vertical stretch
Vertical stretch is a term used in mathematics to describe a transformation of a graph or function in which the graph is stretched vertically
Vertical stretch is a term used in mathematics to describe a transformation of a graph or function in which the graph is stretched vertically. This means that the distances between the points on the graph in the vertical direction are increased or decreased.
To understand vertical stretch, let’s consider a basic function, such as f(x) = x^2. This is a graph of a quadratic function that opens upwards, forming a U-shape.
A vertical stretch of the function f(x) = x^2 would be written as f(x) = a * x^2, where “a” is a constant greater than 1. This constant determines the degree of stretch.
If “a” is greater than 1, such as a = 2, the graph of the function will be stretched vertically. This means that the points on the graph will be farther apart in the vertical direction.
For example, let’s compare the graphs of f(x) = x^2 and g(x) = 2x^2:
When x = 1, f(1) = 1^2 = 1, and g(1) = 2(1^2) = 2. The point (1, 1) on the graph of f(x) is mapped to the point (1, 2) on the graph of g(x). Similarly, for other values of x, the corresponding y-values on the graph of g(x) will be twice the y-values on the graph of f(x).
This illustrates how the vertical stretch by a factor of 2 in the function g(x) = 2x^2 stretches the graph vertically, making it taller.
On the other hand, if “a” is a fraction between 0 and 1, such as a = 1/2, the graph will be compressed vertically. This means that the points on the graph will be closer together in the vertical direction.
For example, if we compare the graphs of f(x) = x^2 and h(x) = (1/2) x^2:
When x = 1, f(1) = h(1) = 1^2 = 1. The point (1, 1) on the graph of f(x) is mapped to the point (1, 1) on the graph of h(x). For other values of x, the corresponding y-values on the graph of h(x) will be half the y-values on the graph of f(x).
This demonstrates how the vertical compression by a factor of 1/2 in the function h(x) = (1/2) x^2 compresses the graph vertically, making it shorter.
In summary, a vertical stretch in a function expands or contracts the graph vertically. If the stretch factor “a” is greater than 1, the graph is stretched vertically, making it taller. If the stretch factor “a” is between 0 and 1, the graph is compressed vertically, making it shorter.
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