Horizontal Stretch

Definition

A horizontal stretch is a transformation that alters the width of a graph by expanding it away from the y-axis. This transformation occurs when the input variable (usually x) is multiplied by a factor less than 1, resulting in the graph being stretched horizontally. In the context of secant and cosecant functions, this stretching can significantly impact the appearance of the graphs, including their amplitude and period.

5 Must Know Facts For Your Next Test

1. In a horizontal stretch, if the factor is greater than 1, it compresses the graph towards the y-axis, while a factor between 0 and 1 expands it.
2. For secant and cosecant functions, a horizontal stretch affects their asymptotes and can change how frequently these functions oscillate.
3. The equation for a horizontal stretch can be expressed as $$y = f(kx)$$ where $$k$$ is the stretching factor; if $$0 < k < 1$$, it's a stretch, and if $$k > 1$$, it's a compression.
4. Horizontal stretches do not affect the amplitude of secant or cosecant functions but do alter their period, leading to wider or narrower cycles.
5. When applying a horizontal stretch to secant or cosecant functions, it's essential to adjust your understanding of how quickly the function approaches its asymptotes.

Review Questions

• How does a horizontal stretch affect the graphs of secant and cosecant functions in terms of their period?
  • A horizontal stretch modifies the period of secant and cosecant functions by expanding or compressing the width of their cycles. Specifically, when you apply a horizontal stretch with a factor between 0 and 1, it increases the distance between successive asymptotes, effectively lengthening the period. Conversely, if a factor greater than 1 is applied, it will decrease the distance between these asymptotes, resulting in a shorter period.
• Discuss how horizontal stretches interact with vertical stretches when applied to secant and cosecant functions.
  • When both horizontal and vertical stretches are applied to secant and cosecant functions, they work together to change the graph's appearance in multiple ways. A horizontal stretch affects the width and frequency of cycles, while a vertical stretch impacts their height. Understanding these transformations together allows you to predict how changes will influence features like amplitude and periodicity, leading to more complex graphs that require careful interpretation.
• Evaluate the effects of different stretching factors on the properties of secant and cosecant functions, considering practical examples.
  • Evaluating different stretching factors reveals how they distinctly alter secant and cosecant functions. For instance, applying a factor of 0.5 results in wider oscillations that take longer to complete each cycle, making it crucial for applications requiring specific intervals such as modeling tides or sound waves. In contrast, using a factor of 2 would create more rapid oscillations, which could be relevant in scenarios involving higher frequencies. Recognizing these variations allows for better predictions of real-world behavior based on mathematical models.

Related terms

Amplitude:

The amplitude of a function refers to the maximum distance from the midline of the graph to its peak or trough, indicating how 'tall' or 'short' the waves appear.

Period:

The period is the length of one complete cycle of a periodic function, which defines how often the graph repeats itself.

Vertical Stretch:

A vertical stretch occurs when a graph is stretched away from the x-axis by multiplying the output variable (usually y) by a factor greater than 1, making it taller without altering its horizontal properties.