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!
Consider the function f(x), which has a known point (-5, -3). Suppose this function is transformed as follows:
Before you do anything else, factor the inside of f(x); the result of this is as follows:
You then need to consider the transformations sequentially as follows.
There is a four parameter transformation where a = 1, b = (1/3), h = (-12) and k = (-2).
There is an absolute value transformation, where the coordinate mapping rule is as follows:
(x, y) → (x, |y|)
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:
The result is as follows:
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:
Thus, after all three transformations have been applied the original point (-5, -3) has transformed into (-27, -14).