The graph of \( g(x)=2^{x+5} \) is what type of translation of the graph of \( f(x)=2^{x} \) ? \( \begin{array}{ll}\text { a) } 5 \text { units up } & \text { b) } 5 \text { units left } \\ \text { c) } 5 \text { units down } & \text { d) } 5 \text { units right }\end{array} \)

To determine the type of translation applied to the graph of \( f(x) = 2^x \) to obtain \( g(x) = 2^{x+5} \), let's analyze the transformation step by step. ### Understanding the Transformation: - **Original Function:** \( f(x) = 2^x \) - **Transformed Function:** \( g(x) = 2^{x+5} \) ### Breaking Down the Transformation: 1. **Exponent Addition:** The expression \( x + 5 \) inside the exponent can be rewritten as \( x - (-5) \). - This form resembles the general transformation \( f(x - h) \), which **shifts the graph horizontally**. 2. **Direction of the Shift:** - **Positive \( h \):** \( f(x - h) \) shifts the graph **\( h \) units to the right**. - **Negative \( h \):** \( f(x - (-h)) = f(x + h) \) shifts the graph **\( h \) units to the left**. 3. **Applying to \( g(x) \):** - In \( g(x) = 2^{x+5} \), the transformation corresponds to \( f(x + 5) \), which shifts the graph **5 units to the left**. ### Visual Representation: - **Original Graph \( f(x) = 2^x \):** - Passes through \( (0, 1) \). - **Transformed Graph \( g(x) = 2^{x+5} \):** - Passes through \( (-5, 1) \), illustrating the leftward shift. ### Conclusion: The graph of \( g(x) = 2^{x+5} \) is a **5 units left** translation of the graph of \( f(x) = 2^x \). **Answer:** **b) 5 units left**