Linear Relations

Lesson

A graph is formed by plotting the values of one variable (like $y$`y`) compared to another (like $x$`x`). So a simple way to interpret a graph is to ask something like ‘What will $y$`y` be when $x$`x` is $3$3?’.

However, a good graph will help you to answer more kinds of questions than just this one. An effective graph will clearly add to our understanding of the information that was used to make the graph.

One kind of information that we often want to interpret from a graph is its slope (or slope) at a point. This is a way to describe how the value of one variable changes when the other one is changed.

We often use this when we talk about travelling 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 travelled over periods of time, we are actually talking about speed. That is,

Speed$=$=$\frac{\text{change in Distance}}{\text{change in Time}}$change in Distancechange in Time which is the same as Slope$=$=$\frac{\text{change in }y\text{-axis}}{\text{change in }x\text{-axis}}$change in `y`-axischange in `x`-axis.

The slope of a travel 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 totally 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 between the depot and the first stop than between the first and second stops, because the line between points $A$`A` and $B$`B` is much steeper than the line between points $C$`C` and $D$`D`.

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.

Write a story to describe the graph below.

This graph may look more complicated, but by asking good questions about the context and the slope, we can build a picture of what is happening.

**Think**: What do the axes represent?

The horizontal axis is time, measured in seconds. As time passes we move from left to right. The vertical axis is distance, measured in meters. As the graph gets higher up the vertical axis, the object is further away from the starting position.

So, this graph tells the story of an object that starts to move away and then $200$200 seconds later comes back where it started. By looking at its highest point, it also tells the story of an object that travels $150$150 m away from the starting point before returning.

**Think**: What does the slope during each section represent?

If an object travels further over a fixed period of time, it must be travelling at a higher speed. For this period, the graph would show a steeper increase on the vertical axis. This means that steeper lines indicate a faster moving object. Flat lines tell us that the object is not moving at all.

**Do**: Describe the travel.

Now that we have an understanding of the major features we can start to create a story for this object.

- The object starts to move and takes $20$20 seconds to travel $40$40 m, represented by the line from point A to point B. We can work out the speed of the object in this section, as $\frac{40\text{ m}}{20\text{ s}}$40 m20 s which is $2$2 m/s.
- The object then remains stationary (doesn't move) for $40$40 seconds, represented by the flat line from point $B$
`B`to point $C$`C`. - The object then travels a further $110$110 meters away from the start in $60$60 seconds at a variable speed, reaching the maximum height at point $D$
`D`. We can tell the speed is variable because the line isn’t straight - the slope changes as time goes on. - After this, the object remains stationary for another $40$40 seconds, represented by the flat line from point $D$
`D`to point $E$`E`. - Finally, the object returns to the initial position, taking $40$40 seconds to travel $150$150 m and finishing at $F$
`F`. The speed of the object during this straight section from point $E$`E`to point $F$`F`is$\frac{150\text{ m}}{40\text{ s}}$150 m40 s$=$=$3.75$3.75 m/s.

That was not a very exciting story, though it was very detailed. Let’s try again with less numbers, but more creativity.

A bee leaves its hive, following the precise directions it heard from its friend. It flies steadily to reach a delicious bright blue flower, and spends nearly a whole minute collecting pollen. But then a breeze starts, blowing from the direction of the hive, and knocks the bee off the flower! The wind gets faster and faster, and the bee is carried far away. After a while the bee manages to grab hold of a branch and waits for the wind to die down. After waiting a while the bee flies home as fast as it can.

Try making your own story with graphs like these - you can choose contexts like car chases, roller coasters, other animal stories, or make up a story involving people.

Write a story to describe the graph below.

Ben travels forwards and backwards along a straight line.

The graph shows Ben's distance from his starting point at various times of the day.

When did Ben start his journey?

How far did Ben travel by $11$11 am?

What happened to Ben's speed at $11$11 am?

Ben decreased his speed at $11$11 am.

ABen did not change his speed at $11$11 am.

BBen increased his speed at $11$11 am.

CBen decreased his speed at $11$11 am.

ABen did not change his speed at $11$11 am.

BBen increased his speed at $11$11 am.

CEvaluate Ben's speed between $11$11 am and $1$1 pm.

What distance did Ben travel between $1$1 pm and $2$2 pm?

What is the furthest distance travelled from the starting point?

What is the total distance travelled by Ben from $9$9 am to $4$4 pm?

Describe a situation that would explain the events illustrated by a given graph of a relationship between two variables