Stretch

Definition

Stretch refers to the transformation of a graph that alters its vertical or horizontal dimensions, making it wider or narrower without changing its overall shape. This transformation impacts the behavior of the function, particularly how steep or flat it appears, which is crucial for understanding the properties of the graph and the function it represents.

5 Must Know Facts For Your Next Test

1. A vertical stretch occurs when a function is multiplied by a factor greater than 1, causing the graph to become steeper.
2. Conversely, a vertical compression happens when a function is multiplied by a factor between 0 and 1, making the graph flatter.
3. Horizontal stretches and compressions are achieved by multiplying the x-values by a constant; a factor greater than 1 results in a stretch, while a factor less than 1 leads to compression.
4. The overall shape of a graph is preserved during stretching or compressing, meaning that key points like intercepts remain unchanged.
5. Understanding stretches and compressions helps predict how transformations affect graphs of different functions, allowing for better analysis of their behavior.

Review Questions

• How does applying a vertical stretch affect the key features of a graph?
  • Applying a vertical stretch affects the steepness of the graph, making it taller without changing its x-intercepts. For example, if a function is multiplied by a factor greater than 1, every y-value increases, leading to an increased slope. This can make it easier to identify how quickly the function grows, impacting decisions based on its behavior.
• Compare and contrast vertical stretching and horizontal stretching in terms of their impact on a function's graph.
  • Vertical stretching affects the y-values of the function by making the graph steeper or flatter depending on whether it’s stretched or compressed. In contrast, horizontal stretching changes how wide or narrow the graph appears by altering the x-values. While vertical transformations can impact how quickly a function increases or decreases, horizontal transformations mainly affect the rate at which these changes occur over intervals.
• Evaluate how understanding stretching and compressing can aid in predicting and analyzing real-world applications represented by functions.
  • Understanding stretching and compressing allows us to predict how real-world phenomena change in response to various factors. For instance, in economics, if we model supply and demand with functions, knowing how to apply these transformations can help visualize shifts in market conditions. By adjusting parameters to reflect real situations, we can analyze trends more effectively, leading to better decision-making and predictions based on those models.

Related terms

Compression: Compression is the opposite of stretching; it reduces the size of the graph either vertically or horizontally, making it narrower.

Vertical Shift: A vertical shift is a transformation that moves the graph up or down without altering its shape, affecting the y-values of the function.

Horizontal Shift: A horizontal shift moves the graph left or right, changing the x-values of the function while maintaining its original shape.