Algebra 1 · Grade 8
Reading a Graph: Describing How Two Quantities Change
Quick answer
A graph can be read without calculating anything. Rising means one quantity grows as the other does, falling means it shrinks, and flat means it holds steady. Steepness shows how fast. Reading these features in order turns a graph into a story, and telling a story turns it back into a sketch.
What you'll learn
- Describe where a graph increases, decreases or stays constant
- Interpret steepness as speed of change
- Sketch a graph from a description with no numbers
Three things a graph can do
Read left to right, any section is doing one of three things.
| Section | Means |
|---|---|
| rising | as increases, increases |
| falling | as increases, decreases |
| flat | holds steady while continues |
Flat is the one worth dwelling on. It does not mean nothing is happening — time is still passing, or distance still accumulating. It means the vertical quantity is unchanged. On a distance-time graph, flat means stopped.
Steepness is speed
- steep
- gentle
Both lines rise, so both describe something increasing. The steeper one is increasing faster — more change in for the same step in .
That is exactly slope, read without computing it.
Straight or curved
| Graph | Rate of change |
|---|---|
| straight line | constant |
| curve | changing |
A straight line means the quantity changes by the same amount every step. A curve means the rate itself is changing — steepening if the change is speeding up, flattening if it is slowing.
- steady
- speeding up
The curve starts slower and ends faster. At every point it is still “rising”, but how fast is changing the whole way along.
Turning a graph into a story
Take the features in order, left to right.
A distance-time graph rises steeply, then flattens, then rises gently.
| Section | Reading |
|---|---|
| steep rise | moving away quickly |
| flat | stopped |
| gentle rise | moving away slowly |
So: set off fast, stopped for a while, then continued more slowly.
Note that the flat section is not “went back”. Returning would need the graph to fall, since distance from the start would be decreasing.
Turning a story into a sketch
The same reading, run backwards. Sketching needs no numbers — only the shape.
Water is poured into a tank at a steady rate, then the tap is turned off, then it is drained quickly.
| Stage | Shape |
|---|---|
| filling steadily | straight line rising |
| tap off | flat |
| draining quickly | steep fall |
Three sections, in that order, joined end to end.
Match the number of sections to the number of stages. A story with three stages has three pieces, and one missing is the most common sketching error.
Why qualitative reading is worth practicing
Most graphs are met before any calculation, and often the shape is the whole answer. A report showing sales rising steeply, leveling, then falling has said something complete without a single number being read off.
It is also the check on a calculation. If your working says a quantity is increasing and the graph falls, something is wrong — and that is quicker to spot than to re-derive.
The habit worth building is reading the axes first. “Rising” means nothing until you know what is on each axis: on a distance-time graph a rise means moving away, and on a speed-time graph the same rise means accelerating. Same shape, different story.
Worked examples
Common mistakes
Practice problems
-
A graph rises left to right. Is the quantity increasing or decreasing?
Answer
Increasing
Full solution
As grows, grows.
-
What does a flat section on a distance-time graph mean?
Answer
The object is stopped.
Full solution
Time passes but distance does not change.
-
Two lines rise; one is steeper. Which quantity is changing faster?
Answer
The steeper one
Full solution
Steeper means more change in for the same step in .
-
What does a straight-line graph say about the rate of change?
Answer
It is constant.
Full solution
The same change in happens for every equal step in .
-
A temperature graph falls steadily. Describe what is happening.
Answer
Cooling at a constant rate
Full solution
Falling means decreasing, and straight means the rate is steady.
-
How many sections does a sketch need for a story with four stages?
Answer
Four
Full solution
One piece per stage, joined end to end.
-
On a speed-time graph, what does a flat section mean?
Answer
Constant speed
Full solution
The speed is not changing, so the object keeps moving at the same rate.
-
A graph curves upward, getting steeper. What is happening to the rate of change?
Answer
It is increasing.
Full solution
Each step of produces a larger rise than the one before.
-
A distance-time graph rises, flattens, then falls to zero. Tell the story.
Hint
What does falling distance mean?
Answer
Traveled away, stopped, then returned to the start.
Full solution
Rising means moving away from the start. Flat means stopped. Falling means the distance from the start is decreasing, so the object is coming back.
Reaching zero means it arrived back where it began.
-
Shown a speed-time graph with a flat section, Amir says the car has stopped. Assess his reading.
Hint
What is on the vertical axis?
Answer
No. Flat on a speed-time graph means constant speed.
Full solution
He has applied the distance-time reading to a different graph. There, flat really does mean stopped.
On a speed-time graph the vertical axis is speed, so a flat section means the speed is not changing — the car is traveling at a steady pace.
Stopped would be a flat section at zero, sitting on the horizontal axis. A flat section anywhere above it is constant motion.
This is why reading the axes comes before reading the shape.
Frequently asked questions
What does a rising graph mean?
As the quantity on the horizontal axis increases, the one on the vertical axis increases too. Reading left to right, the line goes up.
What does a flat section mean?
The vertical quantity is not changing while the horizontal one continues. On a distance-time graph, flat means stopped.
What does steepness show?
How fast the change happens. A steeper section means a larger change in y for the same step in x.
What is the difference between a linear and a nonlinear graph?
A linear graph is a straight line, so the rate of change is constant. A curved graph has a rate that is itself changing.
Do I need numbers to describe a graph?
No. Increasing, decreasing, flat, steep and gentle describe the shape, and that is often what a question asks for.
Standards alignment
This lesson covers the following Common Core State Standards for Mathematics.
- CCSS.MATH.CONTENT.8.F.B.5FunctionsDescribe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.