Ordered Pairs On Reflected Exponential Function G(x)

Jul 9, 2025 by ADMIN 53 views

The Transformation and Ordered Pairs of Exponential Functions

In the realm of mathematical functions, transformations play a crucial role in understanding how the graph of a function changes when certain operations are applied. Among the various transformations, reflection across the y-axis holds significant importance, especially when dealing with exponential functions. In this article, we will delve into the concept of reflecting an exponential function across the y-axis and explore how this transformation affects the ordered pairs on the graph of the function. We will use the specific example of the function f(x) = (1/6)(2/5)^x to illustrate the process and determine the ordered pairs that lie on the reflected function g(x).

Before we delve into the transformation, let's first understand the basics of exponential functions. An exponential function is a function of the form f(x) = ab^x, where a is a non-zero constant and b is a positive constant not equal to 1. The constant a represents the initial value of the function, and the constant b is the base, which determines the rate of growth or decay of the function. When b is greater than 1, the function represents exponential growth, and when b is between 0 and 1, the function represents exponential decay.

In our specific case, the function f(x) = (1/6)(2/5)^x is an exponential decay function because the base (2/5) is between 0 and 1. The initial value of the function is 1/6, which means that when x is 0, the function value is 1/6. As x increases, the function value decreases, approaching 0 as a horizontal asymptote.

Reflection across the y-axis is a transformation that creates a mirror image of the original function with respect to the y-axis. Mathematically, reflecting a function f(x) across the y-axis results in a new function g(x), where g(x) = f(-x). This means that the x-coordinate of each point on the graph of the original function is negated, while the y-coordinate remains the same. For example, if the point (x, y) lies on the graph of f(x), then the point (-x, y) lies on the graph of g(x).

To reflect the function f(x) = (1/6)(2/5)^x across the y-axis, we need to replace x with -x in the function's equation. This gives us the reflected function g(x) = (1/6)(2/5)^(-x). We can simplify this expression by using the property that a^(-x) = (1/a)^x. Therefore, g(x) = (1/6)(5/2)^x. Notice that the base of the exponential function has changed from 2/5 to 5/2, which is the reciprocal of 2/5. This transformation effectively changes the decay function into a growth function.

Now that we have the equation for the reflected function g(x) = (1/6)(5/2)^x, we can determine which ordered pairs lie on its graph. An ordered pair (x, y) lies on the graph of g(x) if and only if substituting x into the equation for g(x) yields the value y. In other words, if g(x) = y, then the ordered pair (x, y) is on the graph of g(x).

To determine whether a given ordered pair lies on the graph of g(x), we can substitute the x-coordinate of the ordered pair into the equation for g(x) and check if the result matches the y-coordinate of the ordered pair. If they match, then the ordered pair is on the graph of g(x); otherwise, it is not.

Let's analyze the given options:

Option A: (-3, 4/375) Substitute x = -3 into the equation for g(x): g(-3) = (1/6)(5/2)^(-3) = (1/6)(2/5)^3 = (1/6)(8/125) = 8/750 = 4/375 Since g(-3) = 4/375, the ordered pair (-3, 4/375) lies on the graph of g(x). Thus, Option A is a potential answer.
Option B: (-2, 25/24) Substitute x = -2 into the equation for g(x): g(-2) = (1/6)(5/2)^(-2) = (1/6)(2/5)^2 = (1/6)(4/25) = 4/150 = 2/75 Since g(-2) = 2/75, which is not equal to 25/24, the ordered pair (-2, 25/24) does not lie on the graph of g(x). Thus, Option B is incorrect.
Option C: (2, 25/24) Substitute x = 2 into the equation for g(x): g(2) = (1/6)(5/2)^2 = (1/6)(25/4) = 25/24 Since g(2) = 25/24, the ordered pair (2, 25/24) lies on the graph of g(x). Thus, Option C is also a potential answer.
Option D: (3, 125/48) Substitute x = 3 into the equation for g(x): g(3) = (1/6)(5/2)^3 = (1/6)(125/8) = 125/48 Since g(3) = 125/48, the ordered pair (3, 125/48) lies on the graph of g(x). Thus, Option D is also a potential answer.

In this article, we explored the concept of reflecting an exponential function across the y-axis and how this transformation affects the ordered pairs on the graph of the function. We used the specific example of the function f(x) = (1/6)(2/5)^x to illustrate the process and determined the equation for the reflected function g(x) = (1/6)(5/2)^x. By substituting the x-coordinates of the given ordered pairs into the equation for g(x), we found that the ordered pairs (-3, 4/375), (2, 25/24) and (3, 125/48) lie on the graph of g(x). This analysis demonstrates the importance of understanding function transformations and their impact on the coordinates of points on the graph.

In this comprehensive exploration, we will delve into the fascinating world of function transformations, specifically focusing on the reflection of the function f(x) = (1/6)(2/5)^x across the y-axis. This transformation results in a new function, g(x), and our primary objective is to identify which ordered pair lies on this transformed function. Understanding the behavior of exponential functions and the effects of reflections is crucial in various mathematical and scientific applications. We will break down the process step-by-step, ensuring clarity and a thorough understanding of the underlying concepts.

Understanding the Original Function: f(x) = (1/6)(2/5)^x

Before we can analyze the reflection, it's essential to understand the characteristics of the original function, f(x) = (1/6)(2/5)^x. This is an exponential function, where the variable x appears in the exponent. The general form of an exponential function is f(x) = ab^x, where a is the initial value and b is the base. In our case, a = 1/6 and b = 2/5. Since the base b is between 0 and 1, this function represents exponential decay. This means that as x increases, the value of f(x) decreases, approaching zero. Exponential decay is a fundamental concept in mathematics and is used to model various real-world phenomena, such as radioactive decay, population decline, and the depreciation of assets. The initial value, 1/6, tells us that when x = 0, the function's value is 1/6. This point (0, 1/6) lies on the graph of the function and serves as a crucial reference point. Understanding the decay behavior is vital for predicting the function's values for different inputs and for grasping the impact of transformations. The function's asymptotic behavior, its approach to zero as x increases, is another key characteristic. This behavior influences how the function interacts with transformations like reflections. By thoroughly understanding the original function, we lay a solid foundation for analyzing its reflection across the y-axis.

Reflection Across the y-axis: Creating g(x)

Now, let's explore the critical transformation: reflection across the y-axis. Reflecting a function across the y-axis means creating a mirror image of the function with respect to the y-axis. Mathematically, this transformation is achieved by replacing x with -x in the function's equation. So, to find g(x), the reflection of f(x) across the y-axis, we substitute -x for x in the equation for f(x). This gives us g(x) = (1/6)(2/5)^(-x). To simplify this expression, we can use the property that a^(-x) = (1/a)^x. Applying this property, we get g(x) = (1/6)(5/2)^x. Notice that the base of the exponential function has changed from 2/5 to 5/2. This change is significant because it transforms the function from exponential decay to exponential growth. The reflected function, g(x), now increases as x increases, which is the opposite behavior of the original function, f(x). The reflection across the y-axis has effectively reversed the function's trend. The initial value, 1/6, remains the same, indicating that the y-intercept (0, 1/6) is invariant under this transformation. Understanding the change in the base and the resulting shift from decay to growth is crucial for predicting the behavior of g(x) and identifying ordered pairs that lie on its graph. The mathematical manipulation of the exponent is a key technique to understand and apply in various mathematical contexts.

Identifying Ordered Pairs on g(x) = (1/6)(5/2)^x

The core of our problem lies in identifying which ordered pair lies on the graph of g(x) = (1/6)(5/2)^x. An ordered pair (x, y) lies on the graph of a function if and only if substituting the x-value into the function's equation yields the corresponding y-value. In other words, (x, y) is on the graph if g(x) = y. To determine which of the given options lies on g(x), we will substitute the x-coordinate of each option into the equation for g(x) and check if the result matches the y-coordinate. This process involves direct substitution and evaluation of the exponential function. For each option, we will calculate g(x) and compare it to the given y-value. If they match, the ordered pair lies on the graph of g(x). This method is a straightforward and reliable way to verify whether a point belongs to a function's graph. Let's apply this method to the given options. This method of verification is a fundamental concept in coordinate geometry, used to check if points lie on curves represented by equations. It's a powerful tool for visualizing functions and their properties.

Option A: (-3, 4/375)

To check if the ordered pair (-3, 4/375) lies on g(x), we substitute x = -3 into the equation g(x) = (1/6)(5/2)^x:

g(-3) = (1/6)(5/2)^(-3)

Using the property a^(-n) = 1/a^n, we can rewrite (5/2)^(-3) as (2/5)^3:

g(-3) = (1/6)(2/5)^3

Now, we calculate (2/5)^3: (2/5)^3 = (2^3)/(53) = 8/125

Substitute this back into the equation:

g(-3) = (1/6)(8/125) = 8/(6 * 125) = 8/750

Simplifying the fraction, we get:

g(-3) = 4/375

Since g(-3) = 4/375, the ordered pair (-3, 4/375) lies on the graph of g(x). Therefore, Option A is a valid solution. This calculation demonstrates the use of exponent properties and fraction simplification in evaluating exponential functions. The fact that the calculated y-value matches the given y-value confirms that the point lies on the graph.

Option B: (-2, 25/24)

To determine if the ordered pair (-2, 25/24) lies on g(x), we substitute x = -2 into the equation g(x) = (1/6)(5/2)^x:

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

Again, using the property a^(-n) = 1/a^n, we rewrite (5/2)^(-2) as (2/5)^2:

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

Now, we calculate (2/5)^2: (2/5)^2 = (2^2)/(52) = 4/25

Substitute this back into the equation:

g(-2) = (1/6)(4/25) = 4/(6 * 25) = 4/150

Simplifying the fraction, we get:

g(-2) = 2/75

Since g(-2) = 2/75, which is not equal to 25/24, the ordered pair (-2, 25/24) does not lie on the graph of g(x). Therefore, Option B is not a solution. This calculation reinforces the importance of accurate evaluation of exponents and fractions. The mismatch between the calculated y-value and the given y-value confirms that the point is not on the function's graph.

Option C: (2, 25/24)

To check if the ordered pair (2, 25/24) lies on g(x), we substitute x = 2 into the equation g(x) = (1/6)(5/2)^x:

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

Now, we calculate (5/2)^2: (5/2)^2 = (5^2)/(22) = 25/4

Substitute this back into the equation:

g(2) = (1/6)(25/4) = 25/(6 * 4) = 25/24

Since g(2) = 25/24, the ordered pair (2, 25/24) lies on the graph of g(x). Therefore, Option C is a valid solution. This calculation further demonstrates the evaluation process for exponential functions. The match between the calculated y-value and the given y-value confirms that the point is on the graph.

Option D: (3, 125/48)

To verify if the ordered pair (3, 125/48) lies on g(x), we substitute x = 3 into the equation g(x) = (1/6)(5/2)^x:

g(3) = (1/6)(5/2)^3

Now, we calculate (5/2)^3: (5/2)^3 = (5^3)/(23) = 125/8

Substitute this back into the equation:

g(3) = (1/6)(125/8) = 125/(6 * 8) = 125/48

Since g(3) = 125/48, the ordered pair (3, 125/48) lies on the graph of g(x). Therefore, Option D is a valid solution. This calculation provides another example of the process of evaluating exponential functions and confirming that a point lies on the graph. The precise match between the calculated and given values solidifies the conclusion.

Conclusion: The Ordered Pairs on g(x)

Through a detailed analysis and step-by-step calculations, we have successfully identified the ordered pairs that lie on the graph of g(x) = (1/6)(5/2)^x, which is the reflection of f(x) = (1/6)(2/5)^x across the y-axis. Our findings reveal that the ordered pairs (-3, 4/375), (2, 25/24), and (3, 125/48) all satisfy the equation for g(x) and therefore lie on its graph. Understanding the reflection transformation and its effect on the original function has been crucial in this process. The transformation changed the exponential decay function into an exponential growth function, altering its behavior significantly. By substituting the x-coordinates of the given options into the equation for g(x) and comparing the results with the corresponding y-coordinates, we were able to accurately determine which points belong to the graph. This exercise underscores the importance of function transformations, exponential function properties, and the relationship between equations and their graphs in mathematics. The ability to perform these calculations and interpret the results is essential for solving problems in various mathematical and scientific contexts. The process of substitution and evaluation is a fundamental skill in mathematics, and its application here demonstrates its utility in verifying solutions and understanding function behavior.

The correct ordered pairs on g(x) are A. (-3, 4/375) , C. (2, 25/24) and D. (3, 125/48)