Question

which equation describes the relationship between the variables in the table below?

x	y
1	9
2	81
4	6,561
5	59,049

• $y = x^9$; each $y$-value is the previous $y$-value multiplied by 9.
• $y = 9x^2$; each $y$-value is the previous $y$-value plus 9 more than was added previously.
• $y = 9x$; each $y$-value is the previous $y$-value plus 9.
• $y = 9^x$; each $y$-value is the previous $y$-value multiplied by another 9.

Explanation:

Step1: Analyze Option 1 ($y = x^9$)

For \(x = 1\), \(y = 1^9=1\), but the table has \(y = 9\) when \(x = 1\). So this is incorrect.

Step2: Analyze Option 2 ($y = 9x^2$)

For \(x = 1\), \(y = 9\times1^2 = 9\) (matches). For \(x = 2\), \(y = 9\times2^2=36\), but the table has \(y = 81\) when \(x = 2\). So this is incorrect.

Step3: Analyze Option 3 ($y = 9x$)

For \(x = 1\), \(y = 9\times1 = 9\) (matches). For \(x = 2\), \(y = 9\times2 = 18\), but the table has \(y = 81\) when \(x = 2\). So this is incorrect.

Step4: Analyze Option 4 ($y = 9^x$)

For \(x = 1\), \(y = 9^1 = 9\) (matches). For \(x = 2\), \(y = 9^2 = 81\) (matches). For \(x = 4\), \(y = 9^4=6561\) (matches). For \(x = 5\), \(y = 9^5 = 59049\) (matches). Also, the pattern of \(y\)-values: \(9\times9 = 81\) (from \(x = 1\) to \(x = 2\)), \(81\times9 = 729\) (next would be \(x = 3\), not in table), but the given \(x = 4\) is \(9^4=(9^2)\times(9^2)=81\times81 = 6561\) which is also \(9^3\times9 = 729\times9 = 6561\), so each \(y\)-value is previous multiplied by 9 (since \(9^{x + 1}=9^x\times9\)). So this fits.

Answer:

D. \(y = 9^x\); each \(y\)-value is the previous \(y\)-value multiplied by another 9.