, 0)

Y-intercept (set x = 0):

0 + 3y = 6
y = 2 → (0, 2)

Plot (3, 0) and (0, 2), draw the line.

Writing Equations of Lines

Given Slope and Y-Intercept

Just plug into y = mx + b.

Slope = 3, y-intercept = -2:

y = 3x - 2

Given Slope and One Point

Use point-slope form or substitute to find b.

Slope = 2, through (3, 7):

y = mx + b
7 = 2(3) + b
7 = 6 + b
b = 1

Equation: y = 2x + 1

Given Two Points

Through (1, 5) and (3, 11):

Step 1: Find slope

m = (11 - 5)/(3 - 1) = 6/2 = 3

Step 2: Find b using one point

5 = 3(1) + b
b = 2

Equation: y = 3x + 2

Point-Slope Form

y - y₁ = m(x - x₁)

Slope = 4, through (2, 3):

y - 3 = 4(x - 2)
y - 3 = 4x - 8
y = 4x - 5

Parallel and Perpendicular Lines

Parallel Lines

Same slope, different y-intercept.

y = 2x + 3  and  y = 2x - 1 are parallel

        y
        |   /  /
        |  /  /
        | /  /
        +--+--+--+-- x

Perpendicular Lines

Slopes are negative reciprocals. (Multiply to -1)

y = 2x + 1  and  y = -1/2x + 4 are perpendicular

Slopes: 2 and -1/2
Check: 2 × (-1/2) = -1 ✓

        y
        |   /
        |  /
        | /
        +--+--+--+-- x
           \
            \

Finding Parallel Line Equation

Find a line parallel to y = 3x + 2 through (1, 5):

• Same slope: m = 3
• Through (1, 5): 5 = 3(1) + b → b = 2

Equation: y = 3x + 2 (same line!) Let me redo:

• 5 = 3(1) + b → 5 = 3 + b → b = 2

Wait, that gives the same equation because (1, 5) is on y = 3x + 2!

Try (1, 4):

• 4 = 3(1) + b → b = 1
• Equation: y = 3x + 1

Finding Perpendicular Line Equation

Find a line perpendicular to y = 2x + 3 through (4, 1):

• Perpendicular slope: m = -1/2
• Through (4, 1): 1 = (-1/2)(4) + b → 1 = -2 + b → b = 3

Equation: y = -1/2x + 3

Real-World Linear Functions

Recognizing Linear Relationships

Linear situations have constant rates:

• Earning $15 per hour
• Driving 60 miles per hour
• Paying $0.25 per text message

Example: Cell Phone Plan

"A cell phone plan costs $30 per month plus $0.05 per text."

C(t) = 0.05t + 30

m = 0.05 (cost per text)
b = 30 (monthly base fee)

Questions:

• What does the slope mean? Each text costs $0.05.
• What does the y-intercept mean? Base monthly fee is $30.
• What's the cost for 200 texts? C(200) = 0.05(200) + 30 = $40

Example: Pool Draining

"A pool has 10,000 gallons and drains at 500 gallons per hour."

G(t) = -500t + 10000

m = -500 (losing water)
b = 10000 (starting amount)

When is the pool empty?

0 = -500t + 10000
500t = 10000
t = 20 hours

Example: Temperature Conversion

F(C) = (9/5)C + 32

m = 9/5 = 1.8 (°F per °C)
b = 32 (offset)

Hands-On Activities

Human Number Line

Students stand on a number line and model slope:

• "Everyone go up 2, right 1"
• "Form the line y = 2x"

Slope Field Trip

Find slopes around school:

• Wheelchair ramps
• Stairs
• Playground slides

Calculate rise/run for each!

Match the Scenario

Give stories and equations. Students match:

• "Earn $10/hour starting with $50" → y = 10x + 50
• "Temperature drops 3° each hour from 75°" → y = -3x + 75

Graphing Calculator Exploration

Enter different equations in Y=:

• How does changing m affect the graph?
• How does changing b affect the graph?

Real Data Collection

Measure something with a constant rate:

• Height of a plant over days
• Distance walked over time
• Money spent over number of items bought

Graph the data and find the equation.

Common Mistakes and How to Fix Them

Mistake 1: Rise/Run Reversed

Wrong: m = (x₂ - x₁)/(y₂ - y₁)

Fix: "Rise" is vertical (y), "run" is horizontal (x).
m = (y₂ - y₁)/(x₂ - x₁)

Mistake 2: Sign Errors in Slope

Points: (2, 5) and (4, 1)

Wrong: m = (5-1)/(4-2) = 4/2 = 2

Fix: Keep order consistent!
m = (1-5)/(4-2) = -4/2 = -2
OR
m = (5-1)/(2-4) = 4/-2 = -2

Mistake 3: Graphing Slope Incorrectly

Wrong: For m = 2/3, go right 2, up 3

Fix: Rise is numerator (vertical), run is denominator (horizontal).
For m = 2/3: up 2, right 3

Mistake 4: Confusing Slope and Y-Intercept

In y = 3 + 5x:

Wrong: m = 3, b = 5

Fix: Rewrite in y = mx + b form: y = 5x + 3
m = 5, b = 3

Mistake 5: Vertical Lines Have Slope 0

Wrong: x = 3 has slope 0

Fix: Vertical lines have UNDEFINED slope (dividing by zero).
Horizontal lines (y = 3) have slope 0.

Practice Ideas for Home

Slope Spotting

Find slopes in everyday life:

• What's the slope of your driveway?
• What's the slope of stairs?
• What's a ramp's slope?

Equation Writing

Write equations for scenarios:

• Taxi: $3 plus $2/mile → y = 2x + 3
• Savings: Start with $100, add $25/week → y = 25x + 100

Graph Matching

Given equations and graphs, match them up based on slope and y-intercept.

Prediction Practice

Use a linear equation to predict:

• If y = 3x + 5 represents earnings, how much after 20 hours?
• If a tank drains at y = -100x + 5000, when is it empty?

Connecting to Future Concepts

Systems of Equations

Finding where two lines intersect:

y = 2x + 1
y = -x + 4

Solution: where lines cross

Linear Inequalities

Shading regions:

y > 2x + 1 (shade above the line)

Absolute Value Functions

V-shaped graphs related to linear:

y = |x| is made of two linear pieces

Piecewise Functions

Different linear rules for different domains:

f(x) = { 2x + 1,  x < 0
       { -x + 3,  x ≥ 0

Linear Regression

Finding the "best fit" line for real data.

The Bottom Line

Linear functions are the simplest and most useful functions. The equation y = mx + b tells the whole story:

• m (slope) = rate of change = rise/run
• b (y-intercept) = starting value = where it crosses y-axis

When students can move fluently between equations, graphs, tables, and real-world contexts, they've mastered linear functions. This skill forms the foundation for all the algebra and analysis that follows.

Every line has a story: how fast something is changing (slope) and where it started (y-intercept). Teaching students to read and write these stories prepares them for mathematics, science, and real-world problem-solving throughout their lives.