https://file.aiquickdraw.com/mathbot/file/1726078324457urgwjcjk.jpg

The problem in the image involves triangular numbers, which can be modeled by a quadratic function. This is because triangular numbers are generated by a formula related to the sum of integers, and these sums increase quadratically as the number of terms increases.

The formula for the \(n\)-th triangular number \(T_n\) is:

\[
T_n = \frac{n(n+1)}{2}
\]

From the table:

- For \(x = 1\), \(y = 1\)
- For \(x = 2\), \(y = 3\)
- For \(x = 3\), \(y = 6\)
- For \(x = 4\), \(y = 10\)
- For \(x = 5\), \(y = 15\)

This clearly shows a quadratic relationship between \(x\) and \(y\), meaning the correct answer is **C: quadratic**.

Would you like more details or have any questions?

Here are 5 related questions to expand on the topic:
1. What is the general form of a quadratic equation?
2. How do you derive the formula for triangular numbers?
3. What are the differences between quadratic, linear, and exponential growth?
4. Can you describe other patterns or numbers that follow a quadratic relationship?
5. How can we calculate the sum of the first \(n\) triangular numbers?

**Tip**: Recognizing patterns in number sequences can help determine the type of function involved!

The y-values in the table are called triangular numbers because those numbers of objects can be shaped to form a triangle. What type of function would model the relationship if the number of rows in the triangle, x, is dependent on the number of pennies, y?

Explore how triangular numbers, which represent sums of integers forming triangular shapes, can be modeled by quadratic functions. Learn how to apply the formula T_n = (n(n+1))/2 and understand the relationship between x (rows) and y (pennies). Suitable for Grades 7-9.

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