Simple transformations of y=f(x)
What happens when you perform simple linear transformations on y=f(x)?
The effects on the y-axis are straightforward: multiplying by a constant greater than 1 will stretch the graph in the y direction. A constant smaller than 1 will compress the graph.
Adding a positive constant will raise the graph in the positive y direction, a negative one will lower it.
The effects on the x-axis are the opposite of what you would probably expect. If you multiply x by a constant greater than 1 it will compress the graph in the x-direction, a constant less than 1 will stretch it.
Adding a positive constant will move the graph to the left, in the negative x direction. A negative constant will move it right, in the positive x direction.
The effect of these general form parameters is summarised here.
| x-axis y=f(ax + b) |
a | |a| > 1 | compressed in the x direction |
| |a| < 1 | stretched in the x direction | ||
| a < 0 | flipped left-right | ||
| b | b > 0 | moved to the left | |
| b < 0 | moved to the right | ||
| y-axis y=cf(x) + d |
c | |c| > 1 | stretched in the y direction |
| |c| < 1 | compressed in the y direction | ||
| c < 0 | flipped upside down (reflected in x-axis) | ||
| d | d > 0 | raised by d | |
| d < 0 | lowered by -d |
For example: a function y=f(x) is periodic with period 10. Write down a function of x which is periodic with period 5.
From the table above, to compress a curve in the x-direction you need |a| > 1. We want to compress it by a factor of 2, so the solution is y=f(2x).
You can reason this, without memorising the table, by considering that when x increases by 5 the input the the function must increase by 10. Therefore the input to the function must be 2x.
Exercises [simplefxformsxl]
See also: [simplefxformsml]
Exercises: [simplefxformsxl]
Specifications met: [simplefxformssl]