Graphing Sine and Cosine Functions

      Hey there Pre-Cal buddies! My name's MaYe by the way (ooh it rhymes) and I'm the short red-violet-ish-haired girl sitting at the very front table beside Jim and.....well, just him. But sometimes I consider to be sitting beside Mr. P too since he sometimes stands close to my seat. (umm don't mind the last sentence..so weird)

     Moving on, Mr. Piatek mainly discussed about graphing sine and cosine functions last week and thus, I will be summarizing that for my blog. 

           Graphing sine and cosine functions on a Cartesian plane is also known as "unrolling" the Unit Circle (according to our notes). For each basic sine and cosine function, it has a period of 2π, a set of all real numbers for the domain and an interval if [–1, 1] for the range

Here`s what a basic sine and cosine function look like:

• One period is the length of one cycle that could be written in degrees or in radians. 

period formula for sines &      cosines:

•  If 0 < B < 1, the period of the function will be greater than 2π and the graph will be a horizontal stretching.

•  If b > 1, the period of y = a sin bx would be less than 2π and would result to a horizontal shrinking of y = a sin x graph

• Amplitude is known as the distance from the middle axis to the cos x and sin x function`s highest or lowest point. The amplitude for these functions are |a| 
• *Note: Amplitude should always be positive.

Here are some good examples:

Now, let us go to 

Transformations of sine and cosine functions!!

         ♦  y= a sin b(x-c) +d     &    y= a cos b(x-c) +d  ♦

VERTICAL STREEEEEEEETCHES:

• Amplitude changes from the basic 1 to |a|
• If a is negative, the function is reflected through the horizontal middle axis of the said function.

HORIZONTAL STREEEEEETCHES

• Period changes from 2p to 
• If b is negative, the function is reflected in the y-axis

 Now, for translations:

HORIZONTAL TRANSLATIONS:

• aka the phase shift
• If c ≥ 0, function moves c units to the right
• if c ≤ 0, function moves c units to the left

VERTICAL TRANSLATIONS:

• aka the vertical displacement
• produces a change in the middle axis or the median of the graph
• If d ≥ 0, the function moves d units up
• If d ≤ 0, the function moves d units down

*Note: There is an order when applying transformations!*

1. First, do all horizontal stretches/compressions and also reflections
2. Do all vertical stretches/ compressions and reflections
3. Lastly, do all translations (phase shift and the vertical displacements)

EXAMPLE : Graph y = 3cos(2x - π) + 1

So, based on the function 3cos(2x - π) + 1, the graph should be horizontally stretched by a factor of 2 and vertically stretched by a factor of 3. Also, the period is 2π/2 = π . 

graph of 3cos(2x)

Then, we horizontally translate it to the right by π/2

y = 3cos(2x  π)

Finally, we would vertically translate it by moving the graph up by one. This would be the final answer.  (YAAAAAY!)

y = 3cos(2x - π) + 1

That's the end of my scribe and I hope you guys learned something from it♥ If you guys have further questions, feel free to ask Mr. Piatek or anyone in our class except for me. (lol I might give you some terrible answers.) Heh, kidding. I'll try to help. 

Anyway, we're halfway done our 2nd semester and still no doughnuts? Ugghhh WHY??!  *praying for a miracle atm*