18)+Given+an+equation+of+a+trigonometric+function,+graph+it+by+listing+the+key+values+of+the+graph

Andy Schiffer A.I choose this topic because it is important in understanding trigonometric equations. Understanding how different values effect the equations is an important part of understanding the trigonometric equations as a whole. I enjoyed this section because it made a lot of sense when you figure out what each piece of the equation means. B. When graphing a trigonometric equations each equation has its own parent graph.

YOU NEED A PORTFOLIO PROBLEM AT THE BOTTOM.

sin(X):
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 0 || 1 || 0 || -1 || 0 ||

cos(X):
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 1 || 0 || -1 || 0 || 1 ||

tan(X):
 * X || -90° || -45° || 0° || 45° || 90° ||
 * Y || undefined || -1 || 0 || 1 || undefined ||

csc(X):
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || undefined || 1 || undefined || -1 || undefined ||

sec(X):
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 1 || undefined || -1 || undefined || 1 ||

cot(X):
 * X || -90° || -45° || 0° || 45° || 90° ||
 * Y || 0 || -1 || undefined || 1 || 0 ||

=__ALTERING THE GRAPHS__= Trigonometric Equations can be represented the following way: y=A sin(BX-C)+D. The sin equation can be substituted for any other equation. A=Amplitude B=Period C=Phase Shift D=Vertical Shift

__Amplitude Changes__
When changing the amplitude of equation you use the A value in the equation: y=A sin(BX-C)+D. You then multiply each Y value in the equation by the A value.

EX: y=5 sin (X)

Original New
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 0 || 1 || 0 || -1 || 0 ||
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 0 || 5 || 0 || -5 || 0 ||

__Period Changes__
When changing the period of the equation you use the B value in the equation: y=A sin(BX-C)+D. You then divide each X value in the equation by the B value.

EX: y= sin(3X)

Original
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 0 || 1 || 0 || -1 || 0 ||

New
 * X || 0° || 30° || 60° || 90° || 120° ||
 * Y || 0 || 1 || 0 || -1 || 0 ||

__Phase Shifts__
When changing the phase shift of the equation you use the C value in the equation: y= A sin(BX-C)+D. You then add the C value to each X value in the equation. Note: If there is a period change you must first calculate the X values including the period change before incorporating the phase shift.

EX: y= sin(X+90)

Original
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 0 || 1 || 0 || -1 || 0 ||

New
 * X || -90 ° || 0 °  || 90 °  || 180 °  || 270 °  ||
 * Y || 0 || 1 || 0 || -1 || 0 ||

__Vertical Shifts__
When changing the vertical shift of the equation you use the D value in the equation: y= A sin(BX-C)+D. You then add the D value to each Y value in the equation. Note: If there is an amplitude change you must first calculate the Y values including the amplitude change before incorporating the vertical shift.

EX: y= sin(X)+2

Original


 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 0 || 1 || 0 || -1 || 0 ||

New
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 2 || 3 || 2 || 1 || 2 ||

__Practice Problems__
1: y=5 cos(x)
 * X ||  ||   ||   ||   ||   ||
 * Y ||  ||   ||   ||   ||   ||

2: y=tan(0.5X)
 * X ||  ||   ||   ||   ||   ||
 * Y ||  ||   ||   ||   ||   ||

3: y=3 sin(x+30)
 * X ||  ||   ||   ||   ||   ||
 * Y ||  ||   ||   ||   ||   ||

4: y= cos(6x)+6
 * X ||  ||   ||   ||   ||   ||
 * Y ||  ||   ||   ||   ||   ||

5: y= 2sin(0.25x+60)+10
 * X ||  ||   ||   ||   ||   ||
 * Y ||  ||   ||   ||   ||   ||

__Answers__
1: y=5 cos(x) 5*(Original Y Value)=(New Y Value)
 * X || 0° || 90° || 180° || 270° || 360° ||
 * Y || 5 || 0 || -5 || 0 || 5 ||

2: y=tan(0.5X) (Original X Value) / 0.5=(New X Value)
 * X || -180° || -90° || 0° || 90° || 180° ||
 * Y || undefined || -1 || 0 || 1 || undefined ||

3: y=3 sin(x+30) (Original X Value)-30=(New X Value) 3*(Original Y Value)=(New Y Value)
 * X || -30° || 60° || 150° || 240° || 330° ||
 * Y || 0 || 3 || 0 || -3 || 0 ||

4: y= cos(6x)+6 (Original X Value) / 6=(New X Value) (Original Y Value) + 6=(New Y Value)
 * X || 0° || 15° || 30° || 45° || 60° ||
 * Y || 7 || 6 || 5 || 6 || 7 ||

5: y= 2sin(0.25x-60)+10 REMEMBER TO FACTOR OUT THE 0.25! This is slightly incorrect. ((Original X Value) / 0.25) + 60=(New X Value) ((Original Y Value) * 2) + 10=(New Y Value)
 * X || 60° || 420° || 780° || 1140° || 1500° ||
 * Y || 10 || 12 || 10 || 8 || 10 ||

Portfolio Problems
Create a data table of five key values of the following equations: (1) y=2sin(0.5x-60)+2 (2) y= 5cos(9x+10)-3 (3) y=3tan(0.25x)-4

media type="file" key="trigonometric equations.mp4" width="300" height="300"

Sources http://img.sparknotes.com/figures/A/ad79275cb59e569b790cb945a4ffc553/triggraphs.gif