The document describes various transformations that can be applied to a graph y=f(x) to generate other graphs. It discusses shifting graphs up or down by adding or subtracting a constant a to y, shifting graphs left or right by adding or subtracting a to x, reflecting graphs across the x-axis or y-axis, stretching graphs vertically or horizontally by multiplying f(x) or x by a constant k, and other transformations. These transformations can be used to build new graphs from an original function f(x).

1. (C) Transformations

2. (C) Transformations
Given that the graph y = f(x) can be sketches, then it is possible to build
other sketches through appropriate transformations.

3. (C) Transformations
Given that the graph y = f(x) can be sketches, then it is possible to build
other sketches through appropriate transformations.
 y  f  x   a OR y  a  f  x 
(a is grouped with y: shift up or down by a)

4. (C) Transformations
Given that the graph y = f(x) can be sketches, then it is possible to build
other sketches through appropriate transformations.
 y  f  x   a OR y  a  f  x 
(a is grouped with y: shift up or down by a)
y
y=f(x)
x

5. (C) Transformations
Given that the graph y = f(x) can be sketches, then it is possible to build
other sketches through appropriate transformations.
 y  f  x   a OR y  a  f  x 
(a is grouped with y: shift up or down by a)
y
y  f x  a
a y=f(x)
a
a
x
a

6. (C) Transformations
Given that the graph y = f(x) can be sketches, then it is possible to build
other sketches through appropriate transformations.
 y  f  x   a OR y  a  f  x 
(a is grouped with y: shift up or down by a)
 y  f  x  a  (a is grouped with x: shift left or right by a)
y
y  f x  a
a y=f(x)
a
a
x
a

7. (C) Transformations
Given that the graph y = f(x) can be sketches, then it is possible to build
other sketches through appropriate transformations.
 y  f  x   a OR y  a  f  x 
(a is grouped with y: shift up or down by a)
 y  f  x  a  (a is grouped with x: shift left or right by a)
y
y  f x  a
a y=f(x)
a
b
b
a
x
b
b
a
b
y  f x  b

8.  y   f  x  (reflect f(x) in the x axis)

9.  y   f  x  (reflect f(x) in the x axis)
y
y=f(x)
x

10.  y   f  x  (reflect f(x) in the x axis)
y
y=f(x)
x
y   f x

11.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
y
y=f(x)
x
y   f x

12.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
y
y=f(x)
x
y   f x

13.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
y
y=f(x)
y  f  x 
x
y   f x

14.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)

15.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
y
y=f(x) x 
y f
x

16.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
y
y=f(x) x 
y f
x

17.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
y
y=f(x) x 
y f
x

18.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
y
y=f(x) x 
y f
x

19.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)

20.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
y
y=f(x) x 
y f
x

21.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
y
y=f(x) x 
y f
x

22.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
y
y=f(x) x 
y f
x

23.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
y
y=f(x) x 
y f
x

24.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)

25.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
y
y=f(x)
x

26.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
y y  kf  x , k  1
y=f(x)
x

27.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
y y  kf  x , k  1
y=f(x)
y  kf  x , k  1
x

28.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
y y  kf  x , k  1
y=f(x)
y  kf  x , k  1
x
Note:
•domain remains same
•x intercepts remain same

29.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
 y  f kx  (stretch f(x) horizontally, k<1 shallower,k>1 steeper)

30.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
 y  f kx  (stretch f(x) horizontally, k<1 shallower,k>1 steeper)
y
y=f(x)
x

31.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
 y  f kx  (stretch f(x) horizontally, k<1 shallower,k>1 steeper)
y
y  f kx , k  1 y=f(x)
x

32.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
 y  f kx  (stretch f(x) horizontally, k<1 shallower,k>1 steeper)
y
y  f kx , k  1 y=f(x)
x
y  f kx , k  1

33.  y   f  x  (reflect f(x) in the x axis)
 y  f  x  (reflect f(x) in the y axis)
 y  f x (reflect the part of f(x) where f(x)<0 in the x axis)
 y  f x (reflect the part of f(x) where x>0 in the y axis)
 y  kf  x  (stretch f(x) vertically, k<1 shallower,k>1 steeper)
 y  f kx  (stretch f(x) horizontally, k<1 shallower,k>1 steeper)
y
y  f kx , k  1 y=f(x)
x
Note: y  f kx , k  1
•range remains same
•y intercepts remain same