What is Shifting and Scaling in mathmatic graphs?
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:
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
|acts normally||acts inversely|
To understand these graph translations, let's see what happens in simple steps with just one change at a time.
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.
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.
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.
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.
y = f(x) y = f(2x-1)+1