Reading the Story in a Graph

Lesson 1 of 2: Describing How a Graph Behaves

In this lesson:

• Describe where a graph is increasing, decreasing, or constant
• Tell linear from nonlinear by reading the shape

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Describe a graph as increasing, decreasing, or constant — no exact values
2. Classify a section as linear or nonlinear, and say what the shape means
3. Identify where a graph starts, peaks, and changes behavior

What Happened to the Temperature Today?

A graph of the temperature over one day — no numbers shown:

Without a single number, what can you say about how it changed?

Trace It and Name It

Run your eye along the curve, left to right. It does three kinds of things:

• goes up
• goes down
• stays flat

Those three motions have names: increasing, decreasing, constant.

The Three Behaviors of a Graph

Increasing: output rises. Decreasing: output falls. Constant: output holds.

Reading a Bathtub, Section by Section

Water level in a tub over time — five distinct sections:

Each section is one behavior tied to one real action.

Quick Check: Write the Story

A graph has the same five-section shape — but no bathtub label.

Invent a real-world story where a quantity rises, holds, rises, holds, then falls.

Worked Example: A Hiker's Trip

Distance from the campsite over six hours:

Describe each section — and remember the bottom axis is time, not the trail.

Quick Check: Describe This One

A new graph: a plant's height over several weeks — it rises, then levels off.

Describe it section by section. Which part is increasing? Where does it go constant?

How It Rises, Not Just That It Rises

Both go up. But one rises steadily; the other rises faster and faster.

Linear vs. Nonlinear: Reading the Shape

• Straight = constant rate (same change each step)
• Curve = changing rate (speeding up or slowing down)

Worked Example: Filling a Pool

Water volume in a pool over time:

Linear fill → nonlinear slowdown → constant (full).

Slowing Down — But Still Going

• Tree: still increasing, but the rate decreases (growth slows)
• Coffee: decreasing, and the rate slows as it nears room temp

Predict First: Down Doesn't Mean Negative

A temperature graph falls from 85° to 65° in the evening.

• Is the temperature now negative?
• Is the function decreasing?

Decreasing Is Not the Same as Negative

• Decreasing = getting smaller (the graph slopes down)
• Negative = below zero (the graph is under the axis)

A Curve Is Still a Function

Every vertical line hits the curve once → it's a function. Curves count too.

Quick Check: Classify the Sections

A graph has one straight rising section and one curved, flattening section.

On your own: which is linear, which is nonlinear — and what is each section's rate doing?

Watch Out For These Three Traps

• Decreasing ≠ negative — down-sloping can stay above zero
• A curve is still a function — use the vertical line test
• Steeper means faster change in the vertical-axis quantity — name it

Your Turn: A Complete Graph Read

A four-section graph: rises steeply, levels off (curve), holds flat, then drops.

Describe every section two ways — increasing/decreasing/constant and linear/nonlinear.

Key Takeaways From This Lesson

✓ Read a graph in sections — increasing, decreasing, or constant
✓ Straight = constant rate; curved = changing rate (speeding up or slowing)
✓ A graph tells a story in two layers: direction and shape

Decreasing ≠ negative; the bottom axis is when, not where

What Is Coming Up Next Lesson

You can now read a graph's story off the page.

Next lesson reverses it: given a story in words, you'll sketch the graph yourself — getting direction, shape, and steepness right.