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Quadratic Relationships

7.2.4 Quadratic Relationships

7.2.4 • Quadratic Relationships

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Here are three patterns of dots.

Pattern A

A pattern of dots. Step 0 has no dots. Step 1 has a column of 2 dots. Step 2 has two columns of 2 dots. Step 3 has three columns of 2 dots.

Pattern B

A pattern of dots. Step 0 has 1 dot. Step 1 has a row of 2 dots. Step 2 has 2 rows of dots. In the bottom row are 3 dots and in the top row are 2 dots. The top row is arranged so the upper left space of row 2 is left open. Step 3 has 3 rows of dots. The bottom row has 4 dots. Row 2 has 3 dots and row 3 has 3 dots.

Pattern C

A pattern of dots. Step 0 has 1 dot. Step 1 has a row of 3 dots. Step 2 has 3 rows of 3 dots. Step 3 has 3 rows of 9 dots.
1.

Which pattern shows a quadratic relationship between the step number and the number of dots? Explain how you know algebraically and in complete sentences.

Compare your answer:

Pattern B shows a quadratic relationship.

  • The number of dots is the step number squared plus 1, or n2+1n2+1.
  • Pattern A is growing linearly. It is growing by 2 each time. Pattern C is growing by a factor of 3 each time, so it is growing exponentially.
  • Pattern B is growing by 1, then by 3, and then by 5, which is the pattern we saw in other examples of quadratic relationships in the lesson.