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.
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:
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:
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:
The Absolute Value Function:
The Radical Function:
