Question

Which description matches the function represented by the values in this table?

X | Y
1 | 972
2 | 324
3 | 108
4 | 36
5 | 12

Answer 1

To determine the function represented by the values in the table, we can examine the relationship between the values of \(X\) and \(Y\).

Let's look at the \(Y\) values as \(X\) increases:

- When \(X = 1\), \(Y = 972\)
- When \(X = 2\), \(Y = 324\)
- When \(X = 3\), \(Y = 108\)
- When \(X = 4\), \(Y = 36\)
- When \(X = 5\), \(Y = 12\)

We can calculate the ratios of successive \(Y\) values:

- \( \frac{324}{972} = \frac{1}{3} \)
- \( \frac{108}{324} = \frac{1}{3} \)
- \( \frac{36}{108} = \frac{1}{3} \)
- \( \frac{12}{36} = \frac{1}{3} \)

From these calculations, we can see that each subsequent \(Y\) value is obtained by dividing the previous \(Y\) value by 3. This suggests that the function may be of the form:

\[
Y = \frac{972}{3^{X-1}}
\]

To confirm, let's calculate \(Y\) values using this formula for each \(X\):

- For \(X = 1\): \( Y = \frac{972}{3^{1-1}} = \frac{972}{1} = 972 \)
- For \(X = 2\): \( Y = \frac{972}{3^{2-1}} = \frac{972}{3} = 324 \)
- For \(X = 3\): \( Y = \frac{972}{3^{3-1}} = \frac{972}{3^2} = \frac{972}{9} = 108 \)
- For \(X = 4\): \( Y = \frac{972}{3^{4-1}} = \frac{972}{3^3} = \frac{972}{27} = 36 \)
- For \(X = 5\): \( Y = \frac{972}{3^{5-1}} = \frac{972}{3^4} = \frac{972}{81} = 12 \)

Everything matches. Thus, the function described by the values in this table can be expressed as:

\[
Y = \frac{972}{3^{X-1}}
\]