Which equation represents an exponential function that passes through the point \( (2,36) \) ? \( \begin{array}{l}f(x)=4(3)^{x} \\ f(x)=4(x)^{3} \\ f(x)=6(3)^{x} \\ f(x)=6(x)^{3}\end{array} \)

To find the equation that represents an exponential function passing through the point \( (2,36) \), we can plug in \( x = 2 \) into each function and check which outputs \( 36 \). 1. **For \( f(x) = 4(3)^{x} \)**: \( f(2) = 4(3)^{2} = 4 \cdot 9 = 36 \) ✅ 2. **For \( f(x) = 4(x)^{3} \)**: \( f(2) = 4(2)^{3} = 4 \cdot 8 = 32 \) ✖️ 3. **For \( f(x) = 6(3)^{x} \)**: \( f(2) = 6(3)^{2} = 6 \cdot 9 = 54 \) ✖️ 4. **For \( f(x) = 6(x)^{3} \)**: \( f(2) = 6(2)^{3} = 6 \cdot 8 = 48 \) ✖️ The function \( f(x) = 4(3)^{x} \) is the only one that passes through the point \( (2,36) \). Now, let’s dive into its significance! The concept of exponential functions often arises in nature, economics, and many fields. For example, population growth can often be modeled exponentially, where populations grow steadily and speed up over time, resembling how bacteria multiply! This concept introduces a real-world flair to what might seem like mere math. Understanding the various forms of equations can also help avoid common errors! For instance, it's easy to confuse exponential functions with polynomial ones. Remember, an exponential function has the variable in the exponent, while in polynomials it sits in a base. If you mix them up, your predictions could end up wildly off-base! Stay sharp and double-check which form you’re dealing with!