Understanding and Writing the Equation of a Transformed Square Root Function

Introduction

In the realm of mathematical functions, particularly those involving square roots, transformations can be a powerful tool for understanding and manipulating graphs. A square root function can be altered through various types of transformations, including vertical and horizontal compression, reflection, and translation. This article will guide you through the process of writing an equation for a transformed square root function, focusing on a specific example: a function that has been compressed vertically, horizontally, reflected in the y-axis, and shifted vertically.

Transformation of Square Root Functions

When working with square root functions, such as (y sqrt{x}), it's important to understand how different transformations affect the graph. Here are the transformations we'll cover:

Vertical compression Horizontal compression Reflection in the y-axis Vertical shift

Vertical Compression

A vertical compression by a factor of (a) (where (0 a 1)) means that the function's graph is squeezed vertically, making the function values smaller. For a basic square root function, this can be represented as (y frac{1}{a}sqrt{x}) . In the given example, the factor is ( frac{1}{13} ).

Horizontal Compression

A horizontal compression by a factor of (b) (where (0 b 1)) means that the function's graph is squeezed horizontally, making the input values smaller. For a basic square root function, this can be represented as (y sqrt{abx}) . In the given example, the factor is ( frac{1}{9} ).

Reflection in the y-axis

Reflecting a function in the y-axis means changing the sign of the input variable. For a basic square root function, this can be represented as (y sqrt{-ax}) . In the given example, the factor is (9).

Vertical Shift

Shifting a function vertically involves moving the entire graph up or down. For a basic square root function, this can be represented as (y sqrt{ax} c) . In the given example, the shift is 7 units down, represented as (y sqrt{ax} - 7) .

Combining Transformations

Now that we understand the individual transformations, let's combine them to write the transformed square root function. In the given example, the function has been:

Compressed vertically by a factor of ( frac{1}{13} ) Compressed horizontally by a factor of ( frac{1}{9} ) Reflected in the y-axis Shifted vertically 7 units down

Therefore, the combined transformation can be represented as:

.. (y frac{1}{13}sqrt{-9x} - 7) ..

Note that this transformation changes the domain of the function to non-positive values because of the reflection and the change in the input variable.

Conclusion

Understanding the transformation of square root functions is a valuable skill in algebra and calculus, with applications in various fields such as physics and engineering. By mastering the individual components of function transformations and combining them, you can accurately model and graph complex functions.

Key Points to Remember:

Vertical compression: (y frac{1}{a}sqrt{x}) Horizontal compression: (y sqrt{abx}) Reflection in the y-axis: (y sqrt{-ax}) Vertical shift: (y sqrt{ax} c)

With these tools, you can write the equation of a square root function that has been transformed in various ways.