Beginning with the graph of y=e^x, describe each of the transformations that may be used to graph the function y=6+2e^x. Use complete sentences.

Stretch vertically by factor of 2, move up 6 units

To describe the transformations that can be applied to the graph of y = e^x to obtain the graph of y = 6 + 2e^x, we need to identify each transformation step by step:

1. Vertical Translation:
The first transformation is a vertical translation upward by 6 units. This means that every point on the original graph of y = e^x will be shifted vertically by 6 units in the positive y-direction.

2. Vertical Stretch:
The second transformation is a vertical stretch with a scale factor of 2. This means that the original graph will be stretched vertically by a factor of 2.

It is important to note that these transformations are applied in a specific order. First, the vertical translation is performed, and then the vertical stretch is applied. Therefore, we must apply the vertical translation before the vertical stretch.

To summarize:
Start with the graph of y = e^x.
1. Perform a vertical translation of 6 units upward to obtain the graph of y = 6 + e^x.
2. Apply a vertical stretch with a scale factor of 2 to obtain the final graph of y = 6 + 2e^x.

By following these transformation steps, you will be able to graph the function y = 6 + 2e^x accurately.