Question

Consider the relationship between y = f(x) and y = f(x +/= k).
What is this type of function transformation called? What does it Iook like on a graph?

Answer 1

The relationship between \( y = f(x) \) and \( y = f(x + k) \) or \( y = f(x - k) \) represents a **horizontal transformation** of the function \( f(x) \).

1. **Horizontal Shift**:
- **For \( y = f(x + k) \)**: This represents a **shift to the left** by \( k \) units. Every point on the graph of \( f(x) \) moves \( k \) units left.
- **For \( y = f(x - k) \)**: This represents a **shift to the right** by \( k \) units. Every point on the graph of \( f(x) \) moves \( k \) units right.

### Graphical Representation:
- If you have the graph of \( y = f(x) \), and you create \( y = f(x + k) \), the new graph will appear to be the original graph shifted left by \( k \) units.
- Conversely, for \( y = f(x - k) \), the new graph will shift the original graph to the right by \( k \) units.

### Summary:
- **Type of Transformation**: Horizontal Shift
- **Graphical Appearance**:
- **Left Shift** (for \( k > 0 \)): Points on the graph shift left.
- **Right Shift** (for \( k > 0 \)): Points on the graph shift right.

This transformation does not affect the vertical position or shape of the graph, just the horizontal location of the points along the x-axis.