Reflecting Exponential Functions How To Reflect F(x) = 2(0.35)^x Over The Y-Axis

The question at hand delves into the concept of transformations of functions, specifically reflections across the y-axis. Understanding this transformation is crucial for manipulating and interpreting various mathematical models, especially in the context of exponential functions. Let's break down the given function and explore how reflections impact its equation. Our primary focus is identifying the function that results from reflecting $f(x)=2(0.35)^x$ over the y-axis. To tackle this, we'll dissect the function, understand the transformation rules, and apply them to arrive at the correct answer. This involves a step-by-step analysis, ensuring a clear comprehension of the underlying principles. Remember, a reflection across the y-axis essentially mirrors the function about the vertical axis, swapping the x-coordinates while keeping the y-coordinates (or the function's value) the same for the corresponding reflected point. This understanding forms the bedrock of our approach to solving the problem.

Understanding the Base Function: $f(x) = 2(0.35)^x$

Before we dive into the reflection, let's analyze the base function, f(x) = 2(0.35)^x. This is an exponential function of the form f(x) = a * b^x, where:

• a represents the vertical stretch or compression factor and also the initial value when x = 0.
• b is the base, which determines the growth or decay rate of the function.

In our case, a = 2 and b = 0.35. The coefficient 2 indicates a vertical stretch by a factor of 2, meaning that the y-values of the function are twice what they would be without this coefficient. The base 0.35 is a crucial piece of information, as it's less than 1. This tells us that the function represents exponential decay, meaning the function's value decreases as x increases. We can visualize this by considering a few points. When x = 0, f(0) = 2(0.35)^0 = 2 * 1 = 2. As x increases, (0.35)^x becomes smaller, causing the entire function value to decrease. When x is a large positive number, the function approaches 0. Conversely, as x becomes a large negative number, the function's value increases significantly. The graph of this function would start high on the left side (negative x values) and gradually decrease towards the x-axis as we move to the right (positive x values). This understanding of the base function's behavior is essential for predicting how it will change under various transformations, including reflections.

The Reflection Transformation: Reflecting Over the Y-Axis

The core concept we need to grasp is how a reflection over the y-axis affects a function's equation. A reflection across the y-axis transforms a point (x, y) to (-x, y). This means the x-coordinate changes its sign while the y-coordinate remains the same. Mathematically, if we have a function f(x), its reflection over the y-axis is given by f(-x). In simpler terms, to reflect a function over the y-axis, we replace every x in the function's equation with -x. This substitution is the key to finding the reflected function. The logic behind this transformation stems from the nature of symmetry. Imagine the y-axis as a mirror. For every point on the original function, there's a corresponding point on the reflected function that is the same distance away from the y-axis but on the opposite side. This distance is represented by the absolute value of the x-coordinate. Changing the sign of x effectively swaps the points across the y-axis, thus creating the reflection. Let's consider a simple example, such as f(x) = x^2. Reflecting this over the y-axis gives us f(-x) = (-x)^2 = x^2. In this case, the function is symmetric about the y-axis, so the reflection is the same as the original function. However, this isn't always the case, especially for functions like exponential functions, where the reflection creates a distinctly different graph.

Applying the Reflection to $f(x) = 2(0.35)^x$

Now, let's apply the principle of reflection to our function, f(x) = 2(0.35)^x. To reflect this function over the y-axis, we need to replace every x in the equation with -x. This gives us the new function, which we'll call h(x):

• h(x) = f(-x) = 2(0.35)^(-x)

This is the function that represents the reflection of f(x) over the y-axis. It's crucial to understand how this change affects the function's behavior. The original function, f(x), is an exponential decay function because the base (0.35) is between 0 and 1. Reflecting it over the y-axis essentially flips the decay into growth. As x increases, -x becomes more negative. For the original function, this would have meant approaching zero, but now, with the negative exponent, the function grows exponentially. Let's analyze this further. We can rewrite (0.35)^(-x) as (1/0.35)^x. Since 1/0.35 is greater than 1, the reflected function, h(x), represents exponential growth. This visually confirms our understanding of the reflection. The reflected graph will be a mirror image of the original graph about the y-axis, with the decaying trend transformed into a growing trend. Now that we have the equation for the reflected function, we can compare it to the given options to identify the correct one.

Identifying the Correct Option

We have determined that the reflection of f(x) = 2(0.35)^x over the y-axis is h(x) = 2(0.35)^(-x). Now, let's examine the options provided in the question:

• A. h(x) = 2(0.35)^x
• B. h(x) = -2(0.35)^x
• C. h(x) = 2(0.35)^(-x)
• D. h(x) = 2(-0.35)^(-x)

Comparing our derived equation with the options, we can clearly see that option C, h(x) = 2(0.35)^(-x), matches our result. The other options represent different transformations. Option A is the original function itself, indicating no transformation. Option B represents a reflection over the x-axis (due to the negative sign in front of the function). Option D involves a negative base, which isn't a standard form for exponential functions and wouldn't represent a simple reflection. Therefore, the correct answer is C, which accurately represents the reflection of the given exponential function over the y-axis. This careful comparison highlights the importance of understanding each type of transformation and how they affect the equation of a function. We can confidently select option C as the solution, solidifying our understanding of reflections and exponential functions.

Conclusion

In conclusion, the function that represents a reflection of f(x) = 2(0.35)^x over the y-axis is h(x) = 2(0.35)^(-x). This determination was made by understanding the concept of reflections across the y-axis, which involves replacing x with -x in the original function's equation. We first analyzed the base function, recognizing it as an exponential decay function due to the base being between 0 and 1. Then, we applied the reflection transformation, substituting -x for x, resulting in the function h(x). This transformed function represents an exponential growth function, which is the expected outcome of reflecting a decaying exponential function over the y-axis. By comparing our result with the provided options, we confidently identified option C as the correct answer. This exercise highlights the crucial connection between graphical transformations and algebraic manipulations. Understanding how different transformations affect a function's equation allows us to analyze and manipulate mathematical models effectively. The ability to identify and apply transformations like reflections is a fundamental skill in mathematics, with applications spanning various fields, including physics, engineering, and computer science. This detailed explanation should provide a comprehensive understanding of reflections and their application to exponential functions.