Write an equation of the graph in its final position. The graph of \( y=6^{x} \) is translated 5 units to the left and then 2 units upward. Which of the following is the equation of the graph? A. \( y=6^{x+5}-2 \) C. \( y=6^{x-5}+2 \) D. \( y=6^{x-5}-2 \)

To determine the final position of the graph after the translations, we start with the original function \( y = 6^x \). Translating the graph 5 units to the left means we replace \( x \) with \( x + 5 \), giving us the equation \( y = 6^{x + 5} \). Next, we translate the graph 2 units upwards, which involves adding 2 to the entire function. So, the new equation becomes \( y = 6^{x + 5} + 2 \). Fun fact: Translations in graphs can be thought of as a journey where every transformation is a new destination that changes the position while keeping the same shape! Take note! A common mistake when dealing with translations is reversing the direction of the horizontal translations. Left translations require adding to the \( x \)-value in the equation, while upward movements simply add directly to the function value.