Graph Transformations

Understanding Graph Transformations

  • Graph transformations involve changing the appearance of graphs using different mathematical operations, without changing the underlying function.
  • Transformation can include translations, reflections, expansions, or contractions of the original graph.
  • A transformation that moves a graph up, down, left, or right is a translation. The form f(x) + a translates the graph up or down, while f(x ± a) translates left or right.
  • Reflections flip the graph over an axis. Any function f(x) that becomes -f(x) is reflected in the x-axis, and if x becomes -x, the graph is reflected in the y-axis.
  • A transformation that stretches or shrinks a graph is called a scale transformation. If we transform f(x) to af(x), where a>1, we get a stretch along y-axis. If 0<a<1, it’s a compression along the y-axis.
  • Transforming f(x) to f(ax) provides a stretch along x-axis with factor of 1/a. When a>1 it’s a compression towards y-axis. Else it’s a stretch away from the y-axis.

Plotting Graph Transformations

  • When translating a graph, decide the direction and magnitude of the shift, then move every point accordingly.
  • To reflect a graph, reverse the sign of either the x-coordinate (for y-axis reflection) or the y-coordinate (for x-axis reflection) of each point in the original function.
  • In a scale transformation, multiply the y-coordinate by scale factor ‘a’ in the transformation af(x) to stretch or shrink vertically, and ‘x’ by 1/a in the transformation f(ax), to stretch or shrink horizontally.
  • The order of transformations matters. For example, scaling before translating doesn’t give the same result as translating before scaling.

Multiple Transformations

  • Multiple transformations can be applied to a graph simultaneously. The final transformed graph depends on the sequence and type of transformations applied.
  • It is good practice to apply transformations individually in a set order: reflections, then stretches/shrinks, and finally translations, to help keep track of the changes.
  • It’s important when performing multiple transformations to keep track of each transformation and its effect on the graph.

Inverse Functions and Transformations

  • The plot of the inverse function, denoted as fˉ¹(x), is a reflection of the original function f(x) in the line y = x.
  • When reflecting the graph of a function in the line y = x, the x and y coordinates of each point on the function are reversed in the inverse function.
  • Plotting the inverse function helps reinforce understanding of reflections as a transformation.