Question
Which description matches the function represented by values in this table?
X Y
1 972
2 324
3 108
4 36
5 12
X Y
1 972
2 324
3 108
4 36
5 12
Answers
GPT-4o mini
To analyze the function represented by the values in the table, let's look for patterns in the relationship between the values of \( X \) and \( Y \).
The table provided is:
\[
\begin{array}{|c|c|}
\hline
X & Y \\
\hline
1 & 972 \\
2 & 324 \\
3 & 108 \\
4 & 36 \\
5 & 12 \\
\hline
\end{array}
\]
Now, let's look for a pattern by examining the values of \( Y \):
- When \( X = 1 \), \( Y = 972 \)
- When \( X = 2 \), \( Y = 324 \)
- When \( X = 3 \), \( Y = 108 \)
- When \( X = 4 \), \( Y = 36 \)
- When \( X = 5 \), \( Y = 12 \)
Next, let's check the ratios of consecutive \( Y \) values:
1. \( \frac{972}{324} = 3 \)
2. \( \frac{324}{108} = 3 \)
3. \( \frac{108}{36} = 3 \)
4. \( \frac{36}{12} = 3 \)
We observe that each \( Y \) value is \( \frac{1}{3} \) of the previous \( Y \) value.
This suggests a potential geometric change, specifically, \( Y \) seems to decrease by a factor of 3 as \( X \) increases by 1.
From this observation, we can express the relationship as:
\[
Y = \frac{972}{3^{(X-1)}}
\]
Where:
- When \( X = 1 \), \( Y = \frac{972}{3^{0}} = 972 \)
- When \( X = 2 \), \( Y = \frac{972}{3^{1}} = 324 \)
- When \( X = 3 \), \( Y = \frac{972}{3^{2}} = 108 \)
- When \( X = 4 \), \( Y = \frac{972}{3^{3}} = 36 \)
- When \( X = 5 \), \( Y = \frac{972}{3^{4}} = 12 \)
This confirms the pattern.
Therefore, the function represented by the table is a geometric function where \( Y \) equals \( 972 \) divided by \( 3 \) raised to the power of \( (X-1) \).
The table provided is:
\[
\begin{array}{|c|c|}
\hline
X & Y \\
\hline
1 & 972 \\
2 & 324 \\
3 & 108 \\
4 & 36 \\
5 & 12 \\
\hline
\end{array}
\]
Now, let's look for a pattern by examining the values of \( Y \):
- When \( X = 1 \), \( Y = 972 \)
- When \( X = 2 \), \( Y = 324 \)
- When \( X = 3 \), \( Y = 108 \)
- When \( X = 4 \), \( Y = 36 \)
- When \( X = 5 \), \( Y = 12 \)
Next, let's check the ratios of consecutive \( Y \) values:
1. \( \frac{972}{324} = 3 \)
2. \( \frac{324}{108} = 3 \)
3. \( \frac{108}{36} = 3 \)
4. \( \frac{36}{12} = 3 \)
We observe that each \( Y \) value is \( \frac{1}{3} \) of the previous \( Y \) value.
This suggests a potential geometric change, specifically, \( Y \) seems to decrease by a factor of 3 as \( X \) increases by 1.
From this observation, we can express the relationship as:
\[
Y = \frac{972}{3^{(X-1)}}
\]
Where:
- When \( X = 1 \), \( Y = \frac{972}{3^{0}} = 972 \)
- When \( X = 2 \), \( Y = \frac{972}{3^{1}} = 324 \)
- When \( X = 3 \), \( Y = \frac{972}{3^{2}} = 108 \)
- When \( X = 4 \), \( Y = \frac{972}{3^{3}} = 36 \)
- When \( X = 5 \), \( Y = \frac{972}{3^{4}} = 12 \)
This confirms the pattern.
Therefore, the function represented by the table is a geometric function where \( Y \) equals \( 972 \) divided by \( 3 \) raised to the power of \( (X-1) \).