NON-LINEAR GRAPHS OF FUNCTIONS
Section: Calculus
 
1. INTERPRETING GRAPHS
1.1 THE FIRST GRAPH
In the following window the shape of the bottle is different. 
1.-Have a go yourself at drawing the graph before you see how it is drawn in the window.

2.-Look carefully at the water level after 2,4 and 6 seconds. 

3.-What is the connection between the values of the water level and the time?

4.-What is the difference between the water level at these times? 

5.-The water level now rises more slowly as more time passes. Why is this?

1. 2.   THE SECOND GRAPH
In the following window you can change the shape of the bottle by moving point P. 
1.-Make the neck of the bottle narrower than the base and then look at the different graphs produced by different bottles like this.

2.-Graphs are sometimes convex (like a U) and sometimes concave (like an upside down U), but what does this depend on? Why are some graphs more curved than others? 

 

1. 3.   THE THIRD GRAPH
Look at the following window to quickly review the different shapes of graphs of functions to help you answer the questions above :

1.-Which of the graphs that we have seen would represent the following?

  • "the water level rises more slowly as the time increases".

  • "the water level rises constantly as the time increases".

  • "the function increases more and more quickly".

  • "the water level rises more quickly as the time increases".
1. 4.   THE FOURTH GRAPH
In this window we can see a more complex-shaped bottle. 

1.-Can you guess what the shape of the graph will look like before you turn the tap on?

2.-Try this a few times, changing the shape of the bottle each time by moving P and Q.

3.-Try making bottles that get wider or narrower and that change shape. Describe what the shape of the graph is in each case.

4.-Can you explain the relationship between the shape of the graph and the shape of the bottle? 

 

1. 5. THE FIFTH GRAPH 
Use the window below to find the answer. By now you should be aware that the shape of the bottle is not the same as the graph of the function, but they are related.

1.-In your exercise book try drawing graphs of bottles that you design yourself, e.g. like a goldfish bowl. There are several possibilities and some graphs are far more complicated to draw than others. 

2.-Imagine that the bottles we have seen in this unit already had some water in them before we turned on the tap. What would the graphs look like?

 

                 

 

     
         
  Agustín Muñoz Núñez
 
© Spanish Ministry of Education. Year 2001
 
 

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