Talking about curves
Excerpt: This investigation encourages students to look at curves with turning points and to think about the qualitative behaviour of their gradients. How are the signs of the function and its gradient related?
Problem
The following curve turns once. It changes sign twice and the sign of its gradient changes once, giving rise to four distinct regions, as shown in the image below:
What are the different possibilities of sign combination for a curve which turns exactly once?
How can you be sure that you have found them all?
Suggestion
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Which other curves turn just once?
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How can you sketch a curve where the sign of the gradient changes at some point?
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What happens to your results if you shift the axes up or down?
Prompt cards
You might like to use the following curves to promote discussion. They are available as cards to cut out.
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How many variations on the original curve can you find?
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Do all of these produce a different number of regions?
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Do all of these curves turn exactly once?
Taking it further
What are the different possibilities of sign combination for a curve which turns exactly twice?
How can you be sure that you have found them all?
Mini questions
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How do zeros of the function and zeros of the gradient enter into the analysis?
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Are there any curves which always have the same sign for the function and gradient?
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Can you find algebraic expressions for any of the functions you have used in this problem?
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How does the number of different regions vary for curves with other numbers of turns?
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Is it possible to draw a curve for which there is only one region?