A vertical translation on a graph involves shifting the entire graph of a function up or down without changing its shape.
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Upward Translation: When you increase the function by adding a constant value \( k \) (where \( k > 0 \)), the graph shifts upward. For example, if you have a function \( f(x) \) and you create a new function \( g(x) = f(x) + k \), then every point on the graph of \( f(x) \) moves up by \( k \) units.
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Downward Translation: Conversely, when you decrease the function by subtracting a constant value \( k \) (where \( k > 0 \)), the graph shifts downward. For example, if you create a new function \( g(x) = f(x) - k \), every point on the graph of \( f(x) \) moves down by \( k \) units.
To visualize this:
- If the original graph is at \( y = f(x) \), a shift up will result in the new graph being at \( y = f(x) + k \).
- If the original graph is at \( y = f(x) \), a shift down will result in the new graph being at \( y = f(x) - k \).
In both cases, the horizontal positions of points (i.e., the x-coordinates) do not change; only the vertical positions (y-coordinates) change.
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