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].
