Graph Transformations

• Graph Transformations refer to the shifting, scaling, reflection or rotation of the graph of a function.
• Transformations are typically represented by equations and involve changes to the x and y-values.
• There are four foundational types of transformations: Translations, Dilations, Reflections, and Rotations.
• In a Translation transformation, a constant gets added or subtracted to/from the function’s input (for a horizontal shift) or output (for a vertical shift). For instance, y = f(x - a) shifts the graph of y = f(x) a units to the right.
• A Reflection transformation flips the graph of a function over a line, often the x or y-axis. For instance, y = -f(x) flips the graph of y = f(x) over the x-axis.
• A Dilation or Scaling transformation stretches or compresses the graph of a function by a certain factor. For example, if y = af(x), then the graph y = f(x) stretches vertically by a factor of a.
• In rotation transformations, the graph is spun around a point, typically the origin. However, this is less common at this level of mathematics and will not be covered in detail here.
• Achieving mastery in Graph Transformations involves not only understanding these principles, but also the ability to perform multiple transformations to one function and recognizing the effect of multiple transformations.
• Remember that the order of transformations matter. This is particularly crucial when considering transformations involving both translations and reflections.
• Practice is essential in mastering graph transformations. Use various functions and transform them in different ways, predicting the outcome before sketching the graph.
• Previous knowledge of the Cartesian coordinate system, the behaviour of different function types (like linear, quadratic, etc.) aids the understanding and application process of graph transformations.
• Understanding graph transformations will be pivotal when tackling calculus and further mathematical analysis.