Problem 104 - Precalculus

Question

For the following exercises, start with the graph of \(f(x)=4^{x}\) . Then write a function that results from the given transformation. Reflect \(f(x)\) about the \(x\) -axis

Step-by-Step Solution

Answer

The function after the reflection is \( g(x) = -4^x \).

Step 1: Understanding the Transformation

Reflecting a function about the x-axis means that you invert the sign of the function's output. In mathematical terms, if you have a function \( f(x) \), then reflecting it about the x-axis gives you the new function \( g(x) = -f(x) \).

Step 2: Applying the Transformation

Apply the reflection transformation to the given function \( f(x) = 4^x \). This means you replace \( f(x) \) with \( -f(x) \). Therefore, the reflected function is \( g(x) = -4^x \).

Key Concepts

Reflection

The concept of a reflection in mathematics involves flipping a function or shape across a specific axis, much like looking at a mirror image. When we reflect a function over the x-axis, we change the signs of the y-coordinates of each point on the graph.
In terms of functions, reflecting over the x-axis turns positive outputs into negative outputs and vice versa. To achieve this, we multiply the entire function by -1. The transformation from a function \( f(x) \) to its x-axis reflection \( g(x) \) can be represented as:

• Original function: \( f(x) \)
• Reflected function: \( g(x) = -f(x) \)

By reflecting \( f(x) = 4^x \), you obtain the new function \( g(x) = -4^x \). This transformation results in the downward flip of the graph so all the values that were once positive above the x-axis are now negative below it. The reflection process is a simple and fundamental transformation useful in various mathematics and physics applications.

Graph of Exponential Function

The graph of an exponential function is one of the most striking and commonly seen graphs in mathematics. The general form of an exponential function is \( f(x) = a^x \), where \( a \) is a positive constant.
Exponential functions have some distinctive features:

• Growth or Decay: When \( a > 1 \), the function represents exponential growth, creating a curve that increases rapidly. When \( 0 < a < 1 \), it represents exponential decay, falling sharply.
• Horizontal Asymptote: The line \( y = 0 \) is a horizontal asymptote, meaning that the function approaches but never actually touches this line.
• Y-Intercept: The graph passes through \( (0, 1) \) assuming no vertical shifts.

In the exercise, \( f(x) = 4^x \) exemplifies an exponential growth function. The base, 4, indicates how rapid the growth is, as each increase in \( x \) corresponds to quadrupling the output from the previous point. This creates a curve that gets steeper and rises faster as \( x \) increases. The transformational behavior of these functions allows various real-life phenomena like population growth and radioactive decay to be modeled effectively.

Function Transformation Steps

Transforming functions involves modifying their graphs in various ways such as translations, reflections, stretches, or compressions. These transformations follow systematic steps that you can apply to any given function:

Recognize the Base Function

• Identify the function in its simplest form. For example, in the exercise, it is \( f(x) = 4^x \).

Identify the Transformation

• Determine which transformation is being applied. The problem specifies a reflection about the x-axis.

Apply the Transformation

• Implement the transformation to obtain the new function. Multiply the original function by -1 to reflect it: \( g(x) = -4^x \).

It helps to visualize these transformations on a graph to understand their effects. Understanding and mastering these steps allow you to handle more complex function transformations effectively, aiding in analytical skills and graphical understanding.