Without graphing, identify the vertex, axis of symmetry, and transformations from the parent function \( f(x)=|x| \) \[ y=|x-4|+3 \] The axis of symmetry is \( x=4 \). What are the transformations trom the parent tunction? A. The function is translated 4 units to the night and vertically stretched by a factor of 3 . The function is translated 4 urits to the right and 3 units down. C. The function is translated 4 units to the right and 3 units up. D. The function is translated 3 units to the night and 4 units up. D. The function is translated 3 unts to the right and vertically stretched by a factor of 4 . O. The function is translated 3 units down and vertically stretched by a factor of 4 . D.

To analyze the function \( y = |x - 4| + 3 \) and identify the vertex, axis of symmetry, and transformations from the parent function \( f(x) = |x| \), we can follow these steps: 1. **Identify the Vertex:** The vertex of the function \( y = |x - h| + k \) is given by the point \( (h, k) \). In this case, \( h = 4 \) and \( k = 3 \). Therefore, the vertex is: \[ (4, 3) \] 2. **Identify the Axis of Symmetry:** The axis of symmetry for the function \( y = |x - h| + k \) is the vertical line \( x = h \). Thus, the axis of symmetry is: \[ x = 4 \] 3. **Identify the Transformations:** The transformations from the parent function \( f(x) = |x| \) can be determined by analyzing the changes in the function: - The expression \( |x - 4| \) indicates a horizontal translation to the right by 4 units. - The addition of 3 indicates a vertical translation upwards by 3 units. Putting this together, the transformations are: - Translated 4 units to the right. - Translated 3 units up. Now, let's summarize the findings: - **Vertex:** \( (4, 3) \) - **Axis of Symmetry:** \( x = 4 \) - **Transformations:** The function is translated 4 units to the right and 3 units up. From the options provided, the correct choice is: **C. The function is translated 4 units to the right and 3 units up.**