Describe the Transformations that results in h(x)= f(-x) +5(4 points)

Question

The function f(x) is
and then

Answer 1

Let's break down the transformations that lead to the function \( h(x) = f(-x) + 5 \).

1. Reflection Across the Y-Axis: The transformation \( f(-x) \) reflects the original function \( f(x) \) across the y-axis. This means that for every point \( (x, f(x)) \) on the original graph, the new function will have the corresponding point \( (-x, f(-x)) \).

2. Vertical Shift Upwards: The addition of \( +5 \) in the expression \( h(x) = f(-x) + 5 \) represents a vertical shift of the entire function \( f(-x) \) upward by 5 units. This means that every point \( (x, f(-x)) \) on the modified graph will be transformed to \( (x, f(-x) + 5) \).

Summary of Transformations

• Step 1: Reflect the function \( f(x) \) across the y-axis to obtain \( f(-x) \).
• Step 2: Shift the resulting graph \( f(-x) \) vertically upwards by 5 units to get \( h(x) = f(-x) + 5 \).

These steps provide a clear description of how the function \( h(x) \) is derived from \( f(x) \) through specific transformations.