Graphing Functions Using Transformations

Square Root Functions and Their Transformations

Understanding how to graph functions using transformations is a key skill in College Algebra. This example focuses on the square root function and how horizontal and vertical shifts affect its graph.

• Parent Function: The basic square root function is .

• Given Function:

Key Concepts

• Horizontal Shifts: The expression inside the square root shifts the graph right by units. For , the graph shifts right by 3 units.

• Vertical Shifts: An added or subtracted constant outside the square root, or , shifts the graph up or down by units. In this case, there is no vertical shift.

• Domain: The domain of is , so .

• Range: Since the square root function outputs only non-negative values, the range is .

Graphing Steps

1. Start with the parent graph .

2. Shift the entire graph right by 3 units to obtain .

3. Plot key points:

   • At ,

   • At ,

   • At ,

4. Draw the curve starting at and increasing slowly to the right.

Example Table of Values

Summary

• The graph of is the graph of shifted right by 3 units.

• Domain:

• Range: