Problem 29 - Precalculus with Limits

Question

Transforming the Graph of an Exponential Function In Exercises \(27-30,\) use the graph of \(f\) to describe the transformation that yields the graph of \(g\) . $$f(x)=\left(\frac{7}{2}\right)^{x}, \quad g(x)=-\left(\frac{7}{2}\right)^{-x}$$

Step-by-Step Solution

Answer

The function \(g(x) = -\left(\frac{7}{2}\right)^{-x}\) is the reflection of the function \(f(x) = \left(\frac{7}{2}\right)^x\) about the x-axis and the y-axis.

Step 1: Analyze the Functions

Start by comparing the two functions. \(f(x) = \left(\frac{7}{2}\right)^x\) is a standard exponential function where the base is \(\frac{7}{2}\). On the other hand, the function \(g(x) = -\left(\frac{7}{2}\right)^{-x}\) is a transformation of \(f(x)\).

Step 2: Identify the Transformations

Look at the transformation applied to the base function \(f(x)\). There are two transformations applied to \(f(x)\) to obtain \(g(x)\): 1. Reversal Transformation: The negative sign in front of the base function flips the graph of the base function along the x-axis. 2. Inversion Transformation: Replacing \(x\) with \(-x\) reflects the graph of the function about the y-axis.

Step 3: Describe the Transformations

Describe the graphical transformation in words. Here, the function \(g(x)\) represents the graph of \(f(x)\) after it has been reflected about the x-axis (due to the negative sign) and then reflected about the y-axis (due to \(-x\), instead of \(x\)).

Key Concepts

Exponential Functions

Exponential functions are a fascinating and foundational type of mathematical function featured by their unique growth behavior. An exponential function has the form \( f(x) = a^x \), where \( a \) is a constant called the base and \( x \) is the exponent. This base must be greater than zero. As the variable \( x \) increases, the function values change exponentially, hence the name "exponential."

• Exponential growth occurs when the base \( a \) is greater than 1. The function will increase rapidly as \( x \) becomes larger.
• Exponential decay happens when the base is between 0 and 1, causing the function values to approach zero as \( x \) increases.

Understanding the base and exponent is crucial. In the function \( f(x) = \left(\frac{7}{2}\right)^x \), the base is \( \frac{7}{2} \), which indicates exponential growth since it's greater than 1. The nature of exponential functions allows them to model real-world situations like population dynamics, radioactive decay, and compound interest.

Graph Transformations

Graph transformations are a powerful tool in understanding how certain changes in the function's equation can manipulate its graph. These transformations can include shifts, stretches, compressions, and reflections, which alter the original shape and position of the graph.

• Horizontal transformations affect the input \( x \). For instance, \( f(x - c) \) shifts the graph horizontally by \( c \) units.
• Vertical transformations alter the output. For example, \( f(x) + k \) shifts the graph up by \( k \) units.
• Reflective transformations involve flipping the graph around an axis.

In the context of the function \( g(x) = -\left(\frac{7}{2}\right)^{-x} \), transformation occurs as a result of reflection due to specific adjustments made to the function \( f(x) \). As you learn these concepts, you'll become adept at pinpointing how simple algebraic changes result in distinct visual changes in graphs.

Reflection of Graphs

Reflection in graph transformations is when a graph is flipped over a specific axis. This operation significantly modifies the function's appearance on the graph. There are two main types:

• Reflection over the x-axis: This is achieved by multiplying the function by -1, \( -f(x) \), flipping it vertically.
• Reflection over the y-axis: This occurs when the variable \( x \) is replaced with \( -x \), \( f(-x) \), producing a horizontal flip.

In the given transformation, the function \( g(x) = -\left(\frac{7}{2}\right)^{-x} \) involves both types of reflections. The negative sign before the base \( \left(\frac{7}{2}\right)^{-x} \) indicates a reflection over the x-axis, while replacing \( x \) with \( -x \) signifies a reflection over the y-axis. These reflection transformations collectively illustrate how one can manipulate graphs to achieve the desired outcomes in various applications, enabling you to visualize data and predict patterns effectively.