Scaling

Definition

Scaling refers to the transformation of a function's graph by stretching or compressing it vertically or horizontally. This process affects the function's output values or input values, which can result in a change in the function's steepness and overall shape, thus altering how the function behaves on a graph.

5 Must Know Facts For Your Next Test

1. Scaling can be vertical or horizontal, impacting either the output values or input values of a function respectively.
2. A vertical scaling factor greater than one stretches the graph away from the x-axis, while a factor between zero and one compresses it towards the x-axis.
3. Horizontal scaling is influenced by the reciprocal of the scaling factor applied to the input variable, with factors greater than one compressing the graph and those between zero and one stretching it.
4. The effects of scaling can change not only the appearance of the graph but also important features like intercepts, maximums, and minimums.
5. Combining scaling with other transformations such as translation or reflection can lead to more complex alterations of a function's graph.

Review Questions

• How does vertical scaling affect the steepness and appearance of a function's graph?
  • Vertical scaling affects the steepness of a function's graph by multiplying the output values. If the scaling factor is greater than one, it stretches the graph vertically, making it steeper and taller. Conversely, if the scaling factor is between zero and one, it compresses the graph towards the x-axis, making it flatter. This change in steepness can significantly impact how we understand the behavior of the function at different points.
• In what way does horizontal compression differ from vertical stretching when transforming a function?
  • Horizontal compression differs from vertical stretching in that it alters how input values are processed rather than output values. When applying horizontal compression, we multiply the input variable by a factor greater than one, which makes the graph narrower and leads to quicker changes in output for small changes in input. In contrast, vertical stretching involves multiplying the output values directly, affecting how high or low points on the graph reach relative to their original positions.
• Evaluate how combining scaling transformations with reflections might influence the overall graph of a function.
  • Combining scaling transformations with reflections can create intricate changes in a function's graph. For example, if we first apply a vertical stretch and then reflect it over the x-axis, we can turn peaks into troughs while maintaining their new heights. This combination alters both the shape and position of key features on the graph. Analyzing this interaction helps to deepen our understanding of how functions behave under various transformations and allows us to predict their outcomes more effectively.

Related terms

Vertical Stretch: A transformation that multiplies the output of a function by a factor greater than one, making the graph taller and steeper.

Horizontal Compression: A transformation that multiplies the input of a function by a factor greater than one, resulting in a narrower graph.

Reflection: A transformation that flips the graph of a function over a specified axis, often combined with scaling to create more complex transformations.