Question

multiple transformations of linear functions
choose a transformation to change the graph.
reflection
vertical translation by 5 (units)
enter the transforming function.
f(x)=x
transformation form: y = a f(x) + d

Explanation:

Step1: Identify the base function

The base linear function is \( f(x)=x \).

Step2: Determine the transformation type

The problem involves a vertical translation (shifting up) by 5 units. The transformation form for a vertical shift is \( y = f(x)+D \), where \( D \) is the vertical shift. Here, \( D = 5 \) and there is no vertical stretch (so \( A = 1 \)) and no horizontal transformation (so the horizontal transformation factor is 1, but not relevant here as we are only shifting up).

Step3: Write the transformed function

Using the transformation form \( y=A f(x)+D \), with \( A = 1 \) (no vertical stretch or compression) and \( D = 5 \) (shift up 5 units), and \( f(x)=x \), we get \( y = 1\times x+5=x + 5 \). But in the given transformation form \( y = A f(x)+D \), substituting \( A = 1 \), \( f(x)=x \), and \( D = 5 \), the transforming function is \( f(x)=x \) (wait, no, the question is about the transformation to change the graph. Wait, maybe the base function is \( f(x)=x \), and we are doing a vertical translation up 5 units. So the transformed function is \( y=f(x)+5=x + 5 \). But in the box for the transforming function \( f(x)\), wait, maybe the base function is \( f(x)=x \), and the transformation is vertical translation by 5 units, so the function after transformation (if we consider the transformation form \( y = A f(x)+D \)) with \( A = 1 \), \( D = 5 \), so the transforming function's \( f(x)\) is \( x \), and the transformation is vertical translation by 5 units. Wait, maybe the question is to fill the transforming function as \( f(x)=x \) (the base function) and the vertical translation by 5 units. So the answer for the transforming function \( f(x)\) is \( x \), and the vertical translation is 5 units. But the problem's image shows a box for \( f(x)\) which is to be filled, and the vertical translation by 5 units. So the base function \( f(x)=x \), so we put \( f(x)=x \).

Answer:

The transforming function \( f(x)=\boxed{x} \) (and the vertical translation is by 5 units, but the question about the transforming function is \( f(x)\), so the answer is \( x \)).