14. More about Graphs of Functions transformation effectively? How to memorise the graphs of...

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?


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.


y = f(x)

－ k

Down －



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)



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)



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.


y = f(x)



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.


y = f(x)

Reflection



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.


y = f(x)



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.


y = f(x)



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)



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)


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

1k2

The function representedby the image is

y = f(x)



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



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 －

14. More about Graphs of Functions transformation effectively? How to memorise the graphs of...

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