Short Answer 
The transformations applied to the parent function y = 3 – x include a reflection over the x-axis resulting in y = x – 3, a vertical compression by a factor of 0.4 leading to y = -1.2 + 0.4x, and a horizontal translation 2 units to the right yielding the final function y = -0.4(5 – x). Each transformation modifies the function’s graph while preserving its overall shape.
Step 1: Reflection Over the X-axis
The first transformation involves reflecting the given parent function, which is y = 3 – x, over the x-axis. According to the transformation rules, this is achieved by negating the function: y = – (3 – x). This results in the new function y = -3 + x or simply y = x – 3. This transformation effectively flips the graph vertically across the x-axis.
Step 2: Compression by a Factor of 0.4
Next, the function is compressed by a factor of 0.4. This is represented by adjusting the leading coefficient of the function. According to transformation rules, when the function is multiplied by a factor ‘k’ where k < 1, the graph is compressed vertically. Therefore, the updated function becomes y = -0.4(3 – x), leading to y = -1.2 + 0.4x. This means the new graph appears shorter compared to the previous one.
Step 3: Horizontal Translation to the Right
The final transformation is a horizontal translation of the graph 2 units to the right. This transformation is expressed in the function as subtracting -2 from the input variable (x). The resulting function after this transformation is y = -0.4(3 – (x – 2)), simplifying to y = -0.4(5 – x). This adjustment shifts the entire graph to the right while maintaining its shape.