6.06 Interpret travel graphs

We often use slope when we talk about traveling a distance. Think about a graph that could describe the travel of a school bus from its bus depot to school and then back to the bus depot.

In the graph above, moving from left to right represents the passage of time, and moving up and down represents increasing and decreasing the distance of the bus from the depot.

We can tell when the bus is moving further away from where it started when the distance increases over time (from left to right). Likewise, the bus is moving back towards the depot when the distance decreases over time (from left to right).

We can also tell when the bus is stopped, because at those times the distance does not change. That is, the graph is flat for those times.

So, now we can ask questions like ‘How many times did the bus stop?’. From a quick look at the graph we can see that there are four flat sections in the graph and so the bus must have stopped four times.

Notice that when we compare the distance traveled over periods of time, we are actually talking about speed. That is,

\text{Speed}=\dfrac{\text{change in Distance}}{\text{change in Time}}

is the same as

\text{Slope}=\dfrac{\text{change in }y\text{-axis}}{\text{change in }x\text{-axis}}

The slope of a distance-time graph at any point tells you the speed of the object at that point.

The faster an object is moving, the steeper the graph will be. When the object is moving slower, the graph will be more flat. If the object is not moving at all, the graph will be flat at those times. Whether the graph is going upwards or downwards tells us the direction of travel.

This means that in the above example we can figure out whether the bus is going fast, slow, towards or away, or even not moving at all by looking at the slope at every point along the trip. For example, the bus is going faster in the first part of the graph than in the second, because the first line segment is much steeper than the second line segment.

So, noticing the trends in the graph and its slope prepares us to answer many questions about the moving object that is being described. Always connect your thinking to the context first, to make sure your conclusions make sense.

Examples

Example 1

Write a story to describe the graph below.

Worked Solution

Create a strategy

Think of a story that represents the features of this graph. Increasing parts of the graph mean the height is increasing, flat parts of the graph mean the height is not changing, and decreasing parts of the graph mean the height is decreasing.

Apply the idea

One story that describes this graph is:

"Roald and a group of friends decide to hike up to the top of a nearby hill and take photos from a lookout on the other side. After a day of constant hiking, they stop at a campsite for a night. They then keep hiking and come down the other side of the mountain, where they stop at a lookout."

Another possible story is:

"Gwen and Fred are conducting a science experiment where they need to monitor the height of a plant in certain conditions. Their plant grows fast initially, but after the first week stops growing altogether. A few days later, the growth begins again and then suddenly the plant gets sick and gets shorter before the experiment ends."

Another possible story is:

"An airplane is taking off at sea level. Once the plane reaches a certain altitude, it stops going up and cruises at a constant altitude. The plane then increases in altitude again, followed by decreasing."

Watch question walkthrough

Example 2

For the following situation, sketch a graph that can represent the situation:

Ben starts driving from home towards his destination at 9 am. By 11 am he has driven 150 \text{km}. His speed then slightly decreases and he has driven a total of 250 \text{km} by 1 pm. Between 1 and 2 pm Ben takes a break from driving because he has reached his destination. He gets back on the road to drive home. At 3 pm he is 100 \text{km} from home. At 4 pm he arrives at home.

Worked Solution

Create a strategy

For every sentence of the story, we can sketch a new part of our graph.

Apply the idea

Ben starts driving from home towards his destination at 9 am. By 11 am he has driven 150 \text{km}.

His speed then slightly decreases and he has driven a total of 250 \text{ km} by 1 pm.

Between 1 and 2 pm Ben takes a break from driving because he has reached his destination.

He gets back on the road to drive home. At 3 pm he is 100 \text{ km} from home.

At 4 pm he arrives at home.

Reflect and check

Your graph may look slightly different. We can have slightly different looking sktetches as long as each sentence from the situation is represented.