Translate each graph as specified below. (a) The graph of \( y=|x| \) is shown. Translate it to get the graph of \( y=|x|+3 \). (b) The graph of \( y=|x| \) is shown. Translate it to get the graph of \( y=|x+4| \).

To translate the graphs as specified, we will follow the rules for vertical and horizontal translations of functions. ### (a) Translating \( y = |x| \) to \( y = |x| + 3 \) 1. **Identify the original graph**: The graph of \( y = |x| \) is a V-shaped graph that opens upwards, with its vertex at the origin (0,0). 2. **Understand the translation**: The equation \( y = |x| + 3 \) indicates a vertical translation. Specifically, adding 3 to the function \( |x| \) means we move the entire graph up by 3 units. 3. **Resulting graph**: The vertex of the new graph will be at (0, 3) instead of (0, 0). The shape of the graph remains the same, but it is now shifted upwards. ### (b) Translating \( y = |x| \) to \( y = |x + 4| \) 1. **Identify the original graph**: Again, the graph of \( y = |x| \) is a V-shaped graph with its vertex at the origin (0,0). 2. **Understand the translation**: The equation \( y = |x + 4| \) indicates a horizontal translation. Specifically, replacing \( x \) with \( x + 4 \) means we move the graph to the left by 4 units. 3. **Resulting graph**: The vertex of the new graph will be at (-4, 0) instead of (0, 0). The shape of the graph remains the same, but it is now shifted to the left. ### Summary of Translations - For \( y = |x| + 3 \): The graph is shifted up by 3 units. - For \( y = |x + 4| \): The graph is shifted left by 4 units.