Session 7 Solutions: Nonlinear Functions

A B C D
Homework

Solutions for Session 7, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6| C7 | C8 | C9

Problem C1

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Problem C2

Here is the completed table:


Number of dots on side of square		Total number of dots (square number)
1		1
2		4
3		9
4		16
5		25
6		36
10		100
13		169

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Problem C3

It is neither a linear nor an exponential graph. Its successive outputs do not have the same ratio; therefore, it cannot be an exponential graph. It is certainly not a straight line, because successive outputs do not have the same difference, so it cannot be a linear graph.

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Problem C4

The rule is O = n², where O is the output, the total number of dots, and n is the number of dots on the side of a square. A recursive rule is D_n = D_last + (2n - 1).

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Problem C5

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Problem C6

Here is the completed table.


Number of dots on side of triangle		Total number of dots (triangular number)
1		1
2		3
3		6
4		10
5		15
6		21
9		45
19		190

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Problem C7

As with Problem C3, the graph does not demonstrate exponential behavior, because successive terms do not have the same ratio. It's not a linear graph, either, because it is certainly not a straight line. Actually, the graphs of the table of square numbers and the table of triangular numbers look pretty similar.

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Problem C8

There is more than one answer, but one is O = (n)(n + 1) / 2. The recursive form is easier to find: D_n = D_last + n, because n new dots are added in the nth triangle.

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Problem C9

Both closed-form rules involve multiplying n by itself at some point, and both recursive rules involve adding something linear to n. Compare this to linear functions, which have only a single use of the variable in their closed-form rules, and a constant in the recursive rule.

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