Transformations of Radical Functions - Algebra-2

1. Fundamental Concepts

Definition: Radical functions are functions that involve roots, such as square roots or cube roots. For example, $$f(x) = \sqrt{x}$$ is a radical function.
Transformations: Transformations of radical functions include shifts, reflections, and stretches/shrinks. These transformations can be applied to the basic radical function to create new functions.
Parent Function: The parent function for radical functions is typically $$f(x) = \sqrt{x}$$ .

2. Key Concepts

The general form is , where each parameter functions as follows:

a: Controls vertical reflection and stretch/compression. If (reflection over the x-axis); if (vertical stretch); if (vertical compression).
h: Controls horizontal translation. If (shift right by h units); if (shift left by units) (the sign of h is opposite to the translation direction).
k: Controls vertical translation. If (shift up by k units); if (shift down by units).

3. Examples

Easy: Identify Transformations from the Function Expression

Problem: For the radical function , (1) state its parent function; (2) describe all transformations applied to the parent function to obtain the given function.
Solution:

The parent function is the basic square root function ;
Rewrite the given function to match the general form , which becomes . Thus, the transformations are: a horizontal shift of 2 units to the left and a vertical shift of 1 unit down.

Medium: Graph the Transformed Function and Verify Key Points

Problem: Graph the transformed radical function by using the transformations of its parent function . Label at least 3 key points on the graph and show how these key points are derived from the key points of the parent function.
Solution:

Determine the transformations: Vertical stretch by a factor of 2, horizontal shift of 1 unit to the right, and vertical shift of 3 units up;
Calculate transformed key points using the rule $(x_p + h, a \cdot y_p + k)$ :
- Parent key point → Transformed point: ;
- Parent key point → Transformed point: ;
- Parent key point → Transformed point: ;
Plot the transformed points , , on the coordinate plane and connect them with a smooth curve. The graph starts at and rises steeply (due to vertical stretch).

Hard: Analyze Complex Transformations and Verify a Point on the Graph

Problem: (1) For the radical function , describe all transformations applied to its parent function ; (2) Verify whether the point lies on the graph of the function .
Solution:

Analyze transformations: Rewrite the function as . The transformations are: reflection over the x-axis, vertical stretch by a factor of 3, horizontal shift of 4 units to the left, and vertical shift of 2 units down;
Verify the point: Substitute into the function: . Since the calculated y-value is equal to the y-coordinate of the point , the point lies on the graph.

4. Problem-Solving Techniques

Identify Transformations from Expressions: First find a (to determine reflection and stretch/compression), then find h (to determine horizontal translation), and finally find k (to determine vertical translation).
Graph Using Key Points: List key points of the parent function → Calculate new points using the rule → Plot the new points and connect them with a smooth curve.
Verify Transformation Order: Reverse the transformations in reverse order (undo translation first, then stretch/compression/reflection) to check if the resulting points match the key points of the parent function.
Rewrite Complex Equations: Factor out the coefficient of x inside the radical (e.g., rewrite as ) to clearly identify a, h, and k.