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Function Transformations Explained (a, b, h, k Parameters)

  • Writer: Tyler Buffone
    Tyler Buffone
  • Feb 7
  • 2 min read

Updated: Oct 9

Understanding transformations of functions is a crucial skill for students tackling advanced math concepts. Whether you're shifting, stretching, compressing, or reflecting graphs, mastering these transformations helps you visualize and analyze functions with greater clarity.


Abstract illustration showing smooth, colorful shapes and flowing arrows in orange, green, blue, and cream tones, symbolizing movement, direction, and transformation.

Let f(x) represent any particular function. We then define four transformation parameters: a, b, h, and k, such that:


  • "a" controls the vertical reflection in the x-axis and the vertical stretch and compression.

  • "b" controls the horizontal reflection in the y-axis and the horizontal stretch and compression.

  • "h" controls the horizontal shift - moves the function left and right.

  • "k" controls the vertical shift - moves the function up and down.


With these parameters, we can transform the function f(x) and represent the new, transformed function as follows:


Mathematical equation on a dark background, featuring variables and constants in white and blue text.

We can then say that any coordinate point (x, y) defined on the graph of the original function f(x) will transform into a coordinate point defined on the graph of the new, transformed function. This algebraic transformation occurs as follows:


Math equation on a black background: (x, y) = (x/b + h, ay + k). Variables and operations are in white; constants b and a are blue.

Remark 1: we can also define these four parameters and apply them to any relation, not just to any function.

Remark 2: if any of the four transformation parameters are not visible in the equation of the transformed function, we may assume that:


  • a = 1

  • b = 1

  • h = 0

  • k = 0


When describing these transformations verbally, the following implications hold:

  • If |a| > 1, then there is a vertical stretch by a factor of |a|.

  • If 0 < |a| < 1, then there is a vertical compression by a factor of |a|.

  • If a < 0, then there is a vertical reflection in the x-axis.


  • If |b| > 1, then there is a horizontal compression by a factor of 1 / |b|.

  • If 0 < |b| < 1, then there is a horizontal stretch by a factor of 1 / |b|.

  • If b < 0, then there is a horizontal reflection in the y-axis.


  • If h > 0, then there is a shift to the right by h units.

  • If h < 0, then there is a shift to the left by h units.


  • If k > 0, then there is a shift up by k units.

  • If k < 0, then there is a shift down by k units.


Every single type of relation has its own particular transformation formula. We will provide a sample of common parent functions and their corresponding transformations formulas below.


The Quadratic Function:


Math equation on dark background: y = x² transforming into y = a(b(x-h))² + k. Text is white with some letters in blue.

The Absolute Value Function:


Mathematical equations on a dark background showing transformations of absolute value functions with key variables in white and blue.

The Radical Function:


Math equations on dark background show function transformation: y = √x to y = a√b(x-h) + k. Blue highlights accent variables.


Need more help with Transformations? If you’re in Winnipeg and looking for a tutor, Tutor Advance provides expert one-on-one support!



 
 
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