Shrinking

Definition

Shrinking refers to the transformation of a function that reduces its output values and consequently compresses the graph vertically. This transformation is achieved by multiplying the function by a factor between 0 and 1, leading to a decrease in the height of the graph without altering its horizontal position. Understanding shrinking helps visualize how functions can be adjusted in size, making it crucial for grasping broader concepts of function transformations.

5 Must Know Facts For Your Next Test

1. When a function $f(x)$ is shrunk by a factor of $k$ (where $0 < k < 1$), the new function becomes $g(x) = k imes f(x)$.
2. Shrinking affects all points on the graph equally, meaning that the shape remains the same but with reduced height.
3. If you apply multiple transformations to a function, it's often useful to determine the order in which they are applied, as this can affect the final result.
4. The y-values of points on the graph will be multiplied by the shrinking factor, which directly impacts their distance from the x-axis.
5. Graphically, a shrink makes peaks lower and valleys shallower but doesn't change where those points occur along the x-axis.

Review Questions

• How does shrinking a function impact its overall appearance and what mathematical notation represents this transformation?
  • Shrinking a function impacts its overall appearance by reducing the height of its peaks and making valleys shallower, while keeping the x-coordinates unchanged. Mathematically, this transformation can be represented as $g(x) = k imes f(x)$, where $0 < k < 1$. This shows that each output value of the original function $f(x)$ is multiplied by a factor $k$, resulting in a vertically compressed graph.
• In what ways can shrinking be combined with other transformations, and what should one keep in mind regarding their order?
  • Shrinking can be combined with other transformations like shifting or stretching. When combining these transformations, it's essential to consider their order because transformations do not always commute; for example, shrinking before shifting can yield different results than shifting before shrinking. Understanding how each transformation interacts helps in accurately predicting the final appearance of the graph.
• Evaluate how understanding shrinking transforms your ability to predict behavior in more complex functions and their graphs.
  • Understanding shrinking enhances your ability to predict behavior in more complex functions by providing insight into how changes in scale affect outputs. For instance, if you know how shrinking modifies basic functions, you can apply that knowledge when analyzing combinations of functions or when using them within real-world models. This predictive capability is crucial for tackling complex mathematical problems where transformations play an essential role in graphing or interpreting data.

Related terms

Scaling:

Scaling involves adjusting the size of a graph by either stretching or compressing it, depending on whether the scaling factor is greater than or less than one.

Vertical Stretch: A vertical stretch occurs when a function is multiplied by a factor greater than 1, increasing its output values and making the graph taller.

Transformation: Transformation is a general term for any change made to a function that alters its appearance on a graph, including shifts, stretches, shrinks, and reflections.