Vertical stretch and compression refer to the transformation of a function's graph by multiplying its output values by a constant factor. When a function is vertically stretched, its graph becomes taller and the peaks and valleys of the function increase in distance from the horizontal axis. Conversely, vertical compression makes the graph flatter, decreasing the distance of these points from the horizontal axis. Understanding these transformations helps in analyzing sinusoidal functions and their behaviors.
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A vertical stretch occurs when the output values of a function are multiplied by a factor greater than 1, causing the peaks and troughs to rise or fall more dramatically.
In contrast, a vertical compression happens when output values are multiplied by a factor between 0 and 1, making the graph less steep and reducing the height of peaks and depth of troughs.
The vertical stretch or compression factor directly impacts the amplitude of sinusoidal functions, which is defined as half the distance between the maximum and minimum values.
If a sine function has an original amplitude of 1, applying a vertical stretch by a factor of 3 will result in an amplitude of 3.
In sinusoidal equations, vertical stretch/compression is represented as `y = k * sin(x)` or `y = k * cos(x)`, where |k| indicates the degree of transformation.
Review Questions
How does a vertical stretch affect the amplitude of a sinusoidal function?
A vertical stretch increases the amplitude of a sinusoidal function by multiplying its output values by a factor greater than 1. For example, if we start with an amplitude of 1 and apply a vertical stretch with a factor of 3, the new amplitude becomes 3. This means that the peaks rise higher above the midline and the troughs fall lower below it, enhancing the overall height variation in the graph.
Discuss how you would identify a vertical compression in a given sinusoidal equation and its implications on graph behavior.
To identify a vertical compression in a sinusoidal equation, look for a multiplier between 0 and 1 applied to the sine or cosine function, such as in `y = 0.5 * sin(x)`. This compression reduces both the height of peaks and depth of troughs compared to the parent function. As a result, the graph appears flatter, making it less pronounced in its oscillations and impacting how it interacts with other functions when graphed together.
Evaluate how understanding vertical stretches and compressions can enhance your ability to model real-world phenomena using sinusoidal functions.
Understanding vertical stretches and compressions is crucial for modeling real-world phenomena because many periodic behaviors can be represented using sinusoidal functions. For example, in engineering applications like sound waves or light intensity variations, knowing how to adjust amplitude through vertical transformations allows for accurate simulations. By manipulating these parameters effectively, one can tailor models to fit specific data trends or patterns observed in nature or technology, ultimately leading to better predictions and analyses.
Related terms
Amplitude: The maximum distance between the midline of a sinusoidal function and its maximum or minimum value, affecting the height of the wave.
Period: The length of one complete cycle of a periodic function, determining how frequently the function repeats.
Phase Shift: A horizontal shift left or right for a sinusoidal function, which alters the starting point of the wave.