Question

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?

linear
quadratic
exponential
square root

Explanation:

Step1: Calculate first differences

First differences of \(y\)-values: \(3 - 1=2\), \(6 - 3 = 3\), \(10 - 6=4\), \(15 - 10 = 5\)

Step2: Calculate second differences

Second differences: \(3-2 = 1\), \(4 - 3=1\), \(5 - 4=1\)

Since the second differences are constant (\(1\)), the function is quadratic.

For a linear function, the first differences are constant. For an exponential function, the ratios of consecutive \(y\)-values are constant. For a square - root function, the relationship does not follow the pattern of second - constant differences.

Answer:

quadratic