Tips & Hints

1.) The way i remember transformations is by doing the opposite of what my first instinct tells me to do. For example, if the equation happens to be y= sin (x + 2), the graph of sin x is shifted 2 units to the left. Although i would think that adding 2 to the input would make the graph shift to the right, it actually shifts the graph to the left. This is what i mean by doing the opposite of my first instinct. On the other hand, if the given equation is y= sin (x-2), the graph of sin x shifts 2 units to the right. The way i remember this transformation is if something is added to the input (the x), the graph moves along to the x-axis, either to the right or left. If the entire function is multiplied, for instance, in the case of y= 3sin x, the graph is stretched vertically. The graph will span over double the number in front of the sin x. For instance, in this case, the graph will range from -3 to 3. This is a range of six units in total, which is 2 times 3. Another way to manipulate the graph of a parent function is by affecting the period. The "do the opposite" hint applies to this. For instance, if the equation is sin2x, the period of the graph changes from 2 pi to pi. The graph "speeds up" and goes the same distance in 3.14 units as it would before in 6.28 units. If the equation is y=sin x + 2, then the graph of sin x is moved 2 units upwards. If the equation is y= sin x -2, the graph of sin x simply shifts 2 units downward. This transformation affects the output (the y), and therefore moves along the y-axis, either up or down. The "do the opposite" hint does not apply to this transformation.

2.) The way i understand trigonometry is by knowing the unit circle measurements, both the radians and points. The only trick to this is memorization. All that really needs to be memorized are the measurements in the first quadrant. With this knowledge, one can simply "fold over" the measurements and find the measurements of the proceeding quadrants. The radians in the 2nd quadrant will always be one unit away from being an entire pi. The radians in the 3rd quadrant will always be one unit more than pi. The radians in the 4th quadrant will always be one unit away from being 2 pi. As far as points, they remain the same as the points in the first quadrant. In the 2nd quadrant, the first point (cos x) becomes negative. In the 2nd quadrant, both points are negative (cos & sin). In the 4th quadrant, the last point is negative (sin x). Of course, in the 1st quadrant both points are positive.

The graphs of sin x, cos x, tan x, csc x, sec x, and tan x, as well as their inverses are important in trigonometry. Again, memorization is a key factor. The graphs can be plotted by actually plugging in points or, more quickly, by memorizing how the graph looks, its domain, and its range.

3.) I believe trigonometry is pretty straightforward and all that is really needed to understand it is time and patience, for memorization is a key factor. What worries me about trigonometry is the graphs of the inverses of sin x, cos x, tan x, csc x, sec x, and tan x. I am a little fuzzy on the graphs of these functions, but i know that i just need to spend some more time memorizing the graphs, their domain, and their range in order fully understand them.