LINEAR FUNCTIONS
Section: Calculus
 
1.  INTERPRETING GRAPHS
1.1  THE FIRST GRAPH
The following window shows how a bottle (in red) fills up with water when you turn on the tap.  The whole process of filling up the bottle with water can be described mathematically as a function as at any particular moment it can tell us the level of water in the bottle. The line drawn by point A is the graph of the function. In this unit we are going to look at how to interpret the large amount of information represented on a graph. Thinking along these lines, "a graph is worth more than a thousand words" as long as you know how to read it properly!
1.-Have a go at filling the bottle yourself by clicking on the button and holding it down! 

 

2.-As you can see, the horizontal axis represents the length of time that the tap is turned on for and the vertical axis shows the water level in the bottle. We have started to mark the seconds on the horizontal axis: 1 second, 2 seconds etc.

3.-In this example we can see that the water level is zero when the time is zero and that the graph increases with time.

In maths, we always read graphs from left to right (like reading a book) so we always know if a graph is increasing or decreasing. 

4.-Have a look at the water level at 2,4 and 6 seconds. 

If you click on any point of the graph with the cursor it will give you the horizontal value (time) and vertical value (water level) for that particular point. 

5-What can we say about the relationship between the horizontal and vertical values?

6- How long does it take to fill half the bottle? (Find the answer by moving the mouse over the picture and clicking on the correct point) 

7.-How long does it take to fill a quarter of the bottle? What about three-quarters?

1. 2.   THE SECOND GRAPH
In the following window you can change the shape of the bottle to make it wider or narrower by dragging point R. Change the size a few times and try filling the bottle up each time. Don't change the shape of the bottle while you're filling it up. If you want to try filling a different sized bottle click on the init button and start again.
1.-How does the shape of the function graph change when the bottle is made wider or narrower?

2.-Can you explain why this happens?

3.- When does the graph increase the quickest?

4.-Are you able to explain in your own words, referring to the water level, why some graphs of functions increase quicker than others?

1. 3. THE THIRD GRAPH
The following window will help you answer the questions above if you haven't done so already. The graph is already drawn for us so we don't have to keep filling up the bottle at different widths:

1.-Choose the shape of the bottle. Once the shape of the bottle has been chosen note that the graph always increases. This is because the bottle fills up with time.

2.- However, does the graph rise more quickly or does it rise at the same rate? 

       
           
  Agustín Muñoz Núñez
 
© Spanish Ministry of Education. Year 2001
 
 

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