Additive Property of the Exponential Function

Exploration

A demo of the graph of the exponential function is presented above. Note that there is a horizontal -axis, representing time, and a vertical axis, representing the value of the function. Do the following to explore the symmetry between vertical scaling and translation of the graph above. 1. Grab the blue, horizontally-pointing arrow and move it. This translates the graph of the exponential by a particular value. The change to the function is indicated in the demo. Translate the original graph one unit to the right. 2. Grab the red, vertically-pointing arrow and move it. This stretches the graph vertically by a positive factor. The change to the function is indicated in the demo. Can you estimate the scaling factor that makes the vertically stretched graph and the shifted graph the same? 3. Recall that the exponential function satisfies: The Left-Hand-Side of the equation is the function that is a horizontal translation of by . Multiplication by performs a vertical compression/expansion depending on the sign of . If we translate the graph horizontally by one unit, we could instead compress the original graph vertically to produce the same graph by scaling it by a factor of . Did this agree with what you found earlier in Step 2?