To determine the effect of replacing the graph of [tex]\( y = f(x) \)[/tex] with the graph of [tex]\( y = f(x) - 9 \)[/tex], let's analyze how the alteration in the function affects its graph.
1. Understanding Vertical Shifts:
- When you modify a function by subtracting a constant [tex]\( c \)[/tex] from the entire function, [tex]\( y = f(x) - c \)[/tex], this operation results in a vertical shift.
- Specifically, subtracting a constant [tex]\( c \)[/tex] from [tex]\( f(x) \)[/tex] moves the graph downward by [tex]\( c \)[/tex] units.
2. Applying to the Given Problem:
- Here, the original function [tex]\( y = f(x) \)[/tex] is modified to [tex]\( y = f(x) - 9 \)[/tex].
- This transformation subtracts 9 from every [tex]\( y \)[/tex]-value of the original function.
- Thus, the entire graph of [tex]\( y = f(x) \)[/tex] will be shifted downward by 9 units.
3. Visualizing the Shift:
- Imagine taking the graph of [tex]\( y = f(x) \)[/tex] and moving every point on the graph 9 units straight down.
- This means that for any point [tex]\((x, y)\)[/tex] on the original graph [tex]\( y = f(x) \)[/tex], the corresponding point on the new graph [tex]\( y = f(x) - 9 \)[/tex] would be [tex]\((x, y - 9)\)[/tex].
4. Conclusion:
Based on this analysis, the best description of the effect of replacing the graph of [tex]\( y = f(x) \)[/tex] with [tex]\( y = f(x) - 9 \)[/tex] is:
- The graph of [tex]\( y = f(x) \)[/tex] will shift down 9 units.
Thus, the correct statement is:
- The graph of [tex]\( y=f(x) \)[/tex] will shift down 9 units.
