CA Sp05 O’Brien
Transformations of Functions
I.
Review of Function Library
Constant Function: f(x) = b (horizontal line); Linear Function: f(x) = mx + b (oblique line);
Quadratic Function: f(x) = ax2 + bx +c (parabola); Cubic Function: f(x) = ax3 +bx2 + cx + d (“ziggy”);
Quartic Function: f(x) = ax4 + bx3 + cx2 + dx + e (u or w shape);
Absolute Value Function: f x   x  h  k (v shape); Cube Root Function: f x   3 x  h  k (“lazy ziggy”);


Exponential Function: f(x) = a x – h + k (J shape); Greatest Integer (Step) Function: f(x)  x  h  k (stair steps);
ax  b, x  0
Logarithmic Function: f(x) = log a (x – h) + k (r shape); Piecewise Function: f x   
;
cx  d, x  0
Rational Function: f x  
ax  b
d
, x - ;
cx  d
c
Square Root Functions: f x   x  h  k & f x   a 2  x 2  b (half-parabola & half-circle)
II.
Rigid Transformations
A.
Vertical Shifts
1.
Vertical shift k units up: g(x) = f(x) + k; a constant is added on the “outside”, i.e., to y
3
3
Example: f x  x , gx   f x   5  x  5 a “ziggy” shifted up 5
2.
Vertical shift k units down: g(x) = f(x) - k; a constant is subtracted on the “outside”, i.e., from y
3
3
Example: f x  x , g xf x 6x 6 a “ziggy” shifted down 6
B.
Horizontal Shifts ***counterintuitive***
1.
Horizontal shift h units to the right: g(x) = f(x - h); a constant is subtracted on the “inside:, i.e., from x
2
Example: f x  x , gx f x4 x4
2.
a parabola shifted right 4
Horizontal shift h units to the left: g(x) = f(x + h); a constant is added on the “inside”, i.e., to x
2
Example: f x  x , gx f x5x5
C.
2
2
a parabola shifted left 5
X-axis Reflection
Reflection across the x-axis: g(x) = -f(x); a negative is on the “outside”, i.e., on y
Example: f x  x , gx f x  x a half-parabola reflected across the x-axis
D.
Y-axis Reflection
Reflection across the y-axis: g(x) = f(-x); a negative is on the “inside”, i.e., on x
Example: f x  x , gx f x  x a half-parabola reflected across the y-axis
1
CA Sp05 O’Brien
III.
Non-rigid Transformations
A.
Vertical Stretch
Vertical stretch by a factor of c: gx cf x , c >1
ignoring the sign, a coefficient larger than 1 is on the “outside”, i.e., on y
Example: f x  x , g x2f x 2 x v shape stretched vertically by a factor of 2
B.
Vertical Shrink
Vertical shrink by a factor of c: gx cf x , 0<c <1
ignoring the sign, a coefficient smaller than 1 is on the “outside”, i.e., on y
1
1
1
Example: f x  x , g x f x  x v shape shrunk vertically by a factor of
3
3
3
IV.
Order of Multiple Transformations
To list a sequence of transformations or to graph a function involving more than one transformation, use
the following order:
1. Horizontal shift
V.
2. Stretching or shrinking
3. Reflections
4. Vertical shift
Identifying a sequence of transformations given an equation
3
Example: f x   2x1 5
Identify the basic shape and then list all transformations in their proper order
Shape: “ziggy” [from exponent, 3]; Horiz Shift: 1 left [from x + 1]; Vertical Stretch by a factor of 2 [from coefficient, - 2];
x-axis reflection [from negative on coefficient, -2]; Vertical Shift: 5 down [from - 5 on outside]
VI.
Graphing an equation using transformations
Identify the basic shape. List all transformations in their proper order. Transform the basic graph by
performing the transformations in their proper order.
Example: f x  x 4
4
4
1. Shape: half parabola 
2. Horiz shift: 4 right 
7
7
4
3. No stretch or shrink
VII.
4. y-axis reflection: 
-7
5. No vertical shift
Writing an equation from a transformed graph
Identify the basic shape. List all transformations in their proper order. Modify the basic equation as specified by the
transformations.
Example:
Shape: parabola  fxx
Horizontally shifted 1 right  fxx1
2
2
3
No stretch or shrink
Opens down  x-axis reflection  fx  x1
2
4
2
Vertically shifted up 3  f x    x  1  3
2