Why multiplying a graph by a fraction makes it larger
In short: In mathematics, intuition often fails when we move from arithmetic to geometry. While we expect multiplying by a fraction to shrink a value, doing so to a graph's equation can actually stretch it out. Understanding these transformations reveals the hidden logic of how functions behave under pressure.
In the study of graph transformations, mathematicians distinguish between vertical and horizontal changes based on where the operation is applied. Vertical transformations affect the 'outside' of the function, meaning they are applied to the entire output. For example, if you have a function y = f(x) and you multiply it by a constant 'a', the new equation becomes y = af(x). This results in a vertical stretch or squash where every y-coordinate is multiplied by that constant.
Horizontal transformations, however, are applied directly to the 'x' variables, which leads to some counter-intuitive results. When a number is placed directly with the x-variable, it behaves differently than it does on the outside. For instance, a value of 2 applied horizontally will squash the curve, while a value of 1/2 will stretch it. This inverse relationship is a core concept in coordinate geometry: numbers larger than 1 squash the graph horizontally, while numbers smaller than 1 stretch it.
To visualize these changes, one can take a graph point by point. If a curve is transformed from y = f(x) to y = 3f(x), every y-coordinate doubles or triples in size. Conversely, if the transformation is y = 1/3 f(x), the curve is squashed vertically by a third, meaning a y-coordinate of 3 would move to 1. Mastering these rules allows for the precise manipulation of curves, shifting, reflecting, and scaling shapes across the coordinate plane.
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Sources
- Summary by Intellect · source: Graph Transformations | Graphs | Maths | FuseSchool
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