14. More about Graphs of Functions14. More about Graphs of Functions

transformation effectively?How to memorise the graphs of functions after

O

y

x

(a) Translate the graph of y = f(x) k units upwards.

y = f(x)

y = f(x)+k

k units

Translation

i.e. k is added to the y-coordinate ofeach point of the graph.

The function represented by the image is

y = f(x)

+k

Up +

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

O

y

x

y = f(x)

y = f(x) － k

(a) Translate the graph of y = f(x) k units downwards.

k units

Translation

i.e. k is subtracted from the y-coordinateof each point of the graph.

The function represented by the image is

y = f(x)

－ k

Down －

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

y = f(x )O

y

x

Translation

y1 = y2, x1 = x2+h

+hLeft +

y = f(x)

(x1 , y1)(x2 , y2)

∵ y1 = f(x1)

∴ y2 = f(x2 +h)

y = f(x+h)

h units

(c) Translate the graph of y = f(x) h units to the left.

The function represented by theimage is

y = f(x)

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

O

y

x

Translation

y = f(x)

(x1 , y1) (x2 , y2)

y = f(x －h)

h units

y = f(x )

y1 = y2, x1 = x2 － h

－ hRight －

∵ y1 = f(x1)

∴ y2 = f(x2 － h)

(d) Translate the graph of y = f(x) h units to the right.

The function represented by theimage is

y = f(x)

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

O

y

x

y = f(x)

y = － f(x)

Reflection

－y = f(x)

(a) Reflect the graph of y = f(x) in the x-axis.

i.e. the sign of the y-coordinate of eachpoint of the graph changes.

The function represented by the image is

y = f(x)

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

O

y

x

y = f(x) y = f( － x)

－y = f( x)

(b) Reflect the graph of y = f(x) in the y-axis.

i.e. the sign of the x-coordinate of eachpoint of the graph changes.

The function represented by the image is

y = f(x)

Reflection

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

Dilation

(a) Dilate the graph of y = f(x) vertically.

(i) The graph is enlarged by k1 times vertically, where k1 ＞ 1.

O

y

x

y = f(x)

k1y = f(x)

y = k1f(x),k1 ＞ 1i.e. the y-coordinate of each point of

the graph is multiplied by k1.

The function represented by the image is

y = f(x)

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

Dilation

O

y

x

y = f(x)

y = k2f(x),0 ＜ k2 ＜ 1

(a) Dilate the graph of y = f(x) vertically.

(ii) The graph is contracted to k2 time vertically, where 0 ＜ k2 ＜ 1.

k2y = f(x)

i.e. the y-coordinate of each point of the graph is multiplied by k2.

The function represented by the image is

y = f(x)

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

Dilation

O

y

xy = f(k1x),k1 ＞ 1

y = f(x)

(x1 , y1)(x2 , y2)

y1 = y2, x1 = k1x2

∵ y1 = f(x1)

∴ y2 = f(k1x2)

k1y = f( x)

(b) Dilate the graph of y = f(x) horizontally.

(i) The graph is contracted to time horizontally, where k1 ＞ 1. 1k1

The function representedby the image is

y = f(x)

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

Dilation

O

y

x

y = f(k2x),0 ＜ k2 ＜ 1

y = f(x)(x1 , y1)

(x2 , y2)

y1 = y2, x1 = k2x2

∵ y1 = f(x1)

∴ y2 = f(k2x2)

k2y = f( x)

(b) Dilate the graph of y = f(x) horizontally.

(ii) The graph is enlarged by times horizontally, where 0 ＜ k2 ＜1.

1k2

The function representedby the image is

y = f(x)

How to memorise the graphs of functions after transformation effectively?

14. More about Graphs of Functions14. More about Graphs of Functions

Dilation

The maximum (minimum) value remains unchanged.

O

y

xy = f(k1x)，k1 ＞ 1

y = f(x)

y = f(k2x)，0 ＜ k2 ＜ 1

Enlargement

Contraction

(b) Dilate the graph of y = f(x) horizontally.

14. More about Graphs of Functions14. More about Graphs of Functions

Transformation of graph Transformation of function

1. Translate k units upwards Add k to f(x) externally, i.e. y = f(x)+k

2. Translate k units downwards Subtract k from f(x) externally, i.e. y = f(x) － k

3. Translate k units to the left Add k to f(x) internally, i.e. y = f(x+k)

4. Translate k units to the right Subtract k from f(x) internally, i.e. y = f(x － k)

5. Reflect in the x-axis Add a minus sign to f(x) externally, i.e. y = － f(x)

6. Reflect in the y-axis Add a minus sign to f(x) internally, i.e. y = f( － x)

7. Enlarge by k times vertically (k ＞ 1) Multiply f(x) by k externally, i.e. y = kf(x) , k ＞ 1

8. Contract to k time vertically (0 ＜ k ＜ 1) Multiply f(x) by k externally, i.e. y = kf(x) , 0 ＜ k ＜ 1

9. Contract to time horizontally (k ＞ 1) Multiply f(x) by k internally, i.e. y = f(kx) , k ＞ 1

10. Enlarge by times horizontally (0 ＜ k ＜1)

Multiply f(x) by k internally, i.e. y = f(kx) , 0 ＜ k ＜ 1

1k1k

Transformation Easy Memory Tips:

Down －

Up +

Left +

Right －