Linear Function Modeling — I can use linear functions to...

SECTION 1

Where Did Linear Functions Come From?

People have been solving problems about constant rates for thousands of years. Ancient farmers needed to know how much grain to plant for a certain area of land. Merchants needed to figure out the total cost of many items at the same price. These are all situations where one quantity changes at a steady, constant rate — and that's exactly what linear functions describe.

~1800 BCE

Babylonian Math Tables

Ancient Babylonians created clay tablets with tables showing constant-rate relationships, like the cost of different amounts of grain.

~300 BCE

Euclid's Geometry

The Greek mathematician Euclid studied proportions and ratios. His work laid the groundwork for understanding constant rates.

1637

Descartes and the Coordinate Plane

René Descartes invented the x-y coordinate plane. This let people draw equations as lines and curves for the first time.

Today

Linear Models Everywhere

We use linear functions daily — from ride-share pricing to tracking a runner's distance. Any time something grows at a constant rate, a linear function can model it.

Here's the big question these thinkers were trying to answer: If something changes at a steady rate, how can we predict what will happen next? That's the power of a linear function — it gives you a formula to make predictions.

SECTION 2

Core Principles of Linear Functions

Before we start solving problems, let's lock in a few key ideas. A linear function is a rule that creates a straight line when you graph it. It always has a constant rate of change, meaning the output goes up (or down) by the same amount each time the input goes up by one.

Slope (Rate of Change)

The slope tells you how fast the output changes. It is the amount of change in y for every 1-unit increase in x. Think of it as the 'speed' of the function.

Y-Intercept (Starting Value)

The y-intercept is the output value when the input is 0. It's your starting point — like a flat fee before anything else is added.

y = mx + b

This is the slope-intercept form. Here, m is the slope and b is the y-intercept. Every linear function can be written this way.

Constant Rate

A constant rate means the same amount is added each time. If you earn $10 per hour, after 1 hour you have $10, after 2 hours $20, and so on. The rate never changes.

✦ KEY TAKEAWAY

Think of a linear function like a vending machine. You put in a number of dollars (x), and the machine gives you a result (y). The slope is how much each dollar 'buys,' and the y-intercept is what you get before you even put in any money — like a free sample!

SECTION 3

Seeing a Linear Function on a Graph

A picture is worth a thousand words — especially in math. When you graph a linear function, you always get a straight line. The diagram below shows the function y = 3x + 2. Notice how the line starts at the point (0, 2) on the y-axis. That's the y-intercept. Then it rises 3 units for every 1 unit it moves to the right. That's the slope.

The graph shows y = 3x + 2. The cyan dots are plotted points. The yellow dashed line shows the 'run' (1 unit right), and the pink dashed line shows the 'rise' (3 units up). The slope is 3, and the y-intercept is 2.

Look at how the line goes up from left to right. Every time x increases by 1, y increases by 3. That's the constant rate of change. The line crosses the y-axis at 2. That's the starting value, or y-intercept.

SECTION 4

The Math Behind Linear Models

Every linear function can be written using slope-intercept form. Let's break down the formula and see what each letter means.

SLOPE-INTERCEPT FORM

y = mx + b

y = the output (what you're finding), m = slope (rate of change), x = the input (the value you choose), b = y-intercept (the starting value when x = 0)

To find the slope when you know two points, use the slope formula.

SLOPE FORMULA

m = (y₂ − y₁) ÷ (x₂ − x₁)

Pick any two points on the line. Subtract the y-values (the rise), then divide by the difference in x-values (the run).

Let's see how these formulas connect to real life. Imagine a taxi charges $4 just to get in the car, plus $2 per mile. The function would be: y = 2x + 4. Here, m = 2 (dollars per mile) and b = 4 (the flat fee).

DISTANCE FORMULA (CONSTANT SPEED)

d = r × t

d = distance, r = rate (speed), t = time. This is a linear function where the slope is the speed and the y-intercept is 0 (you start at distance 0).

SECTION 5

Three Types of Linear Models

Linear functions show up in three very common situations: distance problems, cost problems, and constant rate problems. The diagram below compares all three side by side.

All three model types produce straight lines. In distance problems, the slope is speed and the y-intercept is usually 0. In cost problems, the slope is the price per item and the y-intercept is a flat fee. In constant rate problems, the slope is the rate and the y-intercept is the starting value.

Common linear model types and what their parts mean

Model Type	What Slope Means	What Y-Intercept Means	Example
Distance	Speed (miles per hour)	Starting distance (usually 0)	d = 60t (driving 60 mph)
Cost	Price per item or unit	Flat fee or membership cost	C = 3n + 5 ($3/item + $5 shipping)
Constant Rate	Amount gained or lost per unit	Starting amount	y = 15x + 50 (saving $15/week, starting with $50)

SECTION 6

Worked Example: Movie Streaming Cost

Let's work through a real problem step by step. You sign up for a streaming service that charges a $10 monthly fee plus $3 for each movie you rent. How much will you spend if you rent 7 movies this month?

Finding Total Cost

Step 1 — Identify the slope and y-interceptThe cost per movie is $3, so the slope m = 3. The monthly fee is $10, so the y-intercept b = 10.

m = 3, b = 10

Step 2 — Write the equationUsing y = mx + b, we get:

C = 3n + 10

Step 3 — Substitute the input valueYou want to rent 7 movies, so plug in n = 7: C = 3(7) + 10

C = 3 × 7 + 10

Step 4 — SimplifyMultiply first: 3 × 7 = 21. Then add: 21 + 10 = 31.

C = $31

✅ Check Your Work!

You can verify by making a table: 0 movies = $10, 1 movie = $13, 2 movies = $16 … each time the cost goes up by $3. At 7 movies you get $31. ✓

SECTION 7

When Linear Models Work — and When They Don't

Linear functions are super useful, but they don't work for every situation. Let's compare when they're a great fit and when you need a different kind of model.

Strengths and limitations of linear function models

Strengths ✅	Limitations ⚠️
Easy to write and use — just y = mx + b	Only works when the rate of change is constant
Great for predictions — plug in any x to find y	Can't model things that speed up or slow down (like a bouncing ball)
Graphs are simple straight lines, easy to read	Doesn't handle curves, like population growth
Works for distance, cost, temperature, savings, and more	May give unrealistic answers far outside the data range

✦ KEY TAKEAWAY

Think of a linear model like a ruler — it's perfect for drawing straight lines, but it can't draw curves. If the rate of change stays the same, a linear model is your best friend. If the rate changes, you'll need a different tool later on (like quadratic or exponential functions).

SECTION 8

From Linear to Nonlinear: What Comes Next?

You've learned that a linear function has a constant rate of change. But what happens when the rate itself changes? That's where nonlinear functions come in. In future courses, you'll meet exponential functions (where things double repeatedly) and quadratic functions (where things speed up or slow down).

Linear vs. nonlinear functions

Feature	Linear (You Are Here!)	Nonlinear (Coming Soon)
Graph Shape	Straight line	Curved line
Rate of Change	Always the same (constant)	Changes as x changes
Equation Form	y = mx + b	y = ax² + bx + c, y = a × bˣ, and others
Real-World Example	Driving at a steady 50 mph	A ball thrown into the air (slows, stops, falls)

The great news is that everything you learn about linear functions — slope, y-intercept, graphing, writing equations — will help you understand those more advanced models when you get there. You're building a strong foundation right now!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In the equation y = 5x + 12, what does the 5 represent and what does the 12 represent? Give a real-world example that could match this equation.

PROBLEM 2 — BASIC CALCULATION

A bike rental shop charges $8 per hour. Write a linear function for the total cost, and find how much it costs to rent a bike for 5 hours.

PROBLEM 3 — INTERMEDIATE

A cell phone plan costs $25 per month plus $0.10 per text message. Last month, your bill was $40. How many text messages did you send?

PROBLEM 4 — APPLIED

Two friends are saving money. Amir has $30 and saves $8 per week. Brianna has $60 and saves $5 per week. After how many weeks will they have the same amount?

PROBLEM 5 — CRITICAL THINKING

A car is 200 miles from home and drives toward home at 50 miles per hour. Write a linear function for the distance from home over time. What does a negative slope mean here? When does the car arrive home?

SUMMARY

Lesson Summary

A linear function is a rule written as y = mx + b, where m (the slope) is the constant rate of change and b (the y-intercept) is the starting value. Its graph is always a straight line. You can use this form to model distance, cost, and constant rate problems by identifying what the slope and y-intercept mean in the real-world situation.

To solve a problem, write the equation by finding m and b, then substitute your known value and simplify. Linear models work best when the rate of change stays constant. If the rate changes, you'll explore nonlinear functions in future courses. Great work — you now have a powerful tool for making predictions!