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