Mathematics Dictionary

Defintion of y = sin(x) : y - k = a*sin(b*x - h).
- Sinuoidal axis is k. This is the vertical translation
- Amplitude is a. It is vertical stretch.
- Period of function is 2*pi/b. This is horizontal strectch.
- Phase difference is h. This is horizontal traslation.
Example

Go to Begin

TR 18 02. Transformation of y = cos(x) to y - k = a*cos(b*x - h)

Definition

Defintion of y = cos(x) to y - k = a*cos(b*x - h).
- Sinuoidal axis is k. This is the vertical translation
- Amplitude is a. It is vertical stretch.
- Period of function is 2*pi/b. This is horizontal strectch.
- Phase difference is h. This is horizontal traslation.
Example : See Five point sketch in YT 18 03

Go to Begin

TR 18 03. Transformation of y = sin(x) to y - 2 = 3*Sin(2*x + pi/4)

Hint

Find values at the following points : Five point method
- Point 1 : 2*x + pi/4 = 0..... Hence x = -pi/8 and y - 2 = 0.
- Point 2 : 2*x + pi/4 = pi/2.. Hence x = +pi/2 - pi/8 and y - 2 = 3.
- Point 3 : 2*x + pi/4 = pi.... Hence x = +pi - pi/8 and y - 2 = 0.
- Point 4 : 2*x + pi/4 = 3*pi/2.Hence x = +3*pi/2 - pi/8 and y - 2 = -3.
- Point 5 : 2*x + pi/4 = 2*pi.. Hence x = +2*pi - pi/8 and y - 2 = 0.
Put following point on paper
- x = -01*pi/8 and y = +2.
- x = +03*pi/8 and y = +5.
- x = +07*pi/8 and y = +2.
- x = +13*pi/8 and y = -1.
- x = +15*pi/8 and y = +2.
Sketch
- Since the curve is sine curve.
- The range is between -pi/8 and 15*pi/8
- Sinuoidal axis is y = 2.
- Hence we can sketch the curve based on these 5 points.

Highlight

We should remeber five points for sin(x) at x = 0, 90, 180, 270 and 360
- y = sin(000) = +0
- y = sin(090) = +1
- y = sin(180) = -0
- y = sin(270) = -1
- y = sin(360) = +0
- We should know the cosine curve
Sketch a diagram
- Sketch programs 08 01 : y = sin(x) to y - d = a*sin(b*x + c*pi)
- Procedures
  - Click the program to enter contents
  - Click 08 to enter introduction to transformation
  - Open the application program
    
    Select run at current location (No download)
    Slect yes to run
    Click menu
    Click 08 in upper box
    Click 01 in lower box
    Give data : 3, 2, 0.25, 2 (a,b,c,d)
  - Diagram notes
    
    (0,0) of y = sin(x) transformed to (2, -pi/8)
    (0,2*pi) of y = sin(x) tansformed to (2, 15*pi/8)
  Go to Begin
  TR 18 04. Five point method : Sketch cosine curve y - 2 = 3*cos(2*x + pi/4)
  
  Hint
  
  Find values at the following points : Five point method
  
  Point 1 : 2*x + pi/4 = 0..... Hence x = -pi/8 and y - 2 = 3.
  Point 2 : 2*x + pi/4 = pi/2.. Hence x = +pi/2 - pi/8 and y - 2 = 0.
  Point 3 : 2*x + pi/4 = pi.... Hence x = +pi - pi/8 and y - 2 = -3.
  Point 4 : 2*x + pi/4 = 3*pi/2.Hence x = +3*pi/2 - pi/8 and y - 2 = 0.
  Point 5 : 2*x + pi/4 = 2*pi.. Hence x = +2*pi - pi/8 and y - 2 = 3.
  Put following point on paper
  
  x = -01*pi/8 and y = +5.
  x = +03*pi/8 and y = +2.
  x = +07*pi/8 and y = -1.
  x = +13*pi/8 and y = +2.
  x = +15*pi/8 and y = +5.
  Sketch
  
  Since the curve is sine curve.
  The range is between -pi/8 and 15*pi/8
  Sinuoidal axis is y = 2.
  Hence we can sketch the curve based on these 5 points. >/ul>
  Highlight
  
  We should remeber five points for cos(x) at x = 0, 90, 180, 270 and 360
  
  y = Cos(000) = +1
  y = Cos(090) = +0
  y = Cos(180) = -1
  y = Cos(270) = +0
  y = cos(360) = +1
  
  We should know the cosine curve
  Sketch a diagram
  
  Sketch programs 08 02 : y = cos(x) to y - d = a*cos(b*x + c*pi)
  
  Procedures
  
  Click the program to enter contents
  Click 08 to enter introduction to transformation
  Open the application program
  
  Select run at current location (No download)
  Slect yes to run
  Click menu
  Click 08 in upper box
  Click 02 in lower box
  Give data : 3, 2, 0.25, 2 (a,b,c,d)
  
  Diagram notes
  
  (0,1) of y = cos(x) transformed to (5, -pi/8)
  (0,2*pi) of y = sin(x) tansformed to (5, 15*pi/8)
  
  Go to Begin
  
  TR 18 05. Three points and 2 lines : Sketch y = tan(x) between 0 and 2*pi
  
  Hint
  
  We should know 3 points at x = 0, 180 and 360
  
  y = tan(000) = 0
  y = tan(180) = 0
  y = tan(360) = 0
  
  We should know 2 lines at x = 90 and 270
  
  y = tan(90-) = +infinite
  y = tan(90+) = -infinite
  This line 1 at x = 90 and it is asymptote
  y = tan(270-) = +infinite
  y = tan(270+) = -infinite
  This line 2 at x = 270 and it is asymptote
  
  Qucik sketch between line x = 90 and x = 270
  
  The curve from x = 0 to x = 90 is y = 0 to y = +infinite
  
  The curve is increasing and concave upward
  
  The curve from x = 90 to x = 180 is y = -infinite to y = 0
  
  The curve is increasing and concave downward
  
  The curve from x = 180 to x = 270 is y = 0 to y = +infinite
  
  The curve is increasing and concave upward
  
  The curve from x = 270 to x = 360 is y = -infinite to y = 0
  
  The curve is increasing and concave downward
  
  Sketch a diagram
  
  Sketch programs 02 03 y = tan(x)
  Procedures
  
  Click the program to enter contents
  Click 02 to enter graph of functions
  Open the application program
  
  Select run at current location (No download)
  Slect yes to run
  Click menu
  Click 02 in upper box
  Click 03 in lower box
  Give data : No data
  
  Diagram notes
  
  The asymptotes are x = pi/2, 3*pi/2, ....
  Change scale to -4*pi to 4*pi and -8 to 8
  
  1. Click xmax and ymax at left side box
  2. Click replot
  
  Go to Begin
  
  TR 18 06. Three points and 2 lines : Sketch y = sec(x) between 0 and 2*pi
  
  Hint
  
  We should know 3 points at x = 0, 180 and 360
  
  y = sec(000) = +1
  y = sec(180) = -1
  y = sec(360) = +1
  
  We should know 2 lines at x = 90 and 270
  
  y = sec(90-) = +infinite
  y = sec(90+) = -infinite
  This line 1 at x = 90 and it is asymptote
  y = sec(270-) = +infinite
  y = sec(270+) = -infinite
  This line 2 at x = 270 and it is asymptote
  
  Qucik sketch between line x = 90 and x = 270
  
  The curve from x = 0 to x = 90 is y = 1 to y = +infinite
  
  The curve is increasing and concave upward
  
  The curve from x = 90 to x = 180 is y = -infinite to y = -1
  
  The curve is increasing and concave downward
  
  The curve from x = 180 to x = 270 is y = -1 to y = -infinite
  
  The curve is decreasing and concave downward
  
  The curve from x = 270 to x = 360 is y = +infinite to y = 1
  
  The curve is decreasing and concave upward
  
  Sketch a diagram
  
  Sketch programs 02 05 : y = sec(x)
  Procedures
  
  Click the program to enter contents
  Click 02 to enter graph of functions
  Open the application program
  
  Select run at current location (No download)
  Slect yes to run
  Click menu
  Click 02 in upper box
  Click 03 in lower box
  Give data : No data
  
  Diagram notes
  
  The asymptotes are x = pi/2, 3*pi/2, ....
  Change scale to -4*pi to 4*pi and -8 to 8
  
  1. Click xmax and ymax at left side box
  2. Click replot
  
  Go to Begin
  
  TR 18 07. G
  
  Go to Begin
  
  TR 18 08. H
  
  Go to Begin
  
  TR 18 09. I
  
  Go to Begin
  
  TR 18 10. J
  
  Go to Begin
  
  TR 18 11. K
  
  Go to Begin
  
  TR 18 12. L
  
  Go to Begin

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