Sketch the graphs of the following functions:
Question 1: y = sin2 x
Solution:
As we know that,
y = sin2 x =
On shifting the origin at (0, 1/2), we get
X = x and Y =
On substituting these values, we get
The maximum and minimum values of Y are
and respectively and shift it by 1/2 to the up. As the equation in the form of y = – f(x), the graph become inverted of y = f(x)
Question 2: y = cos2 x
Solution:
As we know that,
y = cos2 x =
On shifting the origin at
, we get X = x and Y =
On substituting these values, we get
The maximum and minimum values of Y are
and respectively and shift it by 1/2 to the up.
Question 3: y = sin2
Solution:
To obtain this graph y-0 = sin2
On shifting the origin at (
,0), we get X =
and Y = y – 0 On substituting these values, we get
Y = sin2 X
First we draw the graph of Y = sin2 X and shift it by π/4 to the right.
Question 4: y = tan 2x
Solution:
To obtain this graph y = tan 2x,
First we draw the graph of y = tan x and then divide the x-coordinates of the points where it crosses x-axis by 2.
Question 5: y = 2 tan 3x
Solution:
To obtain this graph y = 2 tan 3x,
First we draw the graph of y = tan x and then divide the x-coordinates of the points where it crosses x-axis by 3.
Stretch the graph vertically by the factor of 2.
Question 6: y = 2 cot 2x
Solution:
To obtain this graph y = 2 cot 2x,
First we draw the graph of y = cot x and then divide the x-coordinates of the points where it crosses x-axis by 2.
Stretch the graph vertically by the factor of 2.
Sketch the graphs of the following functions on the same scale:
Question 7: y = cos 2x, y = cos
Solution:
Graph 1:
y = cos2 x
As we know that,
y = cos2 x =
On shifting the origin at (0, 1/2), we get
X = x and Y =
On substituting these values, we get
The maximum and minimum values of Y are
and respectively and shift it by 1/2 to the up. Graph 2:
To obtain this graph y-0 = cos (2x-
) = cos 2(x- ) On shifting the origin at (π/6, 0), we get
X =
and Y = y – 0 On substituting these values, we get
Y = cos 2X
First we draw the graph of Y = cos 2X and shift it by π/6 to the right.
The graph y = cos2 x and y = cos
are on same axes are as follows:
Question 8: y = sin2 x, y = sin x
Solution:
Graph 1:
y = sin2 x
As we know that,
y = sin2 x =
On shifting the origin at (0,
), we get X = x and Y =
On substituting these values, we get
The maximum and minimum values of Y are
and respectively and shift it by 1/2 to the up. As the equation in the form of y = – f(x), the graph become inverted of y = f(x)
Graph 2:
y = sin x
The graph y = sin2 x and y = sin x are on same axes are as follows:
Question 9: y = tan x, y = tan 2x
Solution:
Graph 1:
y = tan x
Graph 2:
y = tan2 x
The graph y = tan x and y = tan2 x are on same axes are as follows:
Question 10: y = tan 2x, y = tan x
Solution:
Graph 1:
To obtain this graph y = tan 2x,
First we draw the graph of y = tan x and then divide the x-coordinates of the points where it crosses x-axis by 2.
Graph 2:
y = tan x
The graph y = tan 2x and y = tan x are on same axes are as follows: