Reflection Across Y-Axis Exploring Transformations Of F(x) = (1/6)(2/5)^x

Introduction

In the fascinating world of mathematics, functions play a pivotal role in describing relationships between variables. Among the many transformations that functions can undergo, reflection is a fundamental concept that alters the graph of a function in a specific way. In this article, we delve into the reflection of a function across the y-axis, exploring how this transformation affects the function's equation and its corresponding points. We will specifically analyze the function f(x) = (1/6)(2/5)^x and its reflection, g(x), across the y-axis. Our goal is to identify an ordered pair that lies on the graph of g(x), providing a concrete example of how the reflection transformation impacts the coordinates of points on the function.

The concept of function reflection is crucial in various fields, including physics, engineering, and computer graphics. Understanding how functions transform under reflection allows us to model symmetrical phenomena, analyze wave behavior, and manipulate images and objects in digital environments. By grasping the underlying principles of reflections, we can gain a deeper appreciation for the power and versatility of mathematical functions.

Understanding Reflections Across the Y-Axis

A reflection across the y-axis is a transformation that mirrors a function's graph over the vertical y-axis. This transformation essentially flips the graph horizontally, swapping the x-coordinates of points while preserving their y-coordinates. Mathematically, reflecting a function f(x) across the y-axis results in a new function, g(x), defined as g(x) = f(-x). This means that to obtain the reflected function, we simply replace every instance of x in the original function's equation with -x.

The impact of this transformation on the graph is visually significant. Imagine the y-axis as a mirror; the reflected graph appears as the mirror image of the original graph. Points on the original graph that are to the right of the y-axis will be reflected to the left, and vice versa. However, points that lie directly on the y-axis will remain unchanged as they are their own reflections.

To illustrate this concept, consider a simple point (a, b) on the graph of f(x). After reflection across the y-axis, this point will be transformed to (-a, b). Notice that the x-coordinate changes sign, while the y-coordinate remains the same. This pattern holds true for all points on the graph, resulting in a complete horizontal flip of the function.

Analyzing the Function f(x) = (1/6)(2/5)^x

The function f(x) = (1/6)(2/5)^x is an exponential function, characterized by its variable x appearing in the exponent. Exponential functions are known for their rapid growth or decay, depending on the base of the exponent. In this case, the base is 2/5, which is a fraction between 0 and 1. This indicates that f(x) is a decaying exponential function, meaning that its values decrease as x increases.

To understand the behavior of f(x), let's examine some key characteristics. The coefficient 1/6 in front of the exponential term affects the vertical scaling of the function. It compresses the graph vertically, making the values of f(x) smaller compared to the function (2/5)^x. As x approaches positive infinity, f(x) approaches 0, since any fraction between 0 and 1 raised to a large power becomes very small. Conversely, as x approaches negative infinity, f(x) increases without bound.

The graph of f(x) exhibits a typical exponential decay curve. It starts with relatively large values for negative x, gradually decreasing as x increases and approaching the x-axis (the line y = 0) as an asymptote. The y-intercept of the function, where the graph intersects the y-axis, is found by setting x = 0, which gives f(0) = (1/6)(2/5)^0 = 1/6. This point (0, 1/6) is a crucial reference point for understanding the function's behavior.

Determining the Reflected Function g(x)

To find the function g(x), which is the reflection of f(x) = (1/6)(2/5)^x across the y-axis, we apply the transformation rule g(x) = f(-x). This involves replacing every x in the equation of f(x) with -x. Thus, we have:

g(x) = f(-x) = (1/6)(2/5)^(-x)

To simplify this expression, we can use the property of exponents that states a^(-b) = 1/(a^b). Applying this property to the term (2/5)^(-x), we get:

(2/5)^(-x) = 1/((2/5)^x) = (5/2)^x

Therefore, the equation for the reflected function g(x) becomes:

g(x) = (1/6)(5/2)^x

This equation represents an exponential growth function. Notice that the base of the exponent is now 5/2, which is greater than 1. This means that g(x) will increase as x increases, in contrast to f(x), which decreased as x increased. The reflection across the y-axis has effectively transformed a decaying exponential function into a growing exponential function.

The graph of g(x) is a mirror image of the graph of f(x) with respect to the y-axis. It starts with small values for negative x, increasing rapidly as x increases. The y-intercept of g(x) is also (0, 1/6), which is the same as the y-intercept of f(x). This is because the y-axis is the axis of reflection, and points on the axis remain unchanged during the transformation.

Identifying an Ordered Pair on g(x)

Now that we have the equation for the reflected function g(x) = (1/6)(5/2)^x, we can determine which of the given ordered pairs lies on its graph. To do this, we substitute the x-coordinate of each ordered pair into the equation for g(x) and check if the resulting y-coordinate matches the y-coordinate of the given pair.

Let's consider the ordered pairs provided:

A. (-3, 4/375) B. (-2, 25/24) C. (2, 25/24)

For option A, we substitute x = -3 into g(x):

g(-3) = (1/6)(5/2)^(-3) = (1/6)(2/5)^3 = (1/6)(8/125) = 4/375

The calculated y-coordinate matches the given y-coordinate, so the ordered pair (-3, 4/375) lies on the graph of g(x).

For option B, we substitute x = -2 into g(x):

g(-2) = (1/6)(5/2)^(-2) = (1/6)(2/5)^2 = (1/6)(4/25) = 2/75

The calculated y-coordinate does not match the given y-coordinate, so the ordered pair (-2, 25/24) does not lie on the graph of g(x).

For option C, we substitute x = 2 into g(x):

g(2) = (1/6)(5/2)^2 = (1/6)(25/4) = 25/24

The calculated y-coordinate matches the given y-coordinate, so the ordered pair (2, 25/24) lies on the graph of g(x).

Therefore, both ordered pairs (-3, 4/375) and (2, 25/24) lie on the graph of g(x).

Conclusion

In this article, we explored the concept of reflection across the y-axis and its impact on the function f(x) = (1/6)(2/5)^x. By applying the transformation rule g(x) = f(-x), we determined the equation for the reflected function g(x) = (1/6)(5/2)^x. We observed that the reflection across the y-axis transformed a decaying exponential function into a growing exponential function.

Furthermore, we identified ordered pairs that lie on the graph of g(x) by substituting the x-coordinates into the equation and verifying the resulting y-coordinates. We found that both (-3, 4/375) and (2, 25/24) are points on the graph of g(x).

This exploration highlights the importance of understanding function transformations in mathematics. Reflections, along with other transformations such as translations, stretches, and compressions, provide a powerful toolkit for manipulating and analyzing functions. By mastering these concepts, we can gain a deeper insight into the behavior of mathematical models and their applications in various fields of science and engineering.