Problem 51 - College Algebra

Question

Explore and discuss the graphs of \(f(x)=(b)^{x}\) 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 \(b>0\).

Step-by-Step Solution

Answer

The graphs of \(b^x\) and \(\left(\frac{1}{b}\right)^x\) are reflections across the y-axis.

Step 1: Graph the Function f(x)

Start by graphing the function \(f(x) = b^x\) for a constant \(b > 1\). For instance, if \(b=2\), the graph will be an exponentially increasing curve that passes through the point (0,1) because any positive number to the power of 0 is 1. As \(x\) increases, the value of \(f(x)\) grows rapidly.

Step 2: Graph the Function g(x)

Next, graph \(g(x) = \left(\frac{1}{b}\right)^x\) for the same \(b > 1\). Here, \(\left(\frac{1}{b}\right)\) is a fraction less than 1, so the graph shows exponential decay. When \(x = 0\), \(g(x)\) equals 1, and as \(x\) increases, \(g(x)\) decreases towards zero.

Step 3: Compare f(x) and g(x)

Notice the symmetry in their graphs. While \(f(x) = b^x\) demonstrates exponential growth, \(g(x) = \left(\frac{1}{b}\right)^x\) shows exponential decay. The graphs are reflections of each other across the y-axis due to the reciprocal base \(\frac{1}{b}\). This means if \(b^x\) increases, \(\frac{1}{b}^x\) decreases at the same rate and vice versa.

Step 4: Conjecture the Relationship

Formulate a conjecture based on your observations: The graphs of \(f(x) = b^x\) and \(g(x) = \left(\frac{1}{b}\right)^x\) are reflections of each other across the y-axis (vertical line through y-axis) for any real number \(b > 1\). The exponential growth of \(f(x)\) is mirrored by the exponential decay of \(g(x)\).

Key Concepts

Exponential Growth

Exponential growth describes a process where the quantity increases rapidly due to a constant proportionality rate. When graphing an exponential function like \(f(x) = b^x\) with \(b > 1\), the curve rises steeply as \(x\) becomes larger.

In an exponential growth scenario:

• The base \(b\) is greater than 1.
• The function passes through the point \((0, 1)\) because any non-zero number raised to the power of zero is 1.
• As \(x\) increases, \(f(x)\) becomes significantly larger.

Thus, the graph shows a continuous upward trend. Exponential growth is common in real-world phenomena, such as population growth and compound interest, where the rate of growth accelerates over time.

Exponential Decay

Exponential decay is the process where a quantity decreases rapidly over time. This is modeled by functions like \(g(x) = \left(\frac{1}{b}\right)^x\), where \(b > 1\). In this context, \(\frac{1}{b}\) is a fraction representing the base of the decay.

Let's break down its characteristics:

• The base \(\frac{1}{b}\) is a positive number less than 1.
• Just like exponential growth, the function passes through the point \((0, 1)\).
• As \(x\) increases, \(g(x)\) rapidly approaches zero.

Therefore, in exponential decay, the quantity shrinks at a rate proportional to its current value, often observed in radioactive decay, depreciation, and cooling processes.

Graph Interpretation

Interpreting the graphs of exponential functions helps us visualize growth and decay. The graph of \(f(x) = b^x\) is rising and curving upwards, demonstrating exponential growth. On the contrary, \(g(x) = \left(\frac{1}{b}\right)^x\) shows a curve that approaches the x-axis, illustrating exponential decay.

Key features to identify on these graphs include:

• The y-intercept at \((0, 1)\) for any base b.
• For \(f(x) = b^x\), observe the curve steeply accelerating as x increases.
• For \(g(x) = \left(\frac{1}{b}\right)^x\), the curve decreases and flattens out as x increases.

By comparing these graphs, one can understand how changes to the base value affect the rate and direction of change in exponential functions.

Reflections Across the Y-Axis

In the context of \(f(x) = b^x\) and \(g(x) = \left(\frac{1}{b}\right)^x\), these functions are reflections of each other across the y-axis. This means one curve mirrors the other through the vertical line running along the y-axis.

Here's how reflections work:

• If \(b^x\) increases at a certain rate, then \(\left(\frac{1}{b}\right)^x\) decreases at the same rate but in the opposite direction.
• This symmetry arises because \(b\) and \(\frac{1}{b}\) are reciprocal values.
• The reflection property helps in visualizing and understanding the inverse nature of the two functions.

These reflections enable us to predict and confirm the behavior of one function given the other, particularly useful when examining relationships between processes like growth and decay.