Question 75425
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Meaning: The amplitude is the maximum positive number 
of units the graph rises above the x-axis (which is
also the positive number of units it falls below the 
x-axis.)

The period of a function is the pasitive distance
along the x-axis which spans one cycle of the graph.


Rule:


1. Compare graph to {{{f(x) = A*sin(Bx + C)}}} to
   determine A, B, and C
2. Amplitude = {{{abs(A)}}}
3. Period = P = {{{2pi}}}÷{{{B}}}
4. x-coordinate of starting point for basic cycle = S = {{{-C}}}÷{{{B}}} 
5. Mark these five points on the x-axis:
   S, S+P/4, S+P/2, S+3P/4, S+P
6. Plot the 5 points:
   (S,0), (S+P/4,A), (S+P/2,0), (S+3P/4,-A), (S+P,0)
7. Draw a smooth wave though those 5 points
8. Extend graph through as many cycles in both directions
   as desired.

The rule is the same for the cosine graph of 

{{{f(x) = A*cos(Bx + C)}}}

except for step 6, which is

6. Plot the 5 points:
   (S,A), (S+P/4,0), (S+P/2,-A), (S+3P/4,0), (S+P,A)

-------------------------------

Consider the following function 
{{{f(x) = 2*sin(x/4 + pi)}}} 

First write it as

{{{f(x) = 2*sin((1/4)x + pi)}}}

1. Compare graph to {{{f(x) = A*sin(Bx + C)}}} and determine that
   A = 2, B = 1/4, C = {{{pi}}}
2. Amplitude = {{{abs(A)=abs(2)=2}}}
3. Period = P = {{{2pi}}}÷{{{B}}} = {{{2pi}}}÷{{{1/4}}} = {{{2pi}}}·{{{4/1}}} = {{{8pi}}}
4. x-coordinate of starting point for basic cycle = S = {{{-C}}}÷{{{B}}} 
   = {{{-pi}}}÷{{{1/4}}} = {{{pi}}}·{{{4/1}}} = {{{4pi}}} 
5. Mark these five points on the x-axis:
   S, S+P/4, S+P/2, S+3P/4, S+P
   {{{4pi}}}, {{{4pi+(8pi)/4}}}, {{{4pi+(8pi)/2}}}, {{{4pi+(3*8pi)/4}}}, {{{4pi+8pi}}}

   They simplify to:

   {{{4pi}}}, {{{6pi}}}, {{{8pi}}}, {{{10pi}}}, {{{12pi}}}

 6. Plot the 5 points:
   (S,0), (S+P/4,A), (S+P/2,0), (S+3P/4,-A), (S+P,0)
   which are
   (4<font face = "symbol">p</font>,0), (6<font face = "symbol">p</font>,2), (8<font face = "symbol">p</font>,0), (10<font face = "symbol">p</font>,-2), (12<font face = "symbol">p</font>,0)
   and have numerical values for plotting:
   (12.6,0), (18.8,2), (25.1,0), (31.4,-2), (37.7,0) 

7. Draw a smooth wave though those 5 points

{{{graph(300, 200, -50,50,-4,4,-sqrt(x-11.95)/sqrt(x-11.95)*sqrt(38.1-x)/sqrt(38.1-x)*2*sin(x/4+pi))}}}

8. Extend graph through as many cycles in both directions
   as desired.

{{{graph(300, 200, -50,50,-4,4,-2*sin(x/4+pi))}}}

Edwin</pre>