\begin{tabular}{l} Which equation represents the reflected and translated function? \\ \( \begin{array}{l}\text { f(x) }=-|x-2|-1 \\ f(x)=-|x+1|-2 \\ f(x)=|x+2|-1\end{array} \) \\ \hline\( \square \square \square \square \square \square \square \) \\ \hline\( \square \square \)\end{tabular}

To determine which equation represents the reflected and translated function, we need to analyze the transformations applied to the basic absolute value function \( f(x) = |x| \). 1. **Reflection**: The reflection across the x-axis is represented by a negative sign in front of the absolute value function. For example, \( f(x) = -|x| \) is the reflection of \( f(x) = |x| \). 2. **Translation**: The translation involves shifting the graph horizontally and vertically. The general form for translating the function \( f(x) = |x| \) is: - Horizontal translation: \( f(x) = |x - h| \) shifts the graph \( h \) units to the right if \( h > 0 \) and \( |x + h| \) shifts it \( h \) units to the left if \( h < 0 \). - Vertical translation: \( f(x) = |x| + k \) shifts the graph \( k \) units up if \( k > 0 \) and \( k \) units down if \( k < 0 \). Now, let's analyze the given equations: 1. \( f(x) = -|x - 2| - 1 \) - This represents a reflection across the x-axis (due to the negative sign) and a translation 2 units to the right and 1 unit down. 2. \( f(x) = -|x + 1| - 2 \) - This also represents a reflection across the x-axis and a translation 1 unit to the left and 2 units down. 3. \( f(x) = |x + 2| - 1 \) - This does not have a reflection (the absolute value is not negated) and represents a translation 2 units to the left and 1 unit down. From the analysis, the first two equations represent reflected functions, while the third does not. Thus, the equations that represent the reflected and translated function are: - \( f(x) = -|x - 2| - 1 \) - \( f(x) = -|x + 1| - 2 \) The final answer is: - \( \square \) \( f(x) = -|x - 2| - 1 \) - \( \square \) \( f(x) = -|x + 1| - 2 \)