WorkSHEET 2.2 Functions and transformations
1
State the transformations required to change
1
y  2 into each of the following curves.
x
3
a
y
 x  22
b
2
y  2
6
x  12
y
a
Name: _______________________
3
 x  22
A translation of 2 units to the left along
the x-axis, followed by a dilation of
3 units along the y-axis
y  2
b
6
x  12
A translation of 1 unit to the right along
the x-axis, followed by a dilation of
6 units along the y-axis, followed by a
reflection about the x-axis, and then a
shift of 2 units upwards
For each of the following, state:
i
the equations of the asymptotes
ii
the domain
iii
the range.
a
y
1
4  x 
2
1
y
a
i
ii
iii
b
y  4
1
4  x 2
Asymptotes: when 4  x = 0,
i.e. x = 4
and y = 1
Domain (  , 4)  (4, ) or R\{4}
Range ( 1, )
9
x  12
y  4
b
i
ii
iii
Maths Quest 12 Mathematical Methods CAS
1
9
x  12
Asymptotes when undefined:
when x + 1 = 0, i.e. x = 1
and y = 4
Domain (  , 1)  ( 1, )
or R\{1}
Range (  , 4)
1
3
20
Sketch y 
y
a
20
 5 and show clearly:
x  32
the position of the asymptotes
a
Asymptotes when undefined:
when x + 3 = 0,
i.e. x = 3 and y = 5
b
the intercepts with the axes.
b
y-intercept, when x = 0:
20
y
5
2
 0  3
x  32
5
 7 92
1
is translated 5 units
x2
up, so no x-intercept.
The graph of y =
Maths Quest 12 Mathematical Methods CAS
2
4
Sketch y  2 
2
3  2 x 
2
and show clearly:
y  2
a
the position of the asymptotes
a
b
the intercepts with the axes.
b
2
3  2 x 2
Asymptotes: when 3  2x = 0,
i.e. x = 1.5 and y = 2
y-intercept, when x = 0:
2
y  2
2
3  0
 1 79
x-intercept, when y = 0:
2
0  2
3  2 x 2
2
2
3  2 x 2
1  3  2 x 
2
1  3  2x
2 x  2 or 4
x  1 or 2
Maths Quest 12 Mathematical Methods CAS
3
5
State the transformations required to change
y  x into each of the following.
a
y  1 2 x
a
y  1 2 x
Dilation of 2 in the y-direction followed
by upward translation of 1 unit.
b
y  35 9 x
b
y  35 9 x
y  35 x  9
Reflection about the y-axis, followed by
a translation of 9 units to the right,
followed by a dilation of factor 5 along
the y-axis, then a reflection about the
x-axis, and finally an upward translation
of 3 units.
6
For each of the following, state the domain and
the range.
y  4 x 1  2
a
a
b
y  1 2 4  x
Maths Quest 12 Mathematical Methods CAS
b
y  4 x 1  2
Domain [1, ), range [2, )
y  1 2 4  x
Domain (, 4], range (, 1]
4
7
Sketch each of the following and show clearly
any intercepts and endpoints.
1
y
x  4 1
a
2
a
y  12 x  4  1
Domain [4, ), range [1, )
End point (4, 1)
y-intercept, when x = 0:
y  12 0  4  1
y0
x-intercept, when y = 0:
0  12 x  4  1
1  12 x  4
2 x4
4 x4
x0
b
y  2  9  3x
b
y  2  9  3x
Domain (, 3], range (, 2]
End point (3, 2)
y-intercept, when x = 0:
y  2 90
y  1
x-intercept, when y = 0:
0  2  9  3x
9  3x  2
9  3x  4
5  3x
5
2
x  1
3
3
Maths Quest 12 Mathematical Methods CAS
5
8
Sketch each of the following and show clearly
any intercepts with the axes.
a
y  1  3x  2
a
y  1  3x  2
Minimum value when 1  3x = 0
Cusp:  13 , 2 
y-intercept, when x = 0:
y  1 0  2
y 3
x-intercept, when y = 0:
0  1  3x  2

2  1  3x
which is impossible, so no x-intercept.
b
y  2  3  x2
b
y  2  3 x2
Let y = −(3 − x2), which is the same as
y = x2 − 3.
x-intercept, when y = 0:
x2 − 3 = 0
x=± 3
y-intercept, when x = 0:
y = −3
y = −(3 − x2)
To sketch y = − 3  x 2 , reflect the
portion of the parabola for
(−∞, − 3 )  ( 3 , ∞) in the x-axis.
Maths Quest 12 Mathematical Methods CAS
6
y = − 3  x2
To sketch y = 2 − 3  x 2 , translate the
graph of y = − 3  x 2 up 2 units.
The turning point is (0, −1).
The cusps are (− 3 , 2) and ( 3 , 2).
x -intercept, when y  0 :
0  2  3  x2
3  x2  2
3  x 
3  x 
2 2
2
2 2
4
3  x  2
3  2  x2
x   5 and  1
Maths Quest 12 Mathematical Methods CAS
2
7
9
For each of the following, state the domain and
the range.
a
3
a
y   x  2  1
b
10
y
1
1   x  1
2
Find the range of each of the following
relations.
y  x  a, a  0
a
b
y b
c
y
1
, b0
x2
c
, c0
x2
Maths Quest 12 Mathematical Methods CAS
y   x  2  1
3
Domain (, ) or R, range [0, )
y
1
b
1   x  1
For domain, 1 − (x − 1)2 ≠ 0
1 − (x − 1)2 = 0
(x − 1)2 − 1 = 0
x − 1 − 1 = 0, x − 1 + 1 = 0
x = 2, x = 0
Domain (−∞, 0)  (0, 2)  (2, ∞),
range (0, )
a
y  x  a, a  0
Range [0, )
b
c
2
1
, b0
x2
Range (, b)
y b
c
, c0
x2
Range (  , 0)  (0, ) or R\{0}
y
8