Transforming Exponential Functions Reflecting F(x) Over The Y-axis

Jun 17, 2025 by ADMIN 67 views

$The Transformation of Exponential Functions: Reflecting $f(x) = - rac{2}{7}(\frac{5}{3})^x$ over the y-axis to Create g(x)$

In the fascinating realm of mathematical functions, exponential functions hold a special place, characterized by their rapid growth or decay. Understanding how these functions transform, particularly through reflections, is crucial for grasping their behavior and applications. In this article, we delve into the specific transformation of an exponential function, $f(x) = - rac{2}{7}(\frac{5}{3})^x$ , as it's reflected over the $y$ -axis to create a new function, $g(x)$ . Our exploration will involve not only understanding the process of reflection but also identifying points that lie on the transformed function, $g(x)$ .

Understanding Exponential Functions

Before we embark on the journey of transformation, let's first establish a solid foundation by understanding the fundamental nature of exponential functions. An exponential function is generally represented in the form $f(x) = ab^x$ , where 'a' is the initial value (the value of the function when x = 0), 'b' is the base (a positive number not equal to 1), and 'x' is the exponent. The base, 'b', determines whether the function represents exponential growth (if b > 1) or exponential decay (if 0 < b < 1). The coefficient 'a' influences the vertical stretch or compression of the function, and its sign determines whether the function is reflected across the x-axis.

In our specific case, the function $f(x) = - rac{2}{7}(\frac{5}{3})^x$ has an initial value of $- rac{2}{7}$ and a base of $\frac{5}{3}$ . Since the base is greater than 1, this function represents exponential growth. However, the negative coefficient indicates that the function is reflected across the x-axis. This means that instead of increasing above the x-axis, the function will decrease below it as x increases.

The Concept of Reflection over the y-axis

Reflection, in mathematical terms, is a transformation that creates a mirror image of a function or a shape across a line, which we call the line of reflection. When a function is reflected over the y-axis, each point (x, y) on the original function is transformed into the point (-x, y) on the reflected function. In essence, the x-coordinate changes its sign while the y-coordinate remains the same. This transformation results in a function that is horizontally flipped across the y-axis.

Deriving the Transformed Function g(x)

Now, let's apply the concept of reflection over the y-axis to our given function, $f(x) = - rac{2}{7}(\frac{5}{3})^x$ . To obtain the transformed function, $g(x)$ , we replace 'x' with '-x' in the original function's equation. This substitution effectively mirrors the function across the y-axis.

Therefore, the equation for $g(x)$ becomes:

$g(x) = f(-x) = - rac{2}{7}(\frac{5}{3})^{-x}$

To further simplify this expression, we can use the property of exponents that states $a^{-n} = \frac{1}{a^n}$ . Applying this property to our equation, we get:

$g(x) = - rac{2}{7}(\frac{3}{5})^x$

This is the equation of the transformed function, $g(x)$ , which is the reflection of $f(x)$ over the y-axis. Notice that the base of the exponential term has changed from $\frac{5}{3}$ to $\frac{3}{5}$ . This change is a direct consequence of the reflection over the y-axis and indicates a shift from exponential growth to exponential decay.

Identifying Points on g(x)

The next step in our exploration is to identify which points lie on the graph of $g(x)$ . To do this, we can substitute the x-coordinate of each given point into the equation for $g(x)$ and check if the resulting y-coordinate matches the y-coordinate of the given point. If they match, then the point lies on the graph of $g(x)$ .

Let's consider the point $(-7, -10.206)$ . To check if this point lies on $g(x)$ , we substitute $x = -7$ into the equation for $g(x)$ :

$g(-7) = - rac{2}{7}(\frac{3}{5})^{-7}$

Using the property of exponents $a^{-n} = \frac{1}{a^n}$ , we can rewrite this as:

$g(-7) = - rac{2}{7}(\frac{5}{3})^7$

Now, we can calculate the value:

$g(-7) \approx - rac{2}{7} * 65.144 \approx -18.613$

Since the calculated y-coordinate, -18.613, does not match the given y-coordinate, -10.206, the point (-7, -10.206) does not lie on the graph of $g(x)$ .

Conclusion: Understanding Transformations and Their Impact

In this exploration, we've journeyed through the process of reflecting an exponential function over the y-axis. We began by understanding the fundamentals of exponential functions and their characteristics, such as growth, decay, and reflections across the x-axis. We then delved into the concept of reflection over the y-axis, learning how it transforms a function by mirroring it across the vertical axis. By applying this transformation to the given function, $f(x) = - rac{2}{7}(\frac{5}{3})^x$ , we derived the equation for the transformed function, $g(x) = - rac{2}{7}(\frac{3}{5})^x$ .

Furthermore, we learned how to identify points that lie on the graph of $g(x)$ by substituting the x-coordinate of a point into the function's equation and comparing the resulting y-coordinate with the given y-coordinate. This process allows us to verify whether a given point is indeed a part of the transformed function.

The ability to understand and perform transformations on functions is a fundamental skill in mathematics. It allows us to visualize how functions change and adapt under different conditions. Reflections, in particular, provide a powerful tool for understanding symmetry and how functions behave when mirrored across an axis. By mastering these concepts, we gain a deeper appreciation for the elegance and versatility of mathematical functions.

The specific example of reflecting $f(x) = - rac{2}{7}(\frac{5}{3})^x$ over the y-axis to create $g(x)$ serves as a valuable illustration of these principles. It demonstrates how a simple transformation can alter the base of an exponential function, changing its behavior from growth to decay. It also highlights the importance of careful calculation and verification when identifying points on a transformed function. As we continue our mathematical journey, the knowledge and skills gained from this exploration will undoubtedly serve us well in tackling more complex problems and concepts.

To truly master these concepts, it is essential to engage in further practice and exploration. Try reflecting different functions over various axes and identifying points on the transformed functions. Experiment with different types of functions, such as linear, quadratic, and trigonometric functions, to see how transformations affect their graphs and equations. By actively engaging with these concepts, you will develop a deeper understanding of their nuances and applications.

In conclusion, the reflection of exponential functions over the y-axis, as demonstrated by the transformation of $f(x)$ to $g(x)$ , provides valuable insights into the behavior and characteristics of these functions. By understanding the principles of reflection and other transformations, we can unlock a deeper appreciation for the power and beauty of mathematics.