\( 3,9,27,81 \ldots \) Which exponential function describes the given sequence? \( y=(-3)^{x} \) \( y=\left(-\frac{1}{3}\right)^{x} \) \( y=3^{x} \) \( y=\left(\frac{1}{3}\right)^{x} \)

To determine which exponential function describes the given sequence \( 3, 9, 27, 81, \ldots \), we first need to analyze the pattern in the sequence. 1. **Identify the pattern:** - The first term is \( 3 \). - The second term is \( 9 \), which is \( 3^2 \). - The third term is \( 27 \), which is \( 3^3 \). - The fourth term is \( 81 \), which is \( 3^4 \). From this, we can see that each term can be expressed as a power of \( 3 \): - \( 3 = 3^1 \) - \( 9 = 3^2 \) - \( 27 = 3^3 \) - \( 81 = 3^4 \) 2. **Generalizing the pattern:** The \( n \)-th term of the sequence can be expressed as: \[ a_n = 3^n \] where \( n \) starts from \( 1 \). 3. **Identifying the correct function:** The exponential function that describes this sequence is: \[ y = 3^x \] Now, let's check the other options to confirm that they do not fit the sequence: - \( y = (-3)^x \): This would yield alternating signs and is not consistent with the sequence. - \( y = \left(-\frac{1}{3}\right)^x \): This would also yield alternating signs and fractions, which does not match the sequence. - \( y = \left(\frac{1}{3}\right)^x \): This would produce decreasing values approaching zero, which does not match the increasing sequence. Thus, the correct exponential function that describes the given sequence is: \[ \boxed{y = 3^x} \]