Given the parent function \( f(x)=|x| \), which equation would occur if the function is vertically compressed by \( \frac{1}{3} \), shifted down 6 units, and shifted right 5 units? \( f(x)=\frac{1}{3}|x-6|+5 \) \( f(x)=\frac{1}{3}|x-5|-6 \) \( f(x)=\frac{1}{3}|x+5|-6 \) \( f(x)=3|x+5|-6 \)

To find the transformed function from the parent function \( f(x) = |x| \), we need to apply the given transformations step-by-step. First, the vertical compression by \( \frac{1}{3} \) leads to \( f(x) = \frac{1}{3}|x| \). Then we shift it right by 5 units, changing it to \( f(x) = \frac{1}{3}|x - 5| \). Finally, shifting down by 6 units results in \( f(x) = \frac{1}{3}|x - 5| - 6 \). So, the correct transformed function is \( f(x) = \frac{1}{3}|x - 5| - 6 \), which matches the second option. Shifting functions can lead to some common mistakes. Many students mistakenly follow the order of operations incorrectly, resulting in incorrect placements of numbers in their transformation. Remember that horizontal shifts happen inside the function (with \( x \)), while vertical shifts happen outside the function. Keeping this in mind can help you get the transformations just right! For further exploration, you might enjoy reading about transformations in various contexts, such as physics or engineering, where similar principles apply. Understanding how these transformations function within the framework of real-world data can be immensely satisfying and practical. Check out some fun geometry or algebra books that delve into function transformations for more engaging insights!