"Triangular numbers" describe the number of dots needed to make triangles like the ones below. The first triangular number is 1, the second is 3, and so on.
Graph the data in your table using graph paper or a spreadsheet, then describe your graph. How is it different from the linear and exponential graphs you've seen?
Problem C8
Describe a rule relating the number of dots on the side of a triangle (the independent variable) and the total number of dots (the dependent variable).
Problem C9
Describe any similarities and differences between your rules and graphs for the square and triangular numbers. In a way, they both have the same kind of rule. How would you describe it? Note 11
Think about how you would go from one output to the next. What changes? Can you describe these changes with a rule? Close Tip
You can create "figurate numbers" for any polygon shape. Below are pictures of the first few pentagonal and hexagonal numbers.
The growth of figurate numbers is an example of a quadratic function. A quadratic function's formula will always involve squaring the input number: y = 3x2 + 5 is a quadratic function, and y = 3x + 5 is not. In Part D, we will explore the formulas and properties of quadratic functions in more detail.
