The transformation you are describing, where you have \( y = f(x) \) and then modify it to \( y = f(x) \pm k \), is known as a vertical shift or vertical translation.
Explanation:
- Vertical Shift: When you add or subtract a constant \( k \) from the function \( f(x) \), you are effectively shifting the entire graph of the function vertically.
- If \( k > 0 \): The graph shifts upward by \( k \) units. For example, if the original function is \( y = f(x) \) and you change it to \( y = f(x) + k \), the entire graph moves up.
- If \( k < 0 \): The graph shifts downward by \( |k| \) units. For instance, changing \( y = f(x) \) to \( y = f(x) - k \) would move the graph down.
Graphical Representation:
- If you take a simple function, such as \( f(x) = x^2 \), the graph of \( y = x^2 \) is a parabola opening upwards.
- If you transform it to \( y = x^2 + 2 \), the vertex of the parabola moves from (0, 0) to (0, 2).
- Conversely, if you transform it to \( y = x^2 - 2 \), the vertex moves to (0, -2).
In summary, the transformations \( y = f(x) + k \) and \( y = f(x) - k \) result in vertical shifts of the graph of \( f(x) \), either upwards or downwards depending on the sign of \( k \).
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