Analyze and Sketch Function Graphs

SECTION 1

Why Do We Use Graphs to Understand Functions?

Long before calculators existed, people needed ways to understand how things change. A farmer wanted to know how crop growth changed with rainfall. A merchant wanted to see how profit changed with the number of goods sold. Drawing pictures of these relationships turned out to be one of the most powerful ideas in all of math.

The idea of using a graph (a picture that shows how two quantities relate) developed over hundreds of years. Let's look at the key moments.

~300 BCE

Ancient Greek Geometry

Greek mathematicians like Euclid studied shapes and lines, but they didn't yet use coordinate axes to plot relationships.

1637

Descartes Invents the Coordinate Plane

René Descartes created the coordinate plane (the x-y grid). This let people plot points and draw curves to show relationships between numbers.

1700s

Euler Defines Functions

Leonhard Euler introduced the word function and the f(x) notation we still use today. He used graphs to study how quantities depend on each other.

1800s–Today

Graphs Everywhere

Scientists, doctors, economists, and athletes all use graphs daily. From heart-rate monitors to stock charts, reading graphs is a life skill.

Today, you'll learn the same skill those early mathematicians developed: how to look at a graph and describe what's happening between two quantities, and how to sketch a graph from a word description. These are the core ideas behind CCSS.8.F.5.

SECTION 2

Core Principles & Key Vocabulary

Before we dive in, let's build your vocabulary. These four ideas are the building blocks for analyzing any function graph.

Increasing

A function is increasing when the graph goes up as you move from left to right. Think of walking uphill.

Decreasing

A function is decreasing when the graph goes down as you move from left to right. Think of walking downhill.

Linear

A function is linear when its graph is a straight line. The rate of change stays the same.

Nonlinear

A function is nonlinear when its graph is curved. The rate of change is not constant—it speeds up or slows down.

Constant

A function is constant when the graph is a flat horizontal line. The output value doesn't change at all.

✦ KEY TAKEAWAY

Think of a graph like a roller coaster. When the track goes up, the function is increasing. When it goes down, it's decreasing. A straight track means linear, and a curvy track means nonlinear. You're just describing the ride!

SECTION 3

Seeing the Patterns: A Visual Guide

The diagram below shows four different types of function behavior on the same coordinate plane. Study each colored section carefully.

The green line is linear and increasing. The red line is linear and decreasing. The purple curve is nonlinear and increasing. The pink dashed curve is nonlinear and decreasing. The yellow segment is constant.

Notice the key difference between linear and nonlinear. A linear graph makes a straight line—the change from one point to the next is always the same amount. A nonlinear graph curves—the change speeds up or slows down.

💡 Quick Tip

Always read a graph from left to right, just like reading a sentence. "Increasing" means the graph goes up as you move right. "Decreasing" means it goes down as you move right.

SECTION 4

How to Describe a Graph with Math Words

When you analyze a graph, you're telling its story using math vocabulary. Here's a simple framework to follow every time.

Step-by-Step Framework

Step 1: Identify the axes. What does the x-axis represent? What does the y-axis represent?
Step 2: Scan from left to right. Where does the graph go up, go down, or stay flat?
Step 3: Check the shape. Is it a straight line (linear) or a curve (nonlinear)?
Step 4: Describe in sections. Many graphs change behavior. Describe each section separately.

What Does "Qualitatively" Mean?

The word qualitatively (kwah-lih-TAY-tiv-lee) means "describing the overall behavior without exact numbers." You don't need to calculate the slope or find exact coordinates. You just describe the general shape and direction.

QUALITATIVE DESCRIPTION FORMAT

"From x = a to x = b, the function is [increasing / decreasing / constant] and [linear / nonlinear]."

Use this sentence pattern whenever you describe a section of a graph. Replace a and b with the approximate x-values where that section starts and ends.

Rate of Change Clue

You already know rate of change (how fast y changes compared to x). If the rate of change stays the same, the graph is linear. If the rate of change itself changes, the graph is nonlinear. You can spot this visually: straight = linear, curved = nonlinear.

SECTION 5

Sketching a Graph from a Verbal Description

The second half of CCSS.8.F.5 asks you to go in the other direction: someone tells you what happens, and you draw the graph. Let's break down the process with an example scenario.

🏃 Example Scenario

A runner starts slowly, speeds up over the first 3 minutes, then runs at a constant speed for 4 minutes, then slows down and stops over the last 2 minutes. Sketch a graph of the runner's speed versus time.

The graph has three sections. Section A (0–3 min) curves upward as the runner speeds up. Section B (3–7 min) is a flat line at constant speed. Section C (7–9 min) curves downward as the runner slows to a stop.

Tips for Sketching

Label your axes first. Write what each axis represents.
Break the description into sections. Identify each change in behavior.
Use straight lines for linear sections and smooth curves for nonlinear sections.
Connect the sections smoothly. One section should flow into the next without gaps.

SECTION 6

Worked Example: Analyzing a Water Tank Graph

Imagine a graph that shows the water level (in feet) in a tank over 10 hours. Let's walk through how to describe it step by step.

🌊 The Graph Description

From hour 0 to hour 3, the graph is a straight line going up from 2 feet to 8 feet. From hour 3 to hour 6, the graph is a curve that rises slowly from 8 feet to 10 feet. From hour 6 to hour 10, the graph is a straight line going down from 10 feet to 4 feet.

Describe This Graph Qualitatively

Step 1 — Identify the axesThe x-axis is time (hours), from 0 to 10. The y-axis is the water level (feet).

Step 2 — Describe hours 0 to 3The graph is a straight line going upward. So the function is increasing and linear. The water level rises at a steady rate.

0–3 hours: increasing and linear

Step 3 — Describe hours 3 to 6The graph is a curve going upward, but it's flattening out. It's still rising, so the function is increasing and nonlinear. The water is still filling but more slowly.

3–6 hours: increasing and nonlinear

Step 4 — Describe hours 6 to 10The graph is a straight line going downward. So the function is decreasing and linear. The water drains out at a steady rate.

6–10 hours: decreasing and linear

Step 5 — Write the full descriptionPut it all together: "From 0 to 3 hours, the water level increases linearly. From 3 to 6 hours, the water level increases nonlinearly (the rate of filling slows down). From 6 to 10 hours, the water level decreases linearly."

Complete qualitative description ✓

SECTION 7

Comparing Graph Features Side by Side

It's easy to mix up terms at first. This table helps you see the differences clearly.

Summary of qualitative graph features

Feature	What It Looks Like	Real-World Example
Increasing & Linear	Straight line going up ↗	Earning $10 per hour at a job. After 1 hr = $10, 2 hr = $20, 3 hr = $30.
Increasing & Nonlinear	Curve going up, getting steeper ↗↗	A ball rolling downhill. It starts slow and keeps getting faster.
Decreasing & Linear	Straight line going down ↘	A candle burning at the same rate. It shrinks the same amount every hour.
Decreasing & Nonlinear	Curve going down, may flatten ↘	A hot cup of cocoa cooling down. It cools fast at first, then slowly.
Constant	Flat horizontal line →	A parked car's speed: it stays at 0 mph the whole time.

✦ KEY TAKEAWAY

Think of it like describing a hike to a friend. You'd say, "We walked uphill steadily for a while, then the trail leveled off, and then we went downhill really fast on a winding path." That's exactly what qualitative analysis is—telling the story of the graph in words, noting whether each part is going up or down and whether it's straight or curved.

SECTION 8

Connection to High School Math

The skills you're building right now are the foundation for much of high school algebra. Here's a peek at how this connects to what you'll learn next.

How 8th grade graph analysis connects to future math

What You Learn Now (8th Grade)	What You'll Learn Later (Algebra & Beyond)
"The graph is increasing and linear."	You'll calculate the exact slope (like m = 3) and write the equation y = 3x + 1.
"The graph is nonlinear."	You'll learn specific nonlinear functions: quadratics (y = x²), exponentials (y = 2ˣ), and more.
"The graph changes from increasing to decreasing."	You'll find maximum and minimum points. In calculus, you'll use derivatives to find them exactly.
Sketching graphs from descriptions.	You'll graph precise equations using transformations and technology.

The big idea is this: right now, you're learning to see and describe the big picture of a function. Later, you'll add the details—exact equations, precise calculations, and deeper analysis. But the ability to look at a graph and understand its story will always be your first and most important tool.

SECTION 9

Practice Problems

Try these five problems. They start simple and get more challenging. Remember to use the vocabulary you've learned: increasing, decreasing, constant, linear, and nonlinear.

PROBLEM 1 — CONCEPTUAL

If a graph goes from the bottom-left to the top-right in a straight line, is the function increasing or decreasing? Is it linear or nonlinear?

PROBLEM 2 — BASIC CALCULATION

A graph shows temperature in a room over 6 hours. From hour 0 to hour 2, the temperature rises in a straight line from 65°F to 75°F. From hour 2 to hour 6, it stays at 75°F. Describe each section qualitatively.

PROBLEM 3 — INTERMEDIATE

A graph of a car's distance from home over time shows three sections: (1) a curve that rises steeply, (2) a flat horizontal section, and (3) a straight line going downward. Describe each section. What might be happening in real life during each section?

PROBLEM 4 — APPLIED

You're filling a swimming pool with a hose. The water flows fast at first, then the hose pressure drops and it slows down. After 20 minutes, you turn off the hose. Sketch a graph of water volume (y-axis) versus time (x-axis) and label each section.

PROBLEM 5 — CRITICAL THINKING

Can a function be both increasing and decreasing at the same point? Explain your reasoning using what you know about graphs. Then describe a real-world situation where a graph changes from increasing to decreasing.

SUMMARY

Lesson Summary

In this lesson, you learned to analyze and sketch function graphs by describing them qualitatively. The four key behaviors are: increasing (the graph goes up from left to right), decreasing (the graph goes down from left to right), constant (the graph stays flat), and you also identify whether each section is linear (straight) or nonlinear (curved).

To describe a graph, scan from left to right and break it into sections wherever the behavior changes. To sketch a graph from a verbal description, label your axes, identify each section, and connect them smoothly using straight lines for linear parts and curves for nonlinear parts. These skills prepare you for slope, equations of lines, and more advanced function analysis in high school.