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In this lab, consider plotting height of water in a bottle vs. the volume of the water in the bottle. That is, height is on the vertical axis (dependent variable) and volume is on the horizontal axis (independent variable)

Lab Preparation:

1. Explain the meaning of expressing the relationship between height and volume using the function notation \( h(v) \), and include a description of the meaning of the expression \( h(3)=5 \) if volume is measured in cups and height is measured in inches.
   
   
2. Draw a graph of height vs. volume for the bottle shown to the right.

Lab: Complete the two numbered problems assigned to your group.

1. Steepness of the graph is related to the cross-sectional area of the bottle. Explain why a steeper graph corresponds to a narrower bottle and a less steep graph corresponds to a wider bottle, as shown to the right. Make sure that you are talking meaningfully about the rate of change of height with respect to volume by breaking down your explanation in terms of amounts of change in height and amounts of change in volume.
   
   

    

2. Describe what bottle shapes could correspond to a straight line graph. (Be creative to think of all the possibilities!) A linear graph represents a constant rate of change between the two quantities, height and volume. Explain why the bottles you described would have a straight line graph.
   
   

3. 

   The diagram to the right depicts a bottle that is wide at the bottom and narrow at the top (drawn with a solid line). The solid line
   in the graph shows the relationship of height vs. volume for this bottle.
   
   

   To think about the meaning of an average rate of change it is often helpful to introduce an auxiliary situation where the rate is constant. In this case, for the auxiliary situation we can imagine a cylindrical bottle (as drawn with a dotted line) and corresponding linear graph.
   
   

   Use the auxiliary cylindrical bottle and graph to explain the meaning of the average rate of change of height with respect to volume for the original bottle that is wide at the bottom and narrow at the top.
   
   

4. Inflection points correspond to points where the bottle changes from getting narrower to getting wider (or vice-versa). This is because an inflection point on the graph occurs when the graph changes from getting steeper to becoming less steep (or vice-versa). Explain what is happening at the inflection point for a bottle that is narrower in the middle using language about amounts of change.