1. 14. 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 of each point of the graph. The function represented by the image is y = f(x) +k+k Up +

2. How to memorise the graphs of functions after transformation effectively? 14. 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-coordinate of each point of the graph. The function represented by the image is y = f(x) －k－k Down －

3. How to memorise the graphs of functions after transformation effectively? 14. More about Graphs of Functions y = f(x ) O y x Translation y 1 = y 2, x 1 = x 2 +h +h+h Left + y = f(x) (x 1, y 1 ) (x 2, y 2 ) ∵ y 1 = f(x 1 ) ∴ y 2 = f(x 2 +h) y = f(x+h) h units (c)Translate the graph of y = f(x) h units to the left. The function represented by the image is y = f(x)

4. How to memorise the graphs of functions after transformation effectively? 14. More about Graphs of Functions O y x Translation y = f(x) (x 1, y 1 ) (x 2, y 2 ) y = f(x － h) h units y = f(x ) y 1 = y 2, x 1 = x 2 － h －h－h Right － ∵ y 1 = f(x 1 ) ∴ y 2 = f(x 2 － h) (d)Translate the graph of y = f(x) h units to the right. The function represented by the image is y = f(x)

5. How to memorise the graphs of functions after transformation effectively? 14. 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 each point of the graph changes. The function represented by the image is y = f(x)

6. How to memorise the graphs of functions after transformation effectively? 14. 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 each point of the graph changes. The function represented by the image is y = f(x) Reflection

7. How to memorise the graphs of functions after transformation effectively? 14. More about Graphs of Functions Dilation (a) Dilate the graph of y = f(x) vertically. (i) The graph is enlarged by k 1 times vertically, where k 1 ＞ 1. O y x y = f(x) k1k1 y = k 1 f(x), k 1 ＞ 1 i.e. the y-coordinate of each point of the graph is multiplied by k 1. The function represented by the image is y = f(x)

8. How to memorise the graphs of functions after transformation effectively? 14. More about Graphs of Functions Dilation O y x y = f(x) y = k 2 f(x), 0 ＜ k 2 ＜ 1 (a) Dilate the graph of y = f(x) vertically. (ii) The graph is contracted to k 2 time vertically, where 0 ＜ k 2 ＜ 1. k2k2 y = f(x) i.e. the y-coordinate of each point of the graph is multiplied by k 2. The function represented by the image is y = f(x)

9. How to memorise the graphs of functions after transformation effectively? 14. More about Graphs of Functions Dilation O y x y = f(k 1 x), k 1 ＞ 1 y = f(x) (x 1, y 1 ) (x 2, y 2 ) y 1 = y 2, x 1 = k 1 x 2 ∵ y 1 = f(x 1 ) ∴ y 2 = f(k 1 x 2 ) k1k1 y = f( x) (b) Dilate the graph of y = f(x) horizontally. (i) The graph is contracted to time horizontally, where k 1 ＞ 1. 1 k1k1 The function represented by the image is y = f(x)

10. How to memorise the graphs of functions after transformation effectively? 14. More about Graphs of Functions Dilation O y x y = f(k 2 x), 0 ＜ k 2 ＜ 1 y = f(x) (x 1, y 1 ) (x 2, y 2 ) y 1 = y 2, x 1 = k 2 x 2 ∵ y 1 = f(x 1 ) ∴ y 2 = f(k 2 x 2 ) k2k2 y = f( x) (b) Dilate the graph of y = f(x) horizontally. (ii) The graph is enlarged by times horizontally, where 0 ＜ k 2 ＜ 1. 1 k2k2 The function represented by the image is y = f(x)

11. How to memorise the graphs of functions after transformation effectively? 14. More about Graphs of Functions Dilation The maximum (minimum) value remains unchanged. O y x y = f(k 1 x) ， k 1 ＞ 1 y = f(x) y = f(k 2 x) ， 0 ＜ k 2 ＜ 1 Enlargement Contraction (b) Dilate the graph of y = f(x) horizontally.

12. 14. More about Graphs of Functions Transformation of graphTransformation of function 1. Translate k units upwardsAdd 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 leftAdd 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 1 k 1 k Transformation Easy Memory Tips: Down － Up +Left + Right －