Graphing Sine and Cosine functions ( vertical & Horizontal Translation)

/by

Sine and Cosine Functions

Horizontal Translation:

In the standard equation y=asin(bx-c)+d , constant c creates the horizontal translation of basic sine and cosine curves. Comparing y= aSin(bx) with y= aSin(bx-c) , we find that graph of y= aSin(bx-c) completes one cycle from bx-c=0  to bx-c =2π .  By solving for x, we can find the interval for new cycle as,

Function                     left end point                     Right end point

Y= aSin(x)                     x=0                                       x =2π

Y= aSin(bx)                   x=0                                        x=2π/b

Y= asin(bx-c)               x = c/b                                   x = c/b  + 2π/b         

We see that period interval  of old sine function y= aSin(bx)  is [0,2π/b] whereas the period interval for new horizontally translated function would be   \dpi{120} \left [ \frac{c}{b},\frac{c}{b}+\frac{2\pi }{b} \right ]

 

 

This implies that the period of y=aSin(bx-c) is  2π/b  and the graph is shifted by an amount c/b. The number c/b is the phase shift.

 

 

Example1. Find amplitude ,period and phase shift for the following function and analyze its graph in terms of horizontal translation.

  Y= 2Sin(x – π/3)

Solution: Comparing this equation with standard equation

y= aSin(bx-c) we get,

Amplitude(a) = 2

b=1  =>  period =2π/b  =2π/1  = 2π

Phase shift =c/b  = π/3  (right)

By solving the equations,

\dpi{120} x-\frac{\pi }{3}=0 \Rightarrow x=\frac{\pi }{3}

\dpi{120} x-\frac{\pi }{3}=2\pi \Rightarrow x=\frac{\pi }{3}+2\pi =\frac{7\pi }{3}

So we get new interval for one cycle as  \dpi{120} \left [ \frac{\pi }{3},\frac{7\pi }{3} \right ] . That means complete cycle of interval [0,2π ] of  y= 2sinx get shifted to right by π/3 . Now we get its 5 key points by dividing interval \dpi{120} \left [ \frac{\pi }{3},\frac{7\pi }{3} \right ]  into 4 equal parts and they are,

Intercept             max.             intercept                     min.                       intercept

   (π/3,0)          (5π/6,  2)        (4π/3 ,0)               (11π/6 ,-2)                 (7π/3 ,0)

By connecting these points we get the following graph.

 

 

Example2. . Find amplitude ,period ,phase shift  and frequency  for the following function and analyze its graph in terms of horizontal translation.

                      Y= -3cos(2πx+4π)

Solution: Comparing it with standard equation y= aCos(bx-c) we get,               Amplitude(a)= 3

b = 2π    =>   period =2π/b  =2π/2π  = 1

Phase shift =c/b  =-4π/2π  = -2 (left)

Frequency(f) =b/2π  = 1/2π

Solving the equations,

2π x+4π  = 0

2πx = -4π

x =-4π/2π  = -2(left end point)

2π x+4π  = 2π

2π x = -2π

x =-2π/2π  = -1 (right end point)

So we get the interval [-2,-1] for  one cycle of graph. Now we divide this interval into equal 4 parts to get  key points.

Intercept             max.            intercept                 min.                        intercept

   (π/3 ,0)          (5π/6 ,2)        (4π/3,0)               (11π/6,-2)                 (7π/3,0)

Connecting these points we get following graph.

 

Vertical Translation:

This type of transformation is caused by constant d in the equation  y=asin(bx-c)+d.

Graph get shifted up by d units for d>0 and it get shifted down by d units for d<0.

In other words, graph oscillates about the horizontal line y=d instead of  x axis.

 

Example: Sketch the graph of y=2+3cos2x. Also find its amplitude, period ,phase shift, vertical shift  and equation of new central axis.

Solution: Comparing it with standard cosine equation

y= aCos(bx-c)+d we get,

Amplitude(a) = 3

b =2   =>   period =2π/b   = 2π/2  = π

c =0   =>  Phase shift =c/b  =0/b  = 0

d= 2  =>   vertical shift = 2 units up

Equation of new central axis  y= 2

Vertical translation has  the effect on y coordinates  which get increased or decreased  depending on value of d. Here we have the period interval [0,π ].  We divide this interval into 4 equal parts to get key points and they are:

Intercept             max.                  intercept               min.                    intercept

   (0,5)                   (π/4,2)               (π/2,-1)               (3π/4,2)                 (π,5)

Connecting all these points , we get the following graph.

 

 

 

 

 

 

 

 

Practice problems:

Describe the relationship between graphs of f and g  and graph them on same set of axis.

  1. f(x) = sin(2x)

g(x) = 3 +sin(2x)

2.  f(x) = sin(4x)

g(x) = sin(4x+π) +2

3. g(x) is related to a parent function f(x)= sinx or f(x)= cosx

Describe the sequence of transformations from f(x) to g(x)  and sketch the graph of g(x)

g(x)= 2sin(4x-π )-3

g(x) = cos(x-π ) +2

 

 

 

Leave a comment