Final answer:
Replacing t(x) by f(x) + k shifts the graph vertically by k units. The transformation used to produce one graph from another can be identified by observing how the graph has changed in relation to the original graph - vertical shift, horizontal stretch or compression, reflection, or rotation.
Step-by-step explanation:
When replacing t(x) with f(x) + k in a graph, the effect is that the graph is shifted vertically by k units. This means that every point on the graph is moved up or down by k units. For example, if the original graph had a point at (x, y), the new graph would have the same x-coordinate, but the y-coordinate would be y + k.
To identify the transformation used to produce one graph from another, you can observe how the graph has changed in relation to the original graph. If the graph has moved up or down, it has undergone a vertical shift. If the graph has been stretched or compressed horizontally, it has undergone a horizontal stretch or compression. If the graph has been reflected across the x-axis, it has undergone a reflection. If the graph has been rotated 90 degrees counterclockwise, it has undergone a rotation.