Analytic Geometry

Lesson

Distance-time graphs are a way to describe the movement of people or objects. They usually describe a trip that leaves and returns to a point (like a home base).

The vertical axis of a distance-time graph is the distance travelled from a starting point and the horizontal axis is the time taken from the starting point. In most graphs that depict time, time is on the horizontal ($x$`x`) axis.

There are certain important features of a distance-time graph that we can use to interpret the journey being described.

- As the line moves away from the $x$
`x`-axis, the object is moving further away from the "home" point. - As the line moves back towards the $x$
`x`-axis, the object is returning home. - When the line is horizontal, the object is not moving.
- The steeper the line, the greater the speed of the an object (ie. the faster it moves).
- A straight line indicates a steady speed.
- The total distance of the trip is the distance away from and returning home.

Now let's look at some worked examples working with distance-time graphs.

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?

Which graph shows the height of a ball being thrown vertically into the air?

- ABCDABCD

These ideas are easily extended into other areas. So imagine if we replaced distance with height, volume or depth.

Then,

- As the line moves away from the x-axis, then the object is getting taller, more full or deeper.
- As the line moves back towards the x-axis, the object is getting shorter, less full or shallower.
- When the line is horizontal, the objects depth, height, volume (or whatever is being measured on the y-axis) is not changing.
- The steeper the line, the greater the change of the y-axis measurement.
- A straight line indicates a steady change of the y-axis measurement.

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