Graphing Circular Functions

Hi guys this is Gurpreet :) Today I will be summarizing the last lesson of the unit, graphing circular functions!! (we'll save Tan for last -_-)

Although it may seem hard at first, it is pretty easy when you get the hang of it. There are a few things you must first acquire before you can actually begin graphing the function.

- First you need the amplitude (the distance from the middle axis to the highest or lowest point), and the period (the length of a cycle), you can acquire these just by looking at the equation of the function.

Ex. y=asinbx    y=acosbx
      a=amplitude       b=numeral required to find period

      period for sinx, and cosx = 2π/|b|  

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- Sin and Cos Basics -

Ex. y=2sin4x
      amplitude= 2      period= 2π/|4| = π/2
Ex. y=3cos2x
      amplitude= 3      period= 2π/|2| = π
- Now that we have the amplitude and period we can begin graphing, the amplitude is placed on the y-axis (- and +), while the period is divided into four quadrants and placed on the x-axis.
- Once the graph quantities have been set, we can now place the points and graph the functions, but to do this we must remember the pattern of sin and cos functions.


- Patterns To Remember -

•  Sin functions always begin at 0, move up to the highest amplitude, to 0 again, to the lowest amplitude, and then to 0 again.
•  Cos functions always begin at the highest amplitude. move to zero, to the lowest amplitude, back to 0, and then back to the highest amplitude.

(remember to show arrows to state that the function is endless)

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- Sin and Cos Shifting -


Now that we know how to graph basic sin and cos functions, next are functions that shift.
Ex. y=asinb(x +/-h) +/- k    y=acosb(x +/-h) +/- k    
      h= horizontal compression (read opposite)          k= vertical shift

Ex. y=2sin4(x-π/8) + 1  (This means move points π/8 to the right and up 1)
Ex. y=3cos2(x-π/4) - 1  (This means move points π/4 to the right and down 1)


- To graph these functions you ignore the shifting values and graph the function normally. Once graphed, you than shift the function according to the values. (Horizontal compressions and vertical shifts occur last)

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- Tan Basics -

Tan functions are graphed a little differently.

      y=atanbx       period for tanx = π/|b|
Ex. y=2tanx

      a= infinite    b= 1

      period = π/|1| = π

     (Same rules apply for labelling graph)

To graph tan functions you have to find asymptotes, these are values that are undefined. (Ex. π/2,3π/2) The tan functions occur between these asymptotes, and unlike sin and cos functions you only need 3 points to graph these. 

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- Tan Shifting -


Shifting tan functions follow the same rule as shifting sin and cos functions.
Ex. y=atanb(x +/-h) +/- k                                                                                                               h= horizontal compression (read opposite)          k= vertical shift

Ex. y=2tanx + 1 (This means move points up 1)

Once again graph the function while ignoring the shift, and then shift when finished. (Horizontal compressions and vertical shifts occur last)

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There that wasn't so hard :) anyway good luck on the test on friday, I'm going to sleep.      I better get like 100 bonus marks for this!