Question

Which set of data points represents an exponential function?

Answer 1

The set of data points that represents an exponential function is (D) (-1, 1/9), (0, 1), (1, 9), (2, 81), (3, 729).

Data point for an exponential function

An exponential function is characterized by having a constant ratio between its output values for each unit change in the input.

Let's analyze the given data points:

(A) (-1,-9), (0, 0), (1, 9), (2, 18), (3, 27)

(B) (-1, 8), (0, 9), (1, 10), (2, 11), (3, 12)

(C) (-1, 9), (0,0), (1, -9), (2, -18), (3, -27)

(D) (-1, 1/9), (0, 1), (1, 9), (2, 81), (3, 729)

For an exponential function, the output values should have a consistent ratio when the input changes by a constant amount.

Looking at the data sets:

(A) The output values increase by 9 each time the input increases by 1. This shows a constant ratio of 9, indicating an exponential relationship.

(B) The output values increase by 1 each time the input increases by 1. This represents a linear relationship, not an exponential one.

(C) The output values decrease by 9 each time the input increases by 1. This shows a constant ratio of -9, but it's decreasing, which doesn't fit an exponential function.

(D) The output values increase exponentially as the input increases by 1; for each increase in input by 1, the output is multiplied by 9. This represents an exponential relationship.

In an exponential function, the variable in the base is raised to a power. In this case, as the x-values increase, the corresponding y-values increase exponentially. The y-values are increasing at an increasing rate, indicating exponential growth.

Therefore, the set of data points that represents an exponential function is (D) (-1, 1/9), (0, 1), (1, 9), (2, 81), (3, 729).

Answer 2

D

Explanation:

In the data set in option D, the x-values are equally spaced, (i.e.
`\delta x = 1`, however, the x-value are not equally spaced. This rules out the possiblity of the data set in option D being a linear function.

To confirm if it's an exponential function, let's examine if the data set has a "constant multiplying factor", n, that takes us from one y-value to a successive y-value.

Thus:

What do we multiply ⅑ with that will give us the next successive y-value, 1?

Thus, let n be the constant multiplying factor we are looking for.

Therefore:

⅑ × n = 1

Solve for n. Multiply both sides by 9

n = 9.

If we multiply the second y-value which is 9, by 9, wwe will get the next successive y-value, which is 81.

This means the data set has a constant multiplying factor of 9.

Therefore, the data set in option D represents an exponential function.