A vertical stretch is a transformation that affects the y-coordinates of a graph by multiplying them by a factor greater than 1. This transformation makes the graph appear taller and narrower.
Understanding Vertical Stretch
Basic Concept
Imagine you have a graph of a function, say $y = f(x)$. A vertical stretch involves multiplying the entire function by a constant factor, $a$, where $a > 1$. The new function becomes $y = a times f(x)$. For example, if $f(x) = x^2$ and $a = 2$, the vertically stretched function would be $y = 2x^2$
Visualizing the Stretch
Consider the original function $f(x) = x^2$ and its graph, a parabola. When we apply a vertical stretch with $a = 2$, each y-coordinate on the graph is doubled. So, the point (1,1) on $f(x)$ would move to (1,2) on $2f(x)$, and the point (2,4) would move to (2,8). This makes the parabola appear taller and narrower.
Formula Representation
The general formula for a vertical stretch is:
$y = a times f(x)$
where $a$ is the stretch factor. If $a > 1$, the graph stretches vertically. If $0 < a < 1$, the graph compresses vertically, but this is technically called a vertical compression.
Examples
Example 1: Linear Function
Take the linear function $f(x) = x$. If we apply a vertical stretch with $a = 3$, the new function is $y = 3x$. The graph of $y = 3x$ will be three times as tall as the graph of $y = x$
Example 2: Quadratic Function
For the quadratic function $f(x) = x^2$, applying a vertical stretch with $a = 4$ results in $y = 4x^2$. The graph of $y = 4x^2$ will be four times as tall as the original graph of $y = x^2$
Applications
Vertical stretches are commonly used in various fields such as physics, engineering, and economics to model situations where an increase in amplitude or intensity is observed. For instance, in physics, a vertical stretch can represent an increase in wave amplitude.
Conclusion
Understanding vertical stretches is crucial for modifying and interpreting graphs of functions. By multiplying the y-coordinates by a factor greater than 1, we can make graphs appear taller and narrower. This transformation helps us better understand and visualize changes in various real-world scenarios.