Horizontal Stretch
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
A horizontal stretch refers to the transformation of a graph where the function is "stretched" or "spread out" along the x-axis. This occurs when the input variable x is multiplied by a constant factor less than 1 (but greater than 0) inside the function.
Horizontal Stretch Definition:
If you have a function f(x), and it is transformed to f(kx), where 0 < k < 1, the graph undergoes a horizontal stretch by a factor of 1/k.
In simpler terms, multiplying the input x by a constant k between 0 and 1 "stretches" the graph along the x-axis, making it wider.
Example:
Let’s consider a simple function f(x) = x^2:
The function f(x) = x^2 has a standard U-shaped graph.
If we transform it to f(1/2x) = (1/2x)^2, the graph will be stretched horizontally by a factor of 2. This means the graph will look wider than the original.
Why does this happen?
When you multiply x by a factor less than 1, each value of x needs to change more slowly to reach the same y-value. So, the graph gets "stretched" out, spreading it farther from the y-axis. In other words, for the same y-value, the graph reaches that value at larger x-values.
For example:
In f(x) = x^2, to reach y = 4, x needs to be 2.
In f(1/2x) = (1/2x)^2, to reach y = 4, x needs to be 4. This shows the horizontal stretching because the points reach the same y-value later, with larger values of x.
Summary:
A horizontal stretch happens when the function is transformed by multiplying x by a constant k where 0 < k < 1. This causes the graph to become wider, stretching it along the x-axis.