Reflection Of Exponential Function F(x) = (1/3)(6)^x Across The X-Axis

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In mathematics, understanding transformations of functions is crucial for grasping the behavior of various mathematical models. Among these transformations, reflections play a significant role in altering the orientation of a function's graph. In this article, we will delve into the reflection of an exponential function across the x-axis. Specifically, we will explore the function f(x) = (1/3)(6)^x and determine the table of values for its reflection, g(x), across the x-axis. This exploration will not only enhance our understanding of reflections but also solidify our grasp of exponential functions.

Defining the Function f(x) = (1/3)(6)^x

Let's begin by defining the function f(x) = (1/3)(6)^x. This is an exponential function where the base is 6, and the coefficient is 1/3. To fully understand this function, we need to analyze its properties. Exponential functions are characterized by their rapid growth or decay. In this case, since the base (6) is greater than 1, the function f(x) exhibits exponential growth. As x increases, the value of f(x) increases dramatically. The coefficient 1/3 vertically compresses the function, but the overall exponential growth trend remains.

Key Characteristics of f(x)

  • Exponential Growth: As x increases, f(x) increases rapidly.
  • Vertical Compression: The coefficient 1/3 compresses the function vertically.
  • y-intercept: The y-intercept, where the graph intersects the y-axis, is found by setting x = 0. Thus, f(0) = (1/3)(6)^0 = 1/3. This indicates that the graph of f(x) passes through the point (0, 1/3).
  • Asymptotic Behavior: As x approaches negative infinity, f(x) approaches 0. The x-axis serves as a horizontal asymptote for the function. This means the graph gets arbitrarily close to the x-axis but never actually touches it.

Understanding Exponential Functions

Exponential functions are used to model a variety of real-world phenomena, including population growth, radioactive decay, and compound interest. The general form of an exponential function is f(x) = a * b^x, where:

  • a is the initial value or the y-intercept.
  • b is the base, which determines whether the function grows (b > 1) or decays (0 < b < 1).
  • x is the exponent, representing the independent variable.

In our function, f(x) = (1/3)(6)^x, a = 1/3 and b = 6. The base b = 6 indicates rapid growth, and the coefficient a = 1/3 scales the function vertically. Understanding these parameters is crucial for predicting the behavior of the function and its transformations.

Reflection Across the X-Axis: Defining g(x)

Now, let's consider the reflection of f(x) across the x-axis, which we denote as g(x). A reflection across the x-axis essentially flips the graph of the function over the x-axis. Mathematically, this transformation is achieved by negating the function's output, i.e., g(x) = -f(x). Applying this to our function, we get:

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

This new function, g(x), is also an exponential function, but with a negative coefficient. This negative sign is the key to understanding the reflection across the x-axis. The effect of this reflection is that all the y-values of f(x) are now negated. Points that were above the x-axis are now below it, and vice versa. The shape of the graph remains the same, but its orientation is flipped.

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

  • Exponential Decay in the Negative Direction: As x increases, g(x) decreases rapidly in the negative direction.
  • Vertical Compression and Reflection: The coefficient -1/3 compresses the function vertically and reflects it across the x-axis.
  • y-intercept: The y-intercept of g(x) is found by setting x = 0. Thus, g(0) = -(1/3)(6)^0 = -1/3. The graph of g(x) passes through the point (0, -1/3), which is a reflection of the y-intercept of f(x) across the x-axis.
  • Asymptotic Behavior: As x approaches negative infinity, g(x) approaches 0, similar to f(x). The x-axis remains the horizontal asymptote for g(x), but the function approaches it from the negative side.

Comparing f(x) and g(x)

To better understand the reflection, let's compare the key characteristics of f(x) and g(x):

  • f(x) = (1/3)(6)^x:
    • Exponential growth
    • y-intercept at (0, 1/3)
    • Approaches 0 as x approaches negative infinity
  • g(x) = -(1/3)(6)^x:
    • Exponential decay in the negative direction
    • y-intercept at (0, -1/3)
    • Approaches 0 as x approaches negative infinity

The primary difference between the two functions is the direction of their exponential behavior and the sign of their y-intercepts. f(x) grows exponentially in the positive direction, while g(x) decays exponentially in the negative direction. The y-intercepts are reflections of each other across the x-axis, further illustrating the transformation.

Constructing the Table of Values for g(x)

To create a table of values for g(x), we can start by selecting a few values of x and calculating the corresponding values of f(x) and then g(x). The key is to choose x-values that provide a clear picture of the function's behavior. We will use the relationship g(x) = -f(x) to find the values of g(x).

Selecting x-Values

We will select a range of x-values, including negative, zero, and positive values, to capture the full behavior of the exponential function. Suitable values might include -2, -1, 0, 1, and 2. These values will give us a good representation of the function's growth and decay.

Calculating f(x) Values

First, we calculate the values of f(x) = (1/3)(6)^x for the selected x-values:

  • For x = -2:
    • f(-2) = (1/3)(6)^(-2) = (1/3)(1/36) = 1/108
  • For x = -1:
    • f(-1) = (1/3)(6)^(-1) = (1/3)(1/6) = 1/18
  • For x = 0:
    • f(0) = (1/3)(6)^0 = (1/3)(1) = 1/3
  • For x = 1:
    • f(1) = (1/3)(6)^1 = (1/3)(6) = 2
  • For x = 2:
    • f(2) = (1/3)(6)^2 = (1/3)(36) = 12

Calculating g(x) Values

Next, we calculate the values of g(x) = -f(x) by negating the values of f(x):

  • For x = -2:
    • g(-2) = -f(-2) = -1/108
  • For x = -1:
    • g(-1) = -f(-1) = -1/18
  • For x = 0:
    • g(0) = -f(0) = -1/3
  • For x = 1:
    • g(1) = -f(1) = -2
  • For x = 2:
    • g(2) = -f(2) = -12

Creating the Table of Values

Now, we can create the table of values for f(x) and g(x):

x f(x) g(x)
-2 1/108 -1/108
-1 1/18 -1/18
0 1/3 -1/3
1 2 -2
2 12 -12

This table clearly illustrates the reflection across the x-axis. For each x-value, the y-value of g(x) is the negative of the y-value of f(x). This confirms our understanding of the reflection transformation.

Analyzing the Table of Values

The table of values provides a concrete representation of the reflection of f(x) across the x-axis to obtain g(x). By examining the table, we can observe the following:

  • For every x-value, the magnitude of f(x) and g(x) is the same, but their signs are opposite. This is the hallmark of a reflection across the x-axis.
  • As x increases, f(x) increases exponentially, while g(x) decreases exponentially in the negative direction.
  • The y-intercept of f(x) is 1/3, and the y-intercept of g(x) is -1/3, which are reflections of each other across the x-axis.

This analysis reinforces the concept of reflection and its impact on the function's graph. The table of values serves as a practical tool for visualizing and understanding transformations of functions.

Conclusion

In conclusion, we have successfully explored the reflection of the exponential function f(x) = (1/3)(6)^x across the x-axis to obtain g(x) = -(1/3)(6)^x. We defined the functions, analyzed their properties, and constructed a table of values to illustrate the transformation. This exercise highlights the importance of understanding reflections and other transformations in mathematics. By mastering these concepts, we can better understand and model a wide range of mathematical and real-world phenomena. The reflection across the x-axis is a fundamental transformation that provides valuable insights into the behavior of functions and their graphical representations. Understanding these transformations is crucial for problem-solving and advanced mathematical studies.

Final Table of Values

x f(x) g(x)
-2 1/108 -1/108
-1 1/18 -1/18
0 1/3 -1/3
1 2 -2
2 12 -12