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Describe Functional Relationships from Graphs

Students interpret the qualitative features of a function's graph—where it is increasing, decreasing, or constant; where it peaks or levels off—and tell a story that matches the graph. They also sketch a graph that matches a verbal description of a relationship.

Try three of them

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Describe Functional Relationships from Graphs

Grade 8 · Functions · question 1 of 3

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20 rungs. A star at the top.

plant height in cmdays01234567812345678910

The graph shows a plant's height, in centimeters, over several days. How does it change over this time?

Read the line's direction.

How it is taught

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A graph can rise, fall, or stay level as you read left to right.

The story of y is how the curve moves.

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Read it: Describing a graph

Read a graph left to right. This one climbs from 2 to 8 up to x = 4, so it is increasing there. It stays at 8 from 4 to 6, so it is constant. Then it falls to 3, so it is decreasing.

  1. Find the piece of the graph the question names.
  2. Follow that piece from left to right.
  3. Up is increasing, down is decreasing, and level is constant.

Left to right is the way x grows, so following the graph that way shows what y does as x grows.

From 4 to 6 the graph stays at 8: x grows but y does not change, so it is constant.

One graph can do all three, so the answer depends on the piece you read.

A graph is increasing where it climbs, decreasing where it falls, and constant where it stays level.

Reading how high the graph is instead of which way it goes.

From 4 to 6 the graph is high, at 8, but level: it is constant, not increasing.

A speed-vs-time graph from a bike ride

A graph can tell a whole story without any numbers. Let's read one together.

Here's a speed graph from a bike ride. The bottom shows time in minutes, and the side shows how fast the rider is going.

From minute 0 to minute 5, the line climbs upward — that means the speed is increasing as the rider pedals harder.

From minute 5 to minute 15, the line goes flat — the speed stays the same, a steady cruising pace.

From minute 15 to minute 20, the line drops back down — the speed is decreasing as the rider slows to a stop.

A graph tells a storyTime (minutes)Speed051520increasing ↑constant →decreasing ↓Height = speed, not distance traveled!Sped up → cruised → slowed to a stopStory → graph: rise, flat, then fallREAD THE GRAPH↑ → ↓speed vs. time
Grade 8 · Reading a graph's story: increasing, constant, decreasing

Careful: the graph's height shows speed, not how far the rider has traveled. A flat line means constant speed, not standing still.

Put it all together and the graph tells a story: sped up, then cruised, then slowed to a stop.

You can work backwards too — starting from a story like this, you can sketch the rising, flat, and falling parts to match each event.

Every rise, plateau, and drop in a graph is a clue. Read them in order, and you can describe — or even draw — the whole story.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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