Question

7. Sketch one period of the graph of Y = 1- cos (3x) starting with the graph of Y=cosx and performing transformations. Label both axes.

Answer 1

Solution: First we have to plot y=cosx

So we find the y value for a range of x i.e

x	y=cosx
-4π	1
-7 π/2	0
-3 π	-1
-5 π/2	0
-2 π	1
- 3π/2	0
- π	-1
- π/2	0
0	1
π/2	0
π	-1
3π/2	0
2 π	1
5 π/2	0
3 π	-1
7 π/2	0
4 π	1

Corresponding plot of y=cosx is shown here.

2 y-axis (-411, 1) (-21, 1) (0,1) (27, 1) (41, 1) y=cos(x) (-77/2,0) (-57/2,0) (31/2,0) GR/2.0) (1/2,0) (31/2,0) (51/2,0) (71/2,0) 14 -12 12 14 x-axis (-31, -1) (-1,-1) (ot, -1) (3.1, -1)

Now to transform y=cos(x) to y=1-cos(x/3), in first step we transform cos(x) to cos(x/3). Observation can be made from cos(x) plot that period of one cycle is 2π, hence the period for cos(x/3) would be 3 times to that of cos(x) i.e. 3x2 π=6 π. That means the plot of cos(x/3) would be stretches 3 times to that of cos(x) about x-axis. The plot cos(x/3) is shown below.

2 y-axis (0,1) cos(x/3) (-97/2,0) (-31/2, 0) (31/2,0) (91/2,0) -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 x-axis (-3,-1) (31, -1) -2

Now finally we transform cos(x/3) to 1-cos(x/3) by simply subtracting the plot of cos(x/3) to the dc level of 1 along the y axis or by simply adding the plot of –cos(x/3) to dc level to 1. The plot of 1-cos(x/3) is shown below.

y-axis -cos(x/3) x-axis -15 -10 -5 0 5 10 15

2 y-axis -15 -10 -5 0 10 15 x-axis

y-axis 2 1-cos(x/3) -15 -10 -5 0 10 15 x-axis

The comparison of plots cos(x), cos(x/3) and 1-cos(x/3) is shown below.