Question

What does a vertical translation look like on a graph?

Answer 1

A vertical translation on a graph involves shifting the entire graph of a function up or down without changing its shape.

1. **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.

2. **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.