Stretching

Definition

Stretching refers to the transformation of a function that alters its vertical or horizontal dimensions, effectively expanding or compressing its shape. This change in size affects the function's output values and can significantly impact its graphical representation, influencing features such as amplitude, period, and symmetry.

5 Must Know Facts For Your Next Test

1. A vertical stretch occurs when you multiply the function by a constant greater than 1, while a vertical compression happens when the constant is between 0 and 1.
2. When stretching horizontally, you multiply the input variable by a constant; for instance, if you replace 'x' with 'kx' where 'k < 1', it stretches the graph horizontally.
3. The direction of stretching (vertical vs. horizontal) can be determined by whether you are altering the 'y' values or 'x' values of the function.
4. Stretching affects the graph's key features such as intercepts, maxima, minima, and asymptotes, depending on how it is applied.
5. Understanding stretching is crucial for analyzing transformations because it can significantly change how a function behaves and interacts with other functions.

Review Questions

• How does stretching a function vertically differ from stretching it horizontally in terms of graphical representation?
  • Stretching a function vertically involves multiplying the output values by a constant factor greater than one, which makes the graph taller. In contrast, stretching horizontally involves modifying the input values; when you multiply them by a factor less than one, it makes the graph wider. This results in different changes to key features such as intercepts and slope, demonstrating that vertical and horizontal stretches have unique effects on how the function appears.
• Analyze how changing the stretching factor impacts the amplitude and period of trigonometric functions like sine and cosine.
  • For trigonometric functions like sine and cosine, a vertical stretch changes the amplitude—the maximum height from the midline—by multiplying the function by a factor greater than one. Conversely, a horizontal stretch affects the period, which is how far along the x-axis one complete cycle occurs. For example, if you stretch horizontally by replacing 'x' with 'kx', where 'k < 1', this increases the length of each cycle, resulting in fewer cycles over the same interval.
• Evaluate how understanding stretching can help in modeling real-world phenomena through functions.
  • Grasping stretching allows for better modeling of real-world situations where changes in size or scale are relevant. For instance, when modeling population growth or waves in physics, knowing how to apply vertical or horizontal stretches helps depict realistic scenarios. By manipulating functions accurately, we can predict behaviors and trends effectively. This understanding also aids in interpreting data more accurately and ensuring that models align with observed outcomes.

Related terms

Scaling:

Scaling is the process of resizing a function by multiplying the output values (y-values) or input values (x-values) by a constant factor.

Vertical Stretch: A vertical stretch occurs when the output of a function is multiplied by a factor greater than one, making the graph taller and affecting its steepness.

Horizontal Stretch: A horizontal stretch happens when the input of a function is multiplied by a factor less than one, which makes the graph wider and alters its period.