Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Functions - Transformations.
Summary of shift and dilation transformations.


 

The following is a summary of the strategies outlined in the video for transformations aiming at shifting and/or dilating curves.

 

  Vertical Horizontal
Shifts Replace y with y - V. Replace x with x - H.
V is the constant by which
the curve is moved vertically.		 H is the constant by which
the curve is moved horizontally.
  V > 0: moves curve up.   H > 0: moves curve to right.
  V < 0: moves curve down.   H < 0: moves curve to left.
   
Dilation Replace y with . Replace y with.
V is the dilation factor from the x axis. H is the dilation factor from the y axis
  V > 1: curve expands up.   H > 1: curve expands out.
  0 < V < 1: curve contracts towards x axis.   0 < H < 1: curve contracts towards y axis.
  V < 0: curve flips around x-axis.   H < 0: curve flips around y-axis.

 

The following table provides examples of transformations together with descriptions.
The f(x) is not the same across all examples do the basic shape is provided:

Transformation Equation Graph
Horizontal shift: y = x3 transformed to
y = (x + 2)3.

The +2 in the brackets moves the curve 2 units to the left.

Vertical shift: y = x3 transformed to
y = x3 + 1
  • moves up 1 (note the +1 is not attached to the x term.
Both shifts: y = x2 transformed to
y = (x - 1)2 - 3
  • moves horizontally 1 to the right
  • moves vertically 3 down.
Dilation y = f(x) transformed to
.

The 2 inside the function shows a horizontal dilation - or stretch) by a factor of 2. The point (-1, 0) becomes (-2, 0).
  y = f(x) transformed to
y = 2f(x) is equivalent to
y ÷ 2 = f(x).
Hence a vertical dilation (a stretch) by a factor of 2 - e.g. the y = -1 goes to y = -2.
(no change to the x intercepts of course).
Mixtures y = f(x) transformed to
y = 1 - f(x):
  • the negative in front of f(x) flips the curve around the x axis (green curve);
  • the +1 moves the whole curve up 1 unit (blue curve).
  y = f(x) is transformed into
y = 1 - 2f(3x):
  • the 3 coefficient for x shrinks the width of f(x) by a factor of 3 (so the curve is narrower) - a division by 1/3;

  • the 2 in front of the function can be thought of as being a division by ½ and so further shrinks the curve horizontally (reducing the x values);
  • the negative sign flips the function around the x axis;

  • the +1 moves the curve up 1 unit.