Shifting and Scaling

Shifting and Scaling in Curves

Terms

What is Shifting and Scaling in mathmatic graphs?

Shift

A translation in which the size and shape of a graph of a function is not changed, but the location of the graph is.

Scale

A translation in which the size and shape of the graph of a function is changed.

Shifting and Scaling can apply on most of the functions and translate them to a new graph without loosing the properties of the old graph. The follwoing are some of common functions:

Constant Function: y=c
Linear Function: y=x
Quadratic Function: y=x^2
Cubic Function: y=x^3
Absolute Value Function: y=|x|
Square Root Function: y=sqrt(x)

The basic graph have the function: y = f(x)

All of the translations can be expressed in the form:

   y = a * f [ b (x-c) ] + d

	Vertical	Horizontal
Scale	a	b
Shift	d	c
	acts normally	acts inversely

To understand these graph translations, let's see what happens in simple steps with just one change at a time.

Shifts

A shift is a rigid translation in that it does not change the shape or size of the graph of the function. All that a shift will do is change the location of the graph. A vertical shift adds/subtracts a constant to/from every y-coordinate while leaving the x-coordinate unchanged. A horizontal shift adds/subtracts a constant to/from every x-coordinate while leaving the y-coordinate unchanged. Vertical and horizontal shifts can be combined into one expression.

Shifts are added/subtracted to the x or f(x) components. If the constant is grouped with the x, then it is a horizontal shift, otherwise it is a vertical shift.

Example:

Vertical shift

y = f(x)

y = f(x)+1
1 is added to all y values to shift up.
Vice versa to shift down.

Horizontal shift

y = f(x)

y = f(x+1)
The height at x on the new graph
is equal to the height at x+1 on the old one.
The new graph is obtained by shifting
the old one 1 to the left. Vice versa to shift right.

Scales (Stretch/Compress)

A scale is a non-rigid translation in that it does alter the shape and size of the graph of the function. A scale will multiply/divide coordinates and this will change the appearance as well as the location. A vertical scaling multiplies/divides every y-coordinate by a constant while leaving the x-coordinate unchanged. A horizontal scaling multiplies/divides every x-coordinate by a constant while leaving the y-coordinate unchanged. The vertical and horizontal scalings can be combined into one expression.

Example:

Vertical scale

y = f(x)

y = 2f(x)
All y-values on the graph are scaled by 2.
In subsituting cf for f,
vertical distances are scaled by c.

Horizontal scale

y = f(x)

y = f(2x)
The height at x on the new graph is
equal to the height at 2x on the old one.
The new graph is obtained by compressing
the old one horizontally by 2.
In substituting cx for x,
horizontal distances are scaled by 1/c.

Question: Putting them all together, do you understand how this translation occur?

y = f(x) y = f(2x-1)+1

Answer


y = f(x)	y = f(x)+1 1 is added to all y values to shift up. Vice versa to shift down.


y = f(x)	y = f(x+1) The height at x on the new graph is equal to the height at x+1 on the old one. The new graph is obtained by shifting the old one 1 to the left. Vice versa to shift right.


y = f(x)	y = 2f(x) All y-values on the graph are scaled by 2. In subsituting cf for f, vertical distances are scaled by c.


y = f(x)	y = f(2x) The height at x on the new graph is equal to the height at 2x on the old one. The new graph is obtained by compressing the old one horizontally by 2. In substituting cx for x, horizontal distances are scaled by 1/c.