1. Graph Transformations (I)f(x), f(x) +/- k, f(x +/- k)See Handout Example The following graph shows y = g(x). Make sketches of (a) y = g(x) + 3 (b) y = g(x) - 4 (c) y = g(x+5) (d) y = g(x-2)

2. X y = g(x) (23, 2) (31, 7) Key points are (23,2) and (31,7). To transform the graph transform the points.

3. X y = g(x) + 3 (23, 5) (31, 10) For y = g(x) + 3 move everything 3 units up. (23,2)  (23,5)(31,7)  (31,10)

4. X y = g(x) - 4 (23, -2) (31, 3) For y = g(x) - 4 move everything 4 units down. (23,2)  (23,-2)(31,7)  (31,3)

5. X y = g(x+5) (18, 2) (26, 7) For y = g(x+5) move everything 5 units left. (23,2)  (18,2)(31,7)  (26,7)

6. X y = g(x-2) (25, 2) (33, 7) For y = g(x-2) move everything 2 units right. (23,2)  (25,2)(31,7)  (33,7)

7. NB: (I) +/- changes outside the brackets slide the graph in the Y direction in the way you would expect. ie + goes up&- goes down. (II) +/- changes inside the brackets slide the graph in the X direction in the opposite way to which you would expect. ie + goes back&- goes forward.

8. ExampleThe graph below shows y = f(x) Make sketches of (I) y = -f(x) (II) y = f(-x) y = f(x) X 12 (14,6) (17,5) Main points are (12,0), (14,6) & (17,5).

9. For y = -f(x) we reflect in the X-axis. (12,0)  (12,0)(14,6)  (14,-6) (17,5)  (17,-5) X 12 (14,-6) (17,-5) y = -f(x) NB: max TP now min TP & min TP now max TP.

10. For y = f(-x) we reflect in the Y-axis. (12,0)  (-12,0)(14,6)  (-14,6) (17,5)  (-17,5) X -12 (-14,6) (-17,5) y = f(-x)

11. ExampleHere is the graph of y = f(x) Make sketches of (I) y = 4 f(x) (II) y = f(3x) X y = f(x) (12,2) (3,-1) Main points are (12,2) & (3,-1).

12. (I) To obtain y = 4 f(x) we stretch the graph vertically by a factor of 4 (ie mult y-coords by 4). X y = 4f(x) (12,8) (3,-4) (12,2)  (12,8)(3,-1)  (3,-4)

13. (II) To obtain y = f(3x) we squash the graph horizontally by a factor of 3 (ie divide x-coords by 3). X y = f(3x) (4,2) (1,-1) (12,2)  (4,2)(3,-1)  (1,-1)

14. Trig Graphs The same transformation rules apply to the basic trig graphs. NB: If f(x) =sinx  then 3f(x) = 3sinx  and f(5x) = sin5x  Think about sin replacing f ! Also if g(x) = cosx  then g(x) –4 = cosx  –4 and g(x+90) = cos(x+90)  Think about cos replacing g !

15. ExampleSketch the graph of y = sinx  - 2 ! If sinx  = f(x) then sinx  - 2 = f(x) - 2 So move the sinx  graph 2 units down. y = sinx  - 2

16. Example Sketch the graph of y = cos(x - 50)  ! If cosx  = f(x) then cos(x - 50)  = f(x - 50) So move the cosx  graph 50 units right. y = cos(x - 50) 

17. Example Sketch the graph of y = 3sinx  ! If sinx  = f(x) then 3sinx  = 3f(x) So stretch the sinx  graph 3X vertically. y = 3sinx 

18. Example Sketch the graph of y = cos4x  ! If cosx  = f(x) then cos4x  = f(4x) So squash the cosx  graph to 1 / 4 size horizontally. y = cos4x 

19. Example Sketch the graph of y = 2sin3x  ! If sinx  = f(x) then 2sin3x  = 2f(3x) So squash the sinx  graph to 1 / 3 size horizontally and also double its height. y = 2sin3x 