Tips and Hints 8]

1. Share how you remember transformations. Do you have any tips or hints that help you to remember?

In order to remember transformations, I first have to memorize what occurs when i change the parent function.

• If I add or subtract a number to the x, then i know that the graph should go either left or right. For example, f(x)=sin(x+1)...Moves ONE unit to the LEFT. I would instinctly believe that the graph would move one unit to the right because of the addition sign, but in reality, it has to move to the left. If you have f(x)=sin(x-1)... then the graph moves ONE unit to the RIGHT (NOT TO THE LEFT!). In order to remember it, I just remember that it moves OPPOSITE to the addition/subtraction sign.


• If I multiply x by a number, then I know that the graph has to shrink horizontally. For example, if I have f(x)=sin2x... then it looks like the graph goes (2x) "faster", compared to its parent function (sinx). It will also compress horizontally because the PERIOD changes to pi instead of the original 2pi. (TO FIND THE PERIOD, REMEMBER THAT YOU DIVIDE THE PERIOD OF THE GRAPH BY THE NUMBER IN FRONT OF THE X). If the number in front of the x is a fraction, then the graph will stretch horizontally instead, since the PERIOD will increase.
  

• If the output has been multiplied by a number, f(x)=2sinx, then the graph stretches vertically. The period still stays the same (if the function has a period). The graph will not be any "faster". If the number in front of the equation is a fraction, then the graph will compress vertically.
• If the number in front of the equation is negative, you just flip the graph. The output has been made negative. You reflect the parent graph across the x axis.

EX. f(x)= -Sinx

• If a number is added to the equation, such as f(x)=sinx+1 (without parenthesis), then the graph will shift up(in this case, up one unit). If a number is subtracted, f(x)=sinx-1, then the graph will shift down (in this case, down one unit).

The only tip I can give you is to MEMORIZE all this information and remembering how the graph will shift, shrink, or compress depending on how the parent function is manipulated.

2. Share how you remember or understand trigonometry. Do you have any tips or hints that help you remember/memorize all those facts?

To understand trigonometry, you have to know that the coordinates on the unit circle are not just random numbers that were made up. They actually MEAN something. Take for example, pi/6. The angle is 60 degrees, since 180 degrees (half of the unit circle) divided by 3 is 30 degrees.

The coordinates are (sqrt of 3/2, 1/2). The x axis is sqrt of 3, divided by the hypotenuse 2, and you get your x coordinate. Same thing for the y coordinate.

You can also remember this by keeping in mind that the longer side of the triangle is sqrt of 3/2 and that the shorter side is 1/2.

For pi/4, just remember that the sides of the triangle are the same (excluding the hypotenuse) and therefore the x and y coordinates are the same. (sqrt 2/2, sqrt 2/2).

The way I remember which coordinate is which for pi/6 and pi/3, (and any other coordinate at "?/6"&"?/3")...

FOR pi/3, I take into consideration that the "/3" comes second, and therefore the 2nd coordinate (y coordinate) has to be sqrt 3/2... (1/2, sqrt 3/2).They will both come SECOND because they include the number "3". Since pi/6 does not have the 3 at the bottom of the fraction like "/3"(the 3 doesn't come second), then the sqrt of 3/2 cannot be the 2nd coordinate (it cannot come second). So the coordinates for pi/6 is (sqrt 3/2, 1/2). The "3" comes FIRST.

This makes sense in my mind, and I tried my best to explain it. Usually people do not understand what i'm talking about.. haha

I think that you should just have the unit circle MEMORIZED and you should only use the tips if you FORGET some of the coordinates.

TO GRAPH: All I do is remember the main coordinates of the Sin, Cos, Tan graphs (What y is at 0, pi/2, pi, 3pi/2, & 2pi). Then I just continue the graph over again since its a new period and looks the same (For a parent graph). For Tan though, I have to memorize where the asymptotes are at as well. For Csc and Sec, I just graph the sin or cos graphs (sin for csc and cos for sec) and at the top or bottom of the curve is where i draw the parabolas. The asymptotes are where the sin or cos graphs cross the x axis. To remember Cot graphs, I just memorized that the asymptotes are at pi and that the "middle" point is at pi/2. Also i remember that the Cot graphs are the tan graphs flipped horizontally. When transformations are involved, I just follow the rules of transformations that I mentioned above.

3. What still confuses you or worries you about trigonometry?

Well when you shift a graph up, it is difficult to find the zeros of the graph. For example, if the regular graph would have the x intercept at pi/4, etc, and you shift the graph up, I just know that the period will be BETWEEN pi/4 and pi/3. I don't know how to find the exact x intercept without a calculator.