Understanding Function Reflection: How To Graph G(x) Of F(x)=(1/3)(6)^x

#mainkeywords Function reflections play a crucial role in understanding transformations in mathematics, and in this comprehensive exploration, we delve into the intricacies of reflecting the exponential function f(x) = (1/3)(6)^x across the x-axis to obtain g(x). To truly grasp this concept, we will dissect the properties of exponential functions, unravel the mechanics of reflections, and meticulously construct a table of values that vividly illustrates the relationship between f(x) and its reflected counterpart, g(x).

Understanding the Exponential Function f(x)

Before we embark on the journey of reflection, it's paramount to establish a firm understanding of the function we're working with. f(x) = (1/3)(6)^x is an exponential function. The exponential function is characterized by its unique property of rapid growth. The variable x appears as an exponent, dictating the rate at which the function increases or decreases. In our case, the base of the exponent is 6, signifying that the function will exhibit exponential growth as x increases. The constant factor (1/3) acts as a vertical scaling factor, compressing the function vertically. To gain a deeper insight into the behavior of f(x), let's evaluate it at a few specific points:

• When x = 0, f(0) = (1/3)(6)^0 = (1/3)(1) = 1/3
• When x = 1, f(1) = (1/3)(6)^1 = (1/3)(6) = 2
• When x = -1, f(-1) = (1/3)(6)^(-1) = (1/3)(1/6) = 1/18

These points provide a glimpse into the exponential nature of f(x), highlighting its swift increase as x moves towards positive values and its gradual approach towards zero as x moves towards negative values.

The Concept of Reflection Across the x-axis

Reflection across the x-axis is a fundamental transformation in coordinate geometry. It involves flipping a function or a shape over the x-axis, creating a mirror image of the original. Mathematically, this transformation is elegantly simple: for any point (x, y) on the original function, its reflection across the x-axis will be the point (x, -y). In essence, the x-coordinate remains unchanged, while the y-coordinate undergoes a sign change. This means that points above the x-axis in the original function will now lie below the x-axis in the reflected function, and vice versa. To visualize this transformation, imagine folding the coordinate plane along the x-axis. The image you see on the other side of the fold represents the reflection.

Deriving the Reflected Function g(x)

Now that we have a firm grasp of reflection across the x-axis, let's apply this transformation to our function f(x). If f(x) represents the original function, then its reflection across the x-axis, denoted as g(x), is obtained by negating the y-values of f(x). In mathematical terms:

g(x) = -f(x)

Substituting the expression for f(x), we get:

g(x) = -(1/3)(6)^x

This equation elegantly captures the essence of the reflection. The negative sign in front of the entire expression ensures that all the y-values of g(x) are the opposites of the y-values of f(x), effectively mirroring the function across the x-axis. The function g(x) will exhibit exponential decay as x increases, mirroring the growth of f(x), but in the opposite direction.

Constructing the Table of Values for g(x)

To solidify our understanding of the relationship between f(x) and g(x), let's construct a table of values that showcases their contrasting behaviors. We'll choose a range of x-values, evaluate both f(x) and g(x) at these points, and then compare the results. This tabular representation will provide a clear visual depiction of how the reflection across the x-axis affects the function's values.

As the table vividly demonstrates, for each x-value, the y-value of g(x) is the negative of the y-value of f(x). This confirms our understanding that g(x) is indeed the reflection of f(x) across the x-axis. The table serves as a powerful tool for visualizing the transformation and appreciating the symmetry between the two functions. For negative values of x, f(x) approaches zero from positive values, while g(x) approaches zero from negative values. As x increases, f(x) grows exponentially, whereas g(x) decreases exponentially, further highlighting the mirrored behavior.

Conclusion

In this comprehensive exploration, we've meticulously examined the concept of reflecting an exponential function across the x-axis. We began by dissecting the properties of f(x) = (1/3)(6)^x, emphasizing its exponential growth and vertical scaling. We then delved into the mechanics of reflection, understanding it as a flipping transformation that negates the y-values of the original function. By applying this principle, we derived the reflected function g(x) = -(1/3)(6)^x. Finally, we constructed a table of values that visually illustrated the mirrored relationship between f(x) and g(x). The table served as a powerful validation of our understanding, showcasing how the y-values of g(x) are the exact opposites of the y-values of f(x). This exploration not only deepened our understanding of function reflections but also reinforced the importance of tabular representations in visualizing transformations and grasping mathematical concepts.

Reflection of functions is a cornerstone concept in mathematics, and the ability to manipulate functions through transformations is a valuable skill. By understanding the impact of reflections, we gain a more profound appreciation for the beauty and elegance of mathematical functions and their graphical representations. The relationship between a function and its reflection is a testament to the symmetry and order that underlies the seemingly complex world of mathematics.

#repair-input-keyword Original Question: The function $f(x)$ is defined as $f(x)=\frac{1}{3}(6)^x$. Which table of values could be used to graph $g(x)$, a reflection of $f(x)$ across the $x$-axis? Rewritten Question: Given the function $f(x) = (1/3)(6)^x$, determine the table of values that represents $g(x)$, which is the reflection of $f(x)$ across the x-axis.

#title Understanding Function Reflection How to Graph g(x) of f(x)=(1/3)(6)^x