Problem 63 - Algebra 2

Question

Sketch the graphs of \(y=\log _{\frac{1}{2}} x\) and \(y=\left(\frac{1}{2}\right)^{x}\) on the same axes. Then describe the relationship between the graphs.

Step-by-Step Solution

Answer

The graphs are reflections across the line \(y = x\).

Step 1: Understanding the Functions

First, recognize that the graphs we need to sketch are: 1) A logarithmic function of the form \(y = \log_{\frac{1}{2}} x\), which is the logarithm with base \(\frac{1}{2}\); and 2) An exponential function \(y = \left(\frac{1}{2}\right)^{x}\), meaning the base \(\frac{1}{2}\) is raised to the power of \(x\).

Step 2: Plot Key Points for \(y = \log_{\frac{1}{2}} x\)

The logarithmic function \(y = \log_{\frac{1}{2}} x\) passes through points such as \((1, 0)\) because \(\log_{\frac{1}{2}} 1 = 0\). For \(x = \frac{1}{2}\), \(y = 1\) because \(\log_{\frac{1}{2}} \frac{1}{2} = 1\). The graph decreases and approaches zero as \(x\) increases.

Step 3: Plot Key Points for \(y = (\frac{1}{2})^{x}\)

For the exponential function \(y = (\frac{1}{2})^{x}\), the point \((0, 1)\) is on the graph since any number to the power of 0 is 1. As \(x\) increases, \((\frac{1}{2})^{x}\) gets closer to 0, and when \(x\) is negative, the function increases because it is equivalent to a reciprocal.

Step 4: Draw the Graphs

Using the key points and understanding from previous steps, sketch \(y = \log_{\frac{1}{2}} x\) as a curve starting from the right, going through \((1, 0)\) and \((\frac{1}{2}, 1)\), approaching the \(x\)-axis as \(x\) increases. Then draw \(y = (\frac{1}{2})^{x}\) starting from the left, going through \((0, 1)\), and approaching the \(x\)-axis from the top.

Step 5: Compare the Graphs

Notice that both the graphs are reflections of each other across the line \(y = x\). This is because the functions \(y = (\frac{1}{2})^{x}\) and \(y = \log_{\frac{1}{2}} x\) are inverses.

Key Concepts

Logarithmic Functions

Logarithmic functions are closely related to exponential functions. They essentially ask the question: "To what power must the base be raised, to result in a given number?" In the function \(y = \log_{\frac{1}{2}} x\), the number \(x\) is the result of raising the base, \(\frac{1}{2}\), to some power, \(y\).
Logarithmic functions have a characteristic graph that differs from exponential functions in several ways:

• The graph of \(y = \log_{\frac{1}{2}} x\) is defined for all positive \(x\) values; it never touches or crosses the x-axis, but the graphs approach it as x increases.
• It passes through the point \((1, 0)\) because any base raised to the power of zero equals one.
• The point \((\frac{1}{2}, 1)\) is on the graph since the base \(\frac{1}{2}\) raised to the power of 1 equals \(\frac{1}{2}\).
• In a logarithmic function where the base is a fraction, the graph decreases as \(x\) increases.

Recognizing these traits helps in plotting and understanding the logarithmic function on a graph.

Exponential Functions

Exponential functions like \(y = \left(\frac{1}{2}\right)^{x}\) are fundamental in understanding exponential growth and decay. The base, \(\frac{1}{2}\), indicates decay since it is less than one.
Key characteristics of exponential functions include their growth or decay rate:

• The graph crosses the y-axis at \((0, 1)\) because any non-zero number raised to the zero power equals one.
• Operands greater than zero in the exponent will result in increasingly smaller outputs, causing decay towards zero as \(x\) increases.
• When \(x\) is negative, the function reflects the properties of the reciprocal, resulting in increasingly larger values.
• The graph does not become negative and approaches the x-axis asymptotically as it moves rightward.

This pattern is crucial for plotting exponential functions and recognizing their inverse nature when compared to logarithmic functions.

Graph Transformations

Understanding graph transformations is essential in analyzing how functions evolve on a graph. Graph transformations involve translation, reflection, stretching, and shrinking of graphs.
For the functions \(y = \log_{\frac{1}{2}} x\) and \(y = \left(\frac{1}{2}\right)^{x}\), the key transformation is reflection.

• The graph of \(y = \left(\frac{1}{2}\right)^{x}\) is a reflection of \(y = \log_{\frac{1}{2}} x\) across the line \(y = x\).
• This reflection happens because logarithmic and exponential functions are inverses.
• Each point on one graph will correspond to a point on the other, such that, if \((a, b)\) is on one graph, \((b, a)\) will be on the inverse graph.
• These transformations help visualize complex interactions and understand inverse relationships.

Recognizing these transformations enhances comprehension of function behavior and their interdependencies.