7.3.4 Is the Pattern a Quadratic Function?

Algebra 17.3.4 Is the Pattern a Quadratic Function?

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A pattern of squares. Step 1 has two columns of squares in two rows, but there are only two squares. In the first column, the square is located in the top row. In the second column, the square is located in the bottom row. It makes the squares look like they are in a diagonal pattern with the bottom left square and the top right square missing. Step 2 has 3 rows and 4 columns of squares. In the top 2 rows, there are two squares in each row stacked on top of each other so they form a 2 by 2 square. The last two spaces in each row do not have a square in them. The last row in step 2 also has 2 squares but they are not in the locations directly under the squares in the previous rows. Instead, they are shifted into the last two spaces in the row. They look offset from the row above. Step 3 has 4 rows and 6 columns of squares. In the top 3 rows, there are three squares in each row stacked on top of each other so they form a 3 by 3 square. The last three spaces in each row do not have a square in them. the last row in step 3 also has 3 squares, but they are not in the locations directly under the squares in the previous rows. Instead, they are shifted into the last three spaces in the row. They look offset from the row above. The overall image for steps 1, 2, and 3 appears as if the last row in each step has been rotated 180 degrees about the top right vertex of the square on the last row.

Write an equation to represent the relationship between the step number and the number of squares in the pattern. Briefly describe how each part of the equation relates to the pattern.

Compare your answer:

$y = n^{2} + n$ or equivalent because there is an $n$ -by- $n$ array of small squares, which gives $n^{2}$ , and then there is a row of $n$ squares in the lower right.

Is the relationship between the step number and the number of squares a quadratic function? Explain how you know.

Compare your answer:

The relationship is a quadratic function because the number of squares could be expressed as $n^{2} + n$ , which is a quadratic expression.

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
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- Publication date: Jun 4, 2025
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