Stretch

Definition

In the context of transformations of trigonometric graphs, a stretch refers to the alteration of a graph's shape in a vertical or horizontal direction, affecting its amplitude or period. This transformation changes the scale of the graph without altering its overall direction or basic form, allowing for a more pronounced or diminished visual representation of the function's behavior. Stretches help in modifying how the graph interacts with the axes, influencing key features like peaks, troughs, and intercepts.

5 Must Know Facts For Your Next Test

1. Vertical stretches are achieved by multiplying the function by a factor greater than 1, increasing the height of peaks and depth of troughs.
2. Horizontal stretches occur when the input variable is multiplied by a factor less than 1, resulting in a wider graph that takes longer to complete one cycle.
3. Stretches can be expressed in mathematical terms; for example, transforming the function $$f(x)$$ to $$af(x)$$ (vertical stretch) or $$f(bx)$$ (horizontal stretch) where $$a$$ and $$b$$ are positive constants.
4. Combining stretches with other transformations like shifts and reflections can create complex behaviors in trigonometric graphs.
5. Understanding how stretches interact with other transformations is key to predicting the resulting shape of a trigonometric function's graph.

Review Questions

• How does a vertical stretch affect the key characteristics of a trigonometric graph?
  • A vertical stretch increases the amplitude of a trigonometric graph by multiplying the function by a factor greater than 1. This change results in higher peaks and deeper troughs while keeping the periodic nature of the function intact. It enhances the visual representation of how far the graph rises or falls from its midline, making oscillations more pronounced.
• Compare and contrast horizontal and vertical stretches in terms of their impact on trigonometric graphs.
  • Horizontal stretches alter the period of a trigonometric function by multiplying the input variable by a factor less than 1, leading to wider oscillations and longer cycles. In contrast, vertical stretches change the amplitude without affecting the period, resulting in taller peaks but maintaining the same cycle length. Both types of stretches modify how the graph behaves but do so in different ways that affect specific features like height and width.
• Evaluate how combining vertical and horizontal stretches can transform a standard sine function into a new trigonometric graph, and what implications this has for real-world applications.
  • Combining both vertical and horizontal stretches can dramatically alter a standard sine function, changing its amplitude and period simultaneously. For instance, if you stretch vertically by a factor of 2 and horizontally by a factor of 0.5, you end up with a sine wave that reaches twice as high and completes its cycles faster. This manipulation is crucial in real-world applications such as sound waves or light waves where amplitude affects volume or brightness, and period influences frequency.

Related terms

Amplitude:

The maximum distance a wave reaches from its central position, which is affected by vertical stretches or compressions.

Period:

The distance between successive points of equal value on a graph, usually influenced by horizontal stretches or compressions.

Compression:

A transformation that reduces the distance between points on a graph, often making it appear more narrow compared to its original shape.