Answer :
Let's examine each of the given equations to determine which one accurately describes the relationship between the variables \( x \) and \( y \) in the table.
Firstly, let's list the equations we need to check:
1. \( y = x^9 \)
2. \( y = 9x^2 \)
3. \( y = 9x \)
4. \( y = 9^x \)
Given:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -1 & 0.1 \\ \hline 1 & 9 \\ \hline 2 & 81 \\ \hline 4 & 6,561 \\ \hline 5 & 59,049 \\ \hline \end{array} \][/tex]
### Checking \( y = x^9 \):
1. For \( x = -1 \):
[tex]\[ y = (-1)^9 = -1 \][/tex]
2. For \( x = 1 \):
[tex]\[ y = 1^9 = 1 \][/tex]
3. For \( x = 2 \):
[tex]\[ y = 2^9 = 512 \][/tex]
4. For \( x = 4 \):
[tex]\[ y = 4^9 = 262144 \][/tex]
5. For \( x = 5 \):
[tex]\[ y = 5^9 = 1953125 \][/tex]
The values do not match the given \( y \)-values exactly.
### Checking \( y = 9x^2 \):
1. For \( x = -1 \):
[tex]\[ y = 9(-1)^2 = 9 \][/tex]
2. For \( x = 1 \):
[tex]\[ y = 9(1)^2 = 9 \][/tex]
3. For \( x = 2 \):
[tex]\[ y = 9(2)^2 = 36 \][/tex]
4. For \( x = 4 \):
[tex]\[ y = 9(4)^2 = 144 \][/tex]
5. For \( x = 5 \):
[tex]\[ y = 9(5)^2 = 225 \][/tex]
The values do not match the given \( y \)-values exactly.
### Checking \( y = 9x \):
1. For \( x = -1 \):
[tex]\[ y = 9(-1) = -9 \][/tex]
2. For \( x = 1 \):
[tex]\[ y = 9(1) = 9 \][/tex]
3. For \( x = 2 \):
[tex]\[ y = 9(2) = 18 \][/tex]
4. For \( x = 4 \):
[tex]\[ y = 9(4) = 36 \][/tex]
5. For \( x = 5 \):
[tex]\[ y = 9(5) = 45 \][/tex]
The values do not match the given \( y \)-values exactly.
### Checking \( y = 9^x \):
1. For \( x = -1 \):
[tex]\[ y = 9^{-1} = \frac{1}{9} \approx 0.1 \][/tex]
2. For \( x = 1 \):
[tex]\[ y = 9^1 = 9 \][/tex]
3. For \( x = 2 \):
[tex]\[ y = 9^2 = 81 \][/tex]
4. For \( x = 4 \):
[tex]\[ y = 9^4 = 6561 \][/tex]
5. For \( x = 5 \):
[tex]\[ y = 9^5 = 59049 \][/tex]
The values match the given \( y \)-values exactly.
### Conclusion:
Hence, the equation that describes the relationship between the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the table is [tex]\( y = 9^x \)[/tex].
Firstly, let's list the equations we need to check:
1. \( y = x^9 \)
2. \( y = 9x^2 \)
3. \( y = 9x \)
4. \( y = 9^x \)
Given:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -1 & 0.1 \\ \hline 1 & 9 \\ \hline 2 & 81 \\ \hline 4 & 6,561 \\ \hline 5 & 59,049 \\ \hline \end{array} \][/tex]
### Checking \( y = x^9 \):
1. For \( x = -1 \):
[tex]\[ y = (-1)^9 = -1 \][/tex]
2. For \( x = 1 \):
[tex]\[ y = 1^9 = 1 \][/tex]
3. For \( x = 2 \):
[tex]\[ y = 2^9 = 512 \][/tex]
4. For \( x = 4 \):
[tex]\[ y = 4^9 = 262144 \][/tex]
5. For \( x = 5 \):
[tex]\[ y = 5^9 = 1953125 \][/tex]
The values do not match the given \( y \)-values exactly.
### Checking \( y = 9x^2 \):
1. For \( x = -1 \):
[tex]\[ y = 9(-1)^2 = 9 \][/tex]
2. For \( x = 1 \):
[tex]\[ y = 9(1)^2 = 9 \][/tex]
3. For \( x = 2 \):
[tex]\[ y = 9(2)^2 = 36 \][/tex]
4. For \( x = 4 \):
[tex]\[ y = 9(4)^2 = 144 \][/tex]
5. For \( x = 5 \):
[tex]\[ y = 9(5)^2 = 225 \][/tex]
The values do not match the given \( y \)-values exactly.
### Checking \( y = 9x \):
1. For \( x = -1 \):
[tex]\[ y = 9(-1) = -9 \][/tex]
2. For \( x = 1 \):
[tex]\[ y = 9(1) = 9 \][/tex]
3. For \( x = 2 \):
[tex]\[ y = 9(2) = 18 \][/tex]
4. For \( x = 4 \):
[tex]\[ y = 9(4) = 36 \][/tex]
5. For \( x = 5 \):
[tex]\[ y = 9(5) = 45 \][/tex]
The values do not match the given \( y \)-values exactly.
### Checking \( y = 9^x \):
1. For \( x = -1 \):
[tex]\[ y = 9^{-1} = \frac{1}{9} \approx 0.1 \][/tex]
2. For \( x = 1 \):
[tex]\[ y = 9^1 = 9 \][/tex]
3. For \( x = 2 \):
[tex]\[ y = 9^2 = 81 \][/tex]
4. For \( x = 4 \):
[tex]\[ y = 9^4 = 6561 \][/tex]
5. For \( x = 5 \):
[tex]\[ y = 9^5 = 59049 \][/tex]
The values match the given \( y \)-values exactly.
### Conclusion:
Hence, the equation that describes the relationship between the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the table is [tex]\( y = 9^x \)[/tex].
