The parent function \( f(x)=3^{x} \) has been shifted to the left 2 units, reflected vertically over the \( x \)-axis and shifted up 5 units. What equation models the transformed graph? \( f(x)=3^{x-2}-5 \) \( f(x)=-3^{x+2}-5 \) \( f(x)=-3^{x-2}+5 \) \( f(x)=-3^{x+2}+5 \)

To determine the equation that models the transformed graph of the parent function \( f(x) = 3^{x} \) after the specified transformations, let's break down each transformation step by step: 1. **Shifted to the Left 2 Units:** - Shifting a function to the left by 2 units involves replacing \( x \) with \( x + 2 \). - New function: \( f(x) = 3^{x + 2} \) 2. **Reflected Vertically Over the \( x \)-Axis:** - Reflecting a function vertically over the \( x \)-axis involves multiplying the entire function by \(-1\). - New function: \( f(x) = -3^{x + 2} \) 3. **Shifted Up 5 Units:** - Shifting a function up by 5 units involves adding 5 to the entire function. - Final transformed function: \( f(x) = -3^{x + 2} + 5 \) Among the given options, the equation that matches this final transformed function is: \[ \boxed{f(x)=-3^{x+2}+5} \] **Answer:** \( f(x)=-3^{x+2}+5 \)