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Problem 13 Medium Difficulty

Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3.

$ y = - 2^{-x} $

Answer

We start with the graph of $y=2^{x}$ (Figure 16 ), reflect it about the $y$ -axis, and then about the $x$ -axis (or just rotate $180^{\circ}$ to handle both reflections) to obtain the graph of $y=-2^{-x}$. In each graph, $y=0$ is the horizontal asymptote.

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Video Transcript

we're making a rough sketch of a function. We basically want to show that we understand the general shape of the basic function, and we understand what transformations are taking place. So when we look at this equation realized that the basic function is y equals two to the X. Make sure you understand that the negative in front of the two is really just multiplying a function by negative one. The negative is not part of the base. Okay, so think about the basic function Y equals two to the X. That's exponential growth. So it would have this basic shape. Now, think about what happens if you multiply the X by a negative. That means we have a reflection across the Y axis. So that would give us something like this, for why equals two to the negative X. Now, think about what happens when we multiply by negative one. That would give us a reflection across the X axis. So our final graph taking the one we just drew and reflecting it across the X axis is going to look like this. Now all of these have a horizontal lassen toad Y equals zero

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