Vertical Stretch

Definition

A vertical stretch occurs when a function is multiplied by a factor greater than one, which causes the graph of the function to stretch away from the horizontal axis. This transformation affects the amplitude of periodic functions, such as sine and cosine, making them taller without changing their frequency or horizontal stretch. The concept of vertical stretch is crucial in understanding how these functions behave and interact with their properties like amplitude and period.

5 Must Know Facts For Your Next Test

1. A vertical stretch is represented mathematically by multiplying a function, such as $$y = f(x)$$, by a constant greater than one, like $$k imes f(x)$$ where $$k > 1$$.
2. Vertical stretches increase the amplitude of sine and cosine functions, leading to higher peaks and lower troughs while maintaining their period.
3. When a function undergoes a vertical stretch, the distance between consecutive maximum and minimum points increases, but the distance between points along the x-axis remains unchanged.
4. The effect of a vertical stretch is visually evident in the graph; for example, applying a vertical stretch by a factor of 3 would triple the height of all points on the graph without altering their x-coordinates.
5. Vertical stretches can be combined with other transformations, such as horizontal shifts and reflections, allowing for complex adjustments to the shape of periodic functions.

Review Questions

• How does a vertical stretch impact the amplitude of periodic functions?
  • A vertical stretch directly increases the amplitude of periodic functions. For instance, if you take the function $$y = ext{sin}(x)$$ and apply a vertical stretch by a factor of 2 to get $$y = 2 imes ext{sin}(x)$$, the amplitude changes from 1 to 2. This means that the peaks of the sine wave will reach twice as high and the troughs will descend twice as low compared to the original function.
• In what ways do vertical stretches differ from horizontal stretches in their effects on periodic functions?
  • Vertical stretches and horizontal stretches affect different aspects of periodic functions. A vertical stretch increases amplitude without changing period; for example, multiplying by 3 makes the peaks three times higher. In contrast, a horizontal stretch alters the frequency and period. For instance, replacing $$x$$ with $$kx$$ (where $$0 < k < 1$$) in $$y = ext{sin}(x)$$ compresses the graph horizontally, effectively increasing how quickly it cycles through its periods.
• Evaluate how combining vertical stretches with other transformations can lead to complex shapes in periodic functions.
  • Combining vertical stretches with other transformations can create intricate shapes in periodic functions that exhibit unique behaviors. For example, if we take $$y = ext{sin}(x)$$ and apply a vertical stretch by a factor of 2 followed by a horizontal shift to the right by 3 units, we transform it into $$y = 2 imes ext{sin}(x - 3)$$. This results in a sine wave that not only has doubled amplitude but also starts its cycle later on the x-axis. This layering effect allows us to manipulate periodic graphs in creative ways to achieve specific results.

Related terms

Amplitude: The maximum distance from the midline of a periodic function to its peak or trough, representing the height of the wave.

Period: The length of one complete cycle of a periodic function, which determines how often the wave repeats itself.

Horizontal Stretch: A transformation that compresses or stretches a function along the horizontal axis by multiplying the input variable by a factor.