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Transforming Coordinates with a, b, h, k and Absolute Value (Step by Step)

  • Writer: Tyler Buffone
    Tyler Buffone
  • Feb 12
  • 2 min read

Updated: Oct 9

Understanding function transformations is essential for mastering algebra and calculus. Rather than working with abstract function notation, this guide will show you how to apply transformations directly to given coordinate points. We'll break down horizontal and vertical shifts, stretches, reflections, and absolute value transformations step by step, helping you confidently determine the new location of any point after a series of transformations. By the end of this post, you'll be able to transform coordinates like a pro!


Abstract digital illustration inspired by the graph of the absolute value function, showing a smooth symmetrical V-shaped form with flowing gradients of blue, teal, and white.

Consider the function f(x), which has a known point (-5, -3). Suppose this function is transformed as follows:


Mathematical equation on a dark background: y = -2 |sqrt((1/3)x + 4) - 2| - 4, written in white handwritten style.

Before you do anything else, factor the inside of f(x); the result of this is as follows:


White handwritten mathematical equation y = -2 |5(1/3(x + 12)) - 2| - 4 on a dark background.

You then need to consider the transformations sequentially as follows.


  1. There is a four parameter transformation where a = 1, b = (1/3), h = (-12) and k = (-2).

  2. There is an absolute value transformation, where the coordinate mapping rule is as follows:


    (x, y) → (x, |y|)


  3. There is another four parameter transformation where a = (-2), b = 1, h = 0, and k = (-4).


Remember, the starting point is (-5, -3).


Transformation 1


Regarding the first transformation, we apply the four parameter coordinate mapping rule:


Math equation on dark background: (x, y) = (x/b + h, ay + K). Letters are in white, with "b" and "a" in blue.

The result is as follows:


Math transformations on a dark background: (-5,-3) becomes (-15-12,-3-2) and finally (-27,-5); green numbers and white text.

Transformation 2


After the completion of transformation 1, the original point (-5, -3) transformed into (-27, -5).


We then apply the absolute value transformation to this point, the result of which is as follows:


(-27, -5) → (-27, |-5|) → (-27, 5)


Transformation 3


After the completion of transformation 2, the point (-27, -5) has been transformed into (-27, 5).


We then apply the second four parameter transformation where a = -2, b = 1, h = 0, and k = -4.


The result of this is as follows:


Math equation on a dark background showing transformation of coordinates: (-27, 5) to (-27, -14). Text in white and green.

Thus, after all three transformations have been applied the original point (-5, -3) has transformed into (-27, -14).



Need help with Pre-Calculus 12 coordinate transformations? If you’re in Winnipeg and looking for a math tutor, Tutor Advance provides expert one-on-one support with step-by-step practice.



 
 
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