Hey everyone. So in this example, we're asked to graph the function, \( y = 2 \cdot \sin(x) - 1 \). Now whenever I'm dealing with these types of problems, I like to start off by solving what I'm familiar with and graphing it, and then building off the graph from there. So what I'm going to do in this problem is I'm going to start by graphing \( y = 2 \cdot \sin(x) \). We're going to ignore the negative one for now, but we'll get to that in a moment. To graph this function, we'll consider the sine of x. What we do is we start here at our graph. We reach a peak at an output of 1, then we dip through \( \pi \), and this pattern continues. However, we can't keep the same pattern for this graph, because we have a 2 here, and 2 changes the amplitude. So we're still going to start at an output of 0, but now our peaks will reach an output of 2, and our valleys will reach an output of -2. Our graph is going to look something like this:
We'll start here at the center of our graph, then we'll reach a peak when we get up here to \( \frac{\pi}{2} \). Our peak is right about there, and then we're going to dip through \( \pi \) on our x-axis, go to \( \frac{3\pi}{2} \), reach a valley, and continue waving to the right. We can also extend this graph to the left. We'll reach a valley at \( -\frac{\pi}{2} \), a peak as we get to \( -\frac{3\pi}{2} \), and continue waving to the left. This is what our sine graph will look like for \( y = 2 \cdot \sin(x) \).
Now, to graph \( y = 2 \cdot \sin(x) - 1 \), this minus one will cause a vertical shift. Since it's a negative value, we'll have a vertical shift down by 1 unit. When we shift down, all of these points will be 1 unit lower. This means this peak right here will move 1 unit down, this valley will move 1 unit down, this peak and this valley will also move 1 unit down. The center, where we started, will actually start at an output of negative one.
So, we can adjust where the peaks and the valleys are going to be, recognizing that the peaks of this graph are actually going to be at positive one when we shift one unit down, and the valleys of this graph are going to be down here at negative 3. Let's go ahead and draw this curve. We'll start here at negative 1, and then we're going to reach our peak right about there when we get to \( \frac{\pi}{2} \). Then we'll cross back down to negative 1 when we get to \( \pi \) on the x-axis, and reach our valley at \( \frac{3\pi}{2} \), which will be at an output of negative 3, and then we're going to keep waving to the right. Likewise, to the left, we'll go down here and reach a valley at \( -\frac{\pi}{2} \) on the x-axis. We'll go back up when we get to \( -\pi \), reach our peak when we get to \( -\frac{3\pi}{2} \), and keep waving as we go to the left. So this is what the graph will look like for \( y = 2 \cdot \sin(x) - 1 \). That concludes this explanation for the graph of this function. Thanks for watching.