Questions: If b is the y-intercept of the graph y=f(x) where b>0 and it is always decreasing, what is the y-intercept and the increase/decrease of the graph of y=-f(-x) after the appropriate transformations?

Solution Steps

To solve this problem, we need to understand the transformations applied to the function \( y = f(x) \). The transformation \( y = -f(-x) \) involves reflecting the graph of \( y = f(x) \) across the y-axis and then across the x-axis. This will affect the y-intercept and the nature of the graph (increasing or decreasing).

1. Y-intercept: The original y-intercept is \( b \). After the transformation, the y-intercept becomes \( -b \) because of the reflection across the x-axis.
2. Nature of the graph: Since the original function is always decreasing, reflecting it across the y-axis and then the x-axis will make it always increasing.

The function \( y = f(x) \) is transformed to \( y = -f(-x) \). This involves two reflections:

• Reflecting across the y-axis: \( y = f(-x) \).
• Reflecting across the x-axis: \( y = -f(-x) \).

The original y-intercept of \( y = f(x) \) is \( b \). After the transformation, the y-intercept becomes \( -b \) due to the reflection across the x-axis.

The original function \( y = f(x) \) is always decreasing. Reflecting it across the y-axis and then the x-axis changes its nature to always increasing.

Final Answer