Comparing Properties of Functions in Different...

Section 1

Where Did the Idea of "Functions" Come From?

People have been looking for patterns in numbers for thousands of years. Long before anyone used the word function, ancient civilizations were building tables of values — like tax records or star charts — that showed how one quantity depended on another. Over time, mathematicians developed new ways to describe those patterns: words, tables, equations, and graphs. Here are some key moments in that story.

~300 BCE

Babylonian Tables

Babylonian scribes carved clay tablets with rows of numbers showing how areas changed with the length of a side. These were some of the earliest "input → output" tables in history.

1637

Descartes & the Coordinate Plane

French mathematician René Descartes invented the coordinate plane, letting people draw graphs of relationships for the first time. Now you could see a function, not just calculate it.

1694

Leibniz Names "Function"

Gottfried Leibniz used the Latin word functio to mean a quantity that changes depending on a point on a curve. This was the birth of the word we still use today.

1837

Dirichlet's Modern Definition

Peter Dirichlet said a function is any rule that assigns exactly one output to each input. This is basically the definition you learned in class!

Today

Multiple Representations

Modern math education emphasizes switching between equations, graphs, tables, and verbal descriptions. Comparing functions across these different forms is a core skill in 8th-grade mathematics.

So here's the big question this lesson answers: If two functions are shown in different ways — say, one is an equation and the other is a table — how can you still compare them? That's exactly what you'll learn to do.

Section 2

Core Principles: The Four Representations

A function is a rule that takes each input and gives exactly one output. You can describe the same function in four different ways. Each way has its own strengths. Let's explore them.

Algebraic (Equation)

An equation like y = 2x + 3 tells you the exact rule. You plug in any value for x and calculate y. Equations are precise and powerful for predicting outputs far into the future.

Graphical (Graph)

A picture on a coordinate plane. You can see the shape, steepness, and direction of a function at a glance. Graphs are great for spotting trends and comparing two functions visually.

Numerical (Table)

A table shows specific input-output pairs. You can read exact values and look for patterns in how the output changes from row to row.

Verbal (Description)

Words that describe what the function does, like "A gym charges $25 per month plus a $10 sign-up fee." You need to translate the words into math ideas.

When you're comparing two functions, each one might be shown in a different form. For example, Function A might be a graph while Function B is a table. Your job is to figure out the same set of properties for both functions so you can compare them fairly.

✦ Key Takeaway

Think of it like comparing two athletes from different sports. One has basketball stats and the other has soccer stats. To compare them fairly, you need to look at shared qualities — like speed, endurance, or teamwork. With functions, the "shared qualities" are properties like rate of change, y-intercept, and initial value. You extract these from whatever representation you're given.

Section 3

Seeing It: One Function, Four Ways

Let's look at the same function — y = 2x + 1 — shown in all four representations. Notice how every version gives you the same information, just in a different form.

Four representations of the function y = 2x + 1: equation, graph, table, and verbal description shown together.

Look at the diagram above. All four boxes describe the exact same function. The equation tells you the rule. The graph shows you the line. The table lists specific points. The verbal description tells you the real-world story. When you compare two functions, they might each be shown in any one of these four forms — and your job is to pull out the same information from each one.

The key properties you'll usually compare are: the rate of change (how fast the output grows), the y-intercept (the output when the input is 0), and sometimes specific function values (like "what is y when x = 5?").

Section 4

How to Extract Properties from Each Representation

No matter how a function is shown, you can always find these important properties. Here's how to find them in each representation.

Rate of Change (Slope)

rate of change = (y₂ − y₁) ÷ (x₂ − x₁)

Pick any two points. Subtract the y-values and divide by the difference in x-values.

From an equation: If the function is written as y = mx + b, the rate of change is just m (the number in front of x), and the y-intercept is b. For example, in y = 3x − 4, the rate of change is 3 and the y-intercept is −4.

From a table: Pick two rows. Subtract the y-values and divide by the difference in x-values. If x goes from 2 to 4 and y goes from 7 to 13, the rate of change is (13 − 7) ÷ (4 − 2) = 6 ÷ 2 = 3. To find the y-intercept, look for the row where x = 0. If that row isn't there, use the rate of change to work backward.

From a graph: The rate of change is the steepness of the line. Count how many units up (rise) and how many units right (run) between two clear points. The y-intercept is where the line crosses the y-axis (the vertical axis).

Y-Intercept

y-intercept = b = the value of y when x = 0

This is where the function "starts" — its output at the very beginning.

From a verbal description: Look for phrases like "starts at," "initial amount," or "one-time fee." That's usually the y-intercept. Then look for "per," "each," or "every" — those words signal the rate of change. For example: "A taxi charges $3.00 plus $2.50 per mile" means the y-intercept is 3.00 and the rate of change is 2.50.

Comparing Specific Values

f(x) vs. g(x) at a given x-value

Plug the same x into both functions. Which one gives a bigger output?

✦ Key Takeaway

Think of each representation as a different language — Spanish, French, English, or sign language. They can all say the same thing! Your job is to be a translator: pull out the rate of change and y-intercept no matter what "language" the function is speaking.

Section 5

Side-by-Side: Comparing Two Functions

Let's put this into action. Imagine you're given Function A as a graph and Function B as a table. How do you compare them? You extract the same properties from each and lay them side by side. Here's a visual showing exactly that process.

Side-by-side comparison of Function A (graph) and Function B (table), with extracted properties compared.

In the diagram, Function A (the graph) has a rate of change of 3 and a y-intercept of 2. Function B (the table) has a rate of change of 2 and a y-intercept of 5. Even though they're shown in different ways, we can now say: Function A grows faster (steeper slope), but Function B starts higher (bigger y-intercept). At x = 3, both functions actually give the same output: 11!

Here's a summary of the properties you might be asked to compare:

Property	What It Tells You	How to Find It
Rate of Change (Slope)	How fast the output increases or decreases for each unit increase in input	Equation: m in y = mx + b. Table: (y₂−y₁)÷(x₂−x₁). Graph: rise ÷ run. Words: the "per" amount.
Y-Intercept	The output value when the input (x) is 0 — where the function "starts"	Equation: b in y = mx + b. Table: the y-value when x = 0. Graph: where the line crosses the y-axis. Words: the starting amount or one-time fee.
Specific Values	What does the function output at a particular input?	Equation: plug in x. Table: read the row. Graph: find the y-coordinate at that x. Words: calculate from the description.
Which is Greater?	At a given x, which function has the larger (or smaller) output?	Find both values and compare them directly.
Where Do They Meet?	At what input do both functions give the same output?	Set the two functions equal (or find where their values match in a table or graph).

Section 6

Worked Example: Graph vs. Verbal Description

Let's walk through a complete problem step by step.

ℹ Problem

Function A is described verbally: "A movie streaming service charges a $6 monthly base fee plus $2 per movie rented." Function B is shown on a graph as a straight line passing through the points (0, 4) and (3, 13). Which function has a greater rate of change? Which function has a greater y-intercept? At how many movies rented do both functions cost the same?

Worked Example: Graph vs. Verbal Description

Step 1 — Identify Properties of Function A (Verbal)The description says "$6 monthly base fee" — that's the starting cost, so the y-intercept is 6. It also says "$2 per movie rented" — that's the cost that grows with each movie, so the rate of change is 2. We can write Function A as: y = 2x + 6.

Step 2 — Identify Properties of Function B (Graph)The graph passes through (0, 4) and (3, 13). The y-intercept is the y-value when x = 0, so the y-intercept is 4. For the rate of change, we use the formula: (13 − 4) ÷ (3 − 0) = 9 ÷ 3 = 3. So Function B has a rate of change of 3. We can write Function B as: y = 3x + 4.

Step 3 — Compare Rates of ChangeFunction A's rate of change is 2. Function B's rate of change is 3. Since 3 > 2, Function B has the greater rate of change. This means Function B's cost goes up faster per movie.

Step 4 — Compare Y-InterceptsFunction A's y-intercept is 6. Function B's y-intercept is 4. Since 6 > 4, Function A has the greater y-intercept. This means Function A's base cost is higher.

Step 5 — Find Where They're EqualSet the two equations equal to each other: 2x + 6 = 3x + 4 → 6 − 4 = 3x − 2x → 2 = x. At x = 2 movies, both functions cost the same amount. Plugging back in: 2(2) + 6 = 10 and 3(2) + 4 = 10. Both equal $10. After 2 movies, Function B becomes more expensive because it grows faster.

Section 7

Strengths & Limitations of Each Representation

Each way of showing a function has things it does well and things it doesn't. Understanding these strengths and weaknesses helps you know what to look for — and what might trip you up.

Representation	Strengths	Limitations
Equation	Exact rule — you can find any output for any input. Easy to see slope and y-intercept directly.	Doesn't show the "big picture" shape. Harder to estimate where two functions cross without calculation.
Graph	Great for seeing trends, direction, steepness, and where two functions cross. Very visual.	Hard to read exact values. If the scale is unclear, you might misread coordinates.
Table	Shows exact input-output pairs. Easy to calculate rate of change between rows.	Only shows a few values — you can't see what happens between or beyond the given rows.
Verbal	Explains real-world meaning. Helps you understand what the numbers represent.	Requires translation into math. Wording can be tricky or ambiguous.

✦ Key Takeaway

Think of it like ordering food from a menu in different formats. A picture of the food (graph) lets you see what it looks like, but you can't tell the exact ingredients. A recipe (equation) gives you the exact formula. A nutrition label (table) shows specific values. A friend's description (verbal) gives you the real-world feel. None is "best" — they're each useful for different questions. The power comes from being able to switch between them!

Section 8

Looking Ahead: Beyond Linear Functions

So far, we've mostly been comparing linear functions — functions whose graphs are straight lines. But the skill you're learning here — comparing properties across different representations — works for all kinds of functions. In future math classes, you'll use this same skill with curves, exponentials, and more.

What You Learn Now (8th Grade)	Where It Goes Next (High School)
Compare slopes of two linear functions	Compare rates of change of quadratic, exponential, and other non-linear functions
Find y-intercepts from equations, tables, and graphs	Find x-intercepts (roots), maximums, and minimums of complex functions
Determine which linear function is greater at a point	Analyze systems of equations and inequalities with multiple solutions
Translate verbal descriptions into linear equations	Build mathematical models from real-world data using regression and analysis

The beautiful thing is that the strategy stays the same: identify the key properties, extract them from whatever form you're given, and compare. You're building a skill right now that will help you all the way through high school math and beyond.

Section 9

Practice Problems

Try these five problems on your own. Click "Show Answer" to check your work. They start easy and get harder — give each one a real try before peeking!

PROBLEM 1 — CONCEPTUAL

What are the two most important properties you should compare when looking at two linear functions? Name them and explain what each one tells you in one sentence.

PROBLEM 2 — BASIC

Function A is given by the equation y = 4x + 1. Function B is shown in this table:

x	y
0	3
1	6
2	9
3	12

Which function has a greater rate of change? Which has a greater y-intercept?

PROBLEM 3 — INTERMEDIATE

Function A passes through the points (1, 5) and (4, 14) on a graph. Function B is described verbally: "A dog walker earns $3 for each dog she walks, plus a flat $4 travel fee." Which function has a greater rate of change? What is the y-intercept of Function A?

PROBLEM 4 — APPLIED

Two cell phone plans are being compared. Plan A is described: "$20 per month plus $0.10 per text message." Plan B is shown in this table:

Texts (x)	Cost in $ (y)
0	30
50	33
100	36
200	42

Which plan is cheaper if you send 150 texts per month? At how many texts do both plans cost the same?

PROBLEM 5 — CHALLENGE

Function A is shown in a table where x values are 0, 2, 4, 6 and y values are 10, 16, 22, 28. Function B passes through (0, 24) on a graph and has a slope of 2. For what value of x does Function A first become greater than Function B? Explain why a function with a larger y-intercept doesn't always stay "in the lead."

Summary

Lesson Summary

In this lesson, you learned how to compare two functions even when they're shown in different representations — equations, graphs, tables, or verbal descriptions. The key strategy is always the same: extract the important properties from each representation and then compare them. The most useful properties for linear functions are the rate of change (slope), which tells you how fast the output grows per unit of input, and the y-intercept, which tells you the output value when the input is zero.

You practiced finding the slope from a graph (rise over run), from a table (subtract y-values and divide by the change in x), from an equation (the coefficient of x in y = mx + b), and from a verbal description (the "per" amount). You also learned that a function with a greater y-intercept starts higher, but a function with a greater rate of change will eventually overtake it. Being able to switch between representations and compare functions is one of the most important skills in 8th-grade math — and it's a foundation for everything you'll do with functions in high school.