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Groups: Share answers to Problem C3, and see how many different functions -- and how many kinds of functions -- you've come up with.
Any linear function that answers the question in Problem C3 must be equivalent to the one given in the table. (For example, the expression 2 * (input + 1) - 2 works, but gives the same output everywhere as 2 * input, so it is the same function.) Through any two points (in this case, two input/output pairs) there is exactly one line, so any two linear functions that match those points must be equivalent.
Groups: Discuss the concepts outlined above.
Consider what quadratic functions could answer Problem C3. These are harder to come up with, but in fact there are an infinite number of quadratic functions that fit the two inputs given. The reason for this is subtle: You need two points to find exactly one line. You need three points to find exactly one parabola (quadratic function). Since there are only two points given, you can pick any third point. So you can pick an output for 3 and find a quadratic that works. Picking a different output for 3 will give a different quadratic. Here are just three others that fit the points given:
• | 1 + n (n +1) / 2 (in other words, the outputs are each one more than the triangular number for that input) |
• | 5n - n2 - 2 |
• | 2n2 - 4n + 4 |
Think about whether there are any other kinds of functions that fit the data. Note that a "height of tide" type function like that in Part A could also fit, and is a cyclic function. Here's an example:
Groups: Consider making a poster of the four graphs together. It's easy to see that you could sketch another function through those same two points that doesn't match the other four functions given. In fact, you could sketch infinitely many different functions through those same two points. If you have trouble coming up with algebraic representations of the functions, describe them with sketches, words, or input/output tables.
Note that this table was in no way "rigged." For any finite table, even if it had thousands of entries, there are an infinite number of functions that fit it. Usually, there will be one "simplest" function. But sometimes it's not clear what's simple: Is a quadratic function "simpler" than an exponential function? Is a linear function "simpler" than a cyclic function?
<< back to Part C: Different Functions
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