Hard Problems
Number One
Now we put everything that we have learned together and learn how to do hard problems. Like all of the other graphs, we need to realize that this graph is a sine graph. If you noticed that, then you are right on track. Now that we know what kind of graph we will be drawing, we will now find the period. Just like the others, we put 2pi/B, which in this case is 2pi. Since 2pi is on the top and the bottom, they cancel each other out and the period turns out to be 1. Next, we find the intervals. We put the period over 4 and we see that our intervals will be going up by fourths, since the interval is 1/4. Now that we have found the period and the intervals, we will now get the sinusodial axis and the amplitude. At the end of the equation is a +4, which means that that is our sinusodial axis and that is where our first point, the third point, and the last point will be placed. Now that we have the sinusodial axis, we have to find the amplitude. The equation starts with a 2 in front of the sin, so that means that we will go up 2 and down 2 from our sinusodial axis. Next, we apply what we learned under the medium problems. We look to see if there is anything in parenthesis, which there is. In parenthesis along with the x is -1/2. This means that every dot will be moved to the right by 1/2. So this means that where the sine graph would have started, (0,4), it is now at (1/2,4). Since that dot moved, all of the others will move as well that way the graph can make sense and look right. The other points are at (3/4,6), (1,4), (5/4,2), (3/2,4). If you draw the lines through all of those points, you will get a graph that looks just like the one above.
Number Two
Just like the problem above, we are going to apply everything that we have learned in order to solve and draw this graph. If we look at the equation, we see that it is a cosine graph. First, like we do with every single graph that we get, we try to find the period. As usual we put 2pi/B, where B is 2pi/5. If we put those over each other we get, 2pi/2pi/5. Like we learned under the easy problems, we have to flip and multiply if there is a fraction on top or bottom. If we F&M we get 2pi/1 times 5/2pi. The 2pi's can cancel out out leaving you with 5/1, which means that your period is now 5. We find that our intervals are now going up by 5/4s. Next thing we do is see if there is any sinusodial axis. Since there is a +2 at the end of the equation, that means that our sinusodial axis will be at two. Next, we find the amplitude. There is a two in front of the cosine. This means that we go up two and down two from zero, so our points will be based around zero, two, and four. Next thing that we need to realize is whether or not there is going to be a transformation. We look in the parenthesis and see that there is a -1. This means that every point will now be moved to the right by 1, so the whole graph will move as well. The first original point would have been at (0,4), but its now at (1,4). All of the other points follow and are at (2.25,2), (3.5, 0), (4.75,2), and (6,4).
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