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📈ap pre-calculus review

key term - Horizontal Stretch/Compression

Citation:

Definition

Horizontal stretch and compression refer to the transformations that affect the width of a function's graph, specifically sinusoidal functions like sine and cosine. A horizontal stretch occurs when the graph is stretched away from the y-axis, making it wider, while a horizontal compression occurs when the graph is compressed toward the y-axis, making it narrower. These transformations are essential for understanding how sinusoidal functions can be manipulated to fit different scenarios or data.

5 Must Know Facts For Your Next Test

  1. A horizontal stretch is represented by multiplying the x-value by a factor less than 1, while a horizontal compression uses a factor greater than 1.
  2. When you apply a horizontal stretch or compression, it directly affects the period of the sinusoidal function, with stretches increasing the period and compressions decreasing it.
  3. The transformation affects how quickly the function completes its cycles; a larger value for horizontal compression means more cycles within a given interval.
  4. Horizontal stretches and compressions keep the amplitude and midline of the function unchanged, focusing solely on modifying width.
  5. Visualizing these transformations can help in graphing sinusoidal functions accurately, especially when dealing with real-world applications.

Review Questions

  • How does a horizontal compression affect the period of a sinusoidal function?
    • A horizontal compression reduces the width of the graph, causing it to complete its cycles more quickly. This means that if you apply a compression factor greater than 1, the period of the sinusoidal function decreases, resulting in more cycles occurring within any given interval. Therefore, understanding this transformation is crucial for predicting how changes will affect the behavior of a sinusoidal graph.
  • Describe how horizontal stretches and compressions maintain other characteristics of a sinusoidal function.
    • Horizontal stretches and compressions specifically target the width of the sinusoidal graph without altering its amplitude or midline. This means that even as you manipulate how wide or narrow the graph appears through these transformations, the height of its peaks (amplitude) remains constant. Thus, while you change how rapidly or slowly it oscillates through x-values, core features like height and center point stay intact.
  • Evaluate how combining multiple transformations, including horizontal stretch/compression, alters a sinusoidal function's overall behavior.
    • Combining multiple transformations creates a complex interaction between changes in width, height, and position on the graph. For instance, if you apply both a horizontal compression and a vertical stretch simultaneously, you'll see not only an increase in frequency due to compression but also an increase in amplitude. This layered effect can significantly alter how the function behaves across its domain, making it crucial to understand each transformation's individual impact before predicting the resultant behavior.