When we talk about a vertical stretch in relation to a function, we mean that the graph of the function is being “stretched” vertically. This means that the distance between the points on the graph is increased in the vertical direction.
To understand this concept better, let’s imagine a simple example. Suppose we have a function f(x) = x^2. If we graph this function, we will get a parabola that opens upwards. Now, if we multiply this function by a positive constant, let’s say 2, we would get g(x) = 2x^2.
The graph of g(x) would still be a parabola that opens upwards, but it would be “stretched” vertically. This means that the points on the graph are further apart from each other in the vertical direction compared to the original function f(x).
In practical terms, a vertical stretch can be visualized as if we were pulling the graph of the function upwards, making it taller and thinner. The overall shape of the graph remains the same, but it becomes stretched vertically.
To further illustrate this concept, let’s consider another example. Suppose we have the function h(x) = sin(x), which represents a sine wave. If we multiply this function by a positive constant, let’s say 3, we would get k(x) = 3sin(x).
The graph of k(x) would still be a sine wave, but it would be “stretched” vertically. The peaks and valleys of the wave would be further apart from each other in the vertical direction, making the wave appear taller and thinner.
In general, when we multiply a function by a positive constant greater than 1, we are stretching the graph vertically. The greater the constant, the greater the vertical stretch. On the other hand, if the constant is between 0 and 1, we would have a vertical compression, which would make the graph shorter and wider.
It’s important to note that a vertical stretch or compression does not change the overall shape of the graph. It simply changes the proportions in the vertical direction. This concept is widely used in mathematics and has various applications in fields such as physics, engineering, and economics.
A vertical stretch refers to the stretching of a graph in the vertical direction. It involves increasing the distance between points on the graph in the vertical direction, making the graph appear taller and thinner. This concept is important in understanding the transformations of functions and has practical applications in various fields.