Secondary Four
Mathematics Worksheet
Functions and Graph (II) – The effects of transformation on Functions
A.
Translation
I. Translate (shift) upward or downward
:
P.1
g(x)  f x   k
The figure shows the graph of the function f(x) = x2
Using the given graph of the function y = f(x)
(i)
sketch the new graph y = g(x) obtained after
the given transformation.
(ii)
Write out the new equation of the graph
y = g(x) obtained after the given
transformation.
(a) translating upward 2 units.
(b) translating downward 6 units.
II.
g(x)  f x  h 
The figure shows the graph f ( x )  x  x  2 。
Using the given graph of the function y = f(x)
(i) sketch the new graph y = g(x) obtained after
the given transformation.
(ii) Write out the new equation of the graph
y = g(x) obtained after the given
transformation.
(a)
translating to right 2 units.
Translate (shift) to the right or left :
2
(b)
translating to left 1 units。
e.g. 1
The graph of f(x) = x2 + 2x – 3 is first translated to right to 2 units, and then shift upward to 4
units.
(a) sketch the new graph y = g(x) obtained after the given
transformation.
(b)Write down the function represented by the final graph.
1
P.2
B. Reflection
I. Reflection about the x – axis : y = – f(x)
The figure shows the graph of the function
y = x2 .
Sketch the graph y = – x2 .
II.
Reflection about the y – axis：
y = f( – x)
The figure shows the graph of the function
y  x 2  2x  3 .
(a) Write down the function that reflected
about the y – axis .
(b) Sketch the graph.
e.g. 2
The graph of y = x2 + 4x is first reflected about x – axis , and then reflected about y – axis ,
(a) Write down the new function that obtained after the given transformation.
(b) Sketch the graph.
2
P.3
C. Magnigication
I. Vertical magnification : y = a f(x)
[ a > 1 : enlarge the graph of f(x) to a times about y – axis ;
0 < a < 1 : reduce the graph of f(x) to a times about y – axis ]
The figure show the graph
y = f(x) = x2 – 4 .
Sketch the following the function.
(a) y  3 f x .
(b)
y
1
f x  .
2
( Note ：x – coordinates unchanged)
II.
Horizontal magnification : y = f (ax)
1
[ a > 1 : reduce the graph of f(x) to
times about x – axis ;
a
1
0 < a < 1 : enlarge the graph of f(x) to
times about x – axis ]
a
The figure show the graph
y = f(x) = x2 – 16 .
Sketch the following the function.
(a)
y  f 2 x。
1 
y  f  x 。
2 
(b)
e.g. 3
The figure shows y   x 2  2 .
Sketch the following function.
(a) y   x 2  1
(b)
y  x  1  2
2
3
P.4
e.g. 4
The graph of the function f x   2 x 2  3x  1 is first translated to left 1 units , and then
translated upward 3 units, write down the the new function that obtained after the given
transformation.
e.g. 5
(a)
(b)
(c)
Express y   x 2  4 x  9 in the form y  x  h   k .
2
If the graph y   x 2  4 x  9 is translated to right 3 units, write down the new function
that obtained after the given transformation.
Find the coordinate of the vertex of the new function.y
4