Problem 51 - College Algebra with Corequisite Support

Question

Explore and discuss the graphs of \(\quad F(x)=(b)^{x}\) and \(G(x)=\left(\frac{1}{b}\right)^{x}\). Then make a conjecture about the relationship between the graphs of the functions \(b^{x}\) and \(\left(\frac{1}{b}\right)^{x}\) for any real number \(\quad b>0\).

Step-by-Step Solution

Answer

\( F(x) = b^x \) and \( G(x) = \left(\frac{1}{b}\right)^x \) are reflections about the y-axis.

Step 1: Analyze the function \( F(x) = b^x \)

The function \( F(x) = b^x \) is an exponential function where \( b \) is the base and \( x \) is the exponent. If \( b > 1 \), the graph of \( F(x) \) will rise sharply to the right and approach zero on the left. If \( 0 < b < 1 \), it decreases from left to right.

Step 2: Analyze the function \( G(x) = \left(\frac{1}{b}\right)^x \)

The function \( G(x) = \left(\frac{1}{b}\right)^x \) can be rewritten as \( G(x) = b^{-x} \). This implies a reflection of \( F(x) = b^x \) over the y-axis. For \( b > 1 \), \( G(x) \) decreases as \( x \) increases. For \( 0 < b < 1 \), it increases as \( x \) increases.

Step 3: Compare and conjecture the relationship between \( F(x) \) and \( G(x) \)

For each \( b > 1 \), \( F(x) \) starts near 0 on the left and rises, while \( G(x) \) starts high on the left and decreases. When \( 0 < b < 1 \), the opposite is true. Both functions are thus reflections of each other across the y-axis. The graph of \( b^x \) is a mirror image of \( \left(\frac{1}{b}\right)^x \) about the y-axis for any real \( b > 0 \).

Key Concepts

Function Graphs

Understanding function graphs is crucial when analyzing exponential functions. For an exponential function like \( F(x) = b^x \), the shape of the graph changes based on the value of \( b \), the base of the exponent. If the base \( b \) is greater than 1, the graph will demonstrate exponential growth. This means that as \( x \) increases, the function value grows rapidly, reflecting a steep curve moving upwards to the right. On the other hand, if \( 0 < b < 1 \), the graph illustrates exponential decay. In this case, as \( x \) increases, the function value diminishes, resulting in a curve that falls smoothly toward the right.

The graph passes through the point (0,1) regardless of the base, since any non-zero number raised to the power of zero is 1. These patterns are essential for predicting and understanding the nature of exponential functions through their graphical representations.

Reflection Over Y-axis

Reflection across the y-axis is a transformation that takes a function \( F(x) = f(x) \) and mirrors it to create \( G(x) = f(-x) \). This concept helps explain the relationship between \( F(x) = b^x \) and \( G(x) = \left(\frac{1}{b}\right)^x \). The latter function can also be expressed as \( G(x) = b^{-x} \), indicating it is the reflection of \( F(x) = b^x \) over the y-axis.

When visually inspected, the graph of \( G(x) \) is seen as \( F(x) \)'s mirrored image on the y-axis. For example, if \( b > 1 \), while \( F(x) \) grows from left to right, \( G(x) \)'s values diminish as \( x \) progresses. Conversely, if \( 0 < b < 1 \), \( F(x) \) decays left to right, but \( G(x) \) displays growth as \( x \) moves positively. Understanding this symmetry is crucial for analyzing behaviors in mathematical models and real-world phenomena, which might involve exponential transformations.

Exponential Growth and Decay

Exponential growth and decay represent two fundamental behaviors portrayed by functions expressed in the form of \( y = b^x \). Exponential growth, characterized by \( b > 1 \), describes the increase in a quantity at a rate proportional to its current value. As the function grows, each increment in \( x \) results in larger increases in \( y \), creating a curve that becomes steeper over time.

In contrast, exponential decay occurs when \( 0 < b < 1 \). Here, the function decreases as \( x \) grows, which is illustrated by a curve that gradually flattens, approaching zero but never quite reaching it.

• Real-world examples of exponential growth include population growth and compound interest, where the quantity simultaneously increases with respect to its size, forming ever-accelerating growth.
• Conversely, radioactive decay, cooling of hot objects, and depreciation of assets are examples of exponential decay.

This fundamental understanding of growth and decay allows for predictions that are vital for scientific, financial, and statistical applications.