Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{2} x, \quad g(x)=-3+\log _{2} x$$

Step-by-Step Solution

Answer

The transformation of the graph is a downward shift of 3 units.

Step 1: Identify the Basic Function

Looking at the equation, the original function \(f(x)\) is obviously \(f(x) = \log_2 x\). This is a logarithmic function with base 2.

Step 2: Identify the Transformation

The function \(g(x)\) is given as \(g(x) = -3 + \log_2 x\). Comparing this with the original function \(f(x) = \log_2 x\), it can be noticed that the '-3' is a vertical shift. Since the '-3' is subtracted from the original function, it indicates a vertical shift down by 3 units.

Step 3: Describe the Transformation

Hence, the transformation from \(f(x) = \log_2 x\) to \(g(x) = -3 + \log_2 x\) can be described as a downward shift of the graph of \(f(x) = \log_2 x\) by 3 units.

Key Concepts

Logarithmic Function

A logarithmic function, typically written as \( f(x) = \text{log}_b x \), represents the inverse operation of exponentiation. In simpler terms, if \( b^y = x \), then \( y = \text{log}_b x \). The number \( b \) is called the base of the logarithm and in our problem, it is 2. This means the function increases as x increases, but at a rate that slows down the larger x gets. This relationship is particularly useful in fields like science and engineering, where exponential growth patterns are frequently analyzed and transformed into more manageable logarithmic scales. Understanding the nature of logarithmic functions is crucial when examining their graphical transformations, as they provide us with a way to interpret exponential growth and decay in a linear fashion.

Vertical Shift

A vertical shift in a graph transformation is a movement up or down along the y-axis. In the context of our exercise, the function \( g(x) = -3 + \text{log}_2 x \) showcases a vertical shift from the base logarithmic function. To visualize it, imagine every point on the graph of \( f(x) = \text{log}_2 x \) being moved straight down by 3 units. It’s important to note that vertical shifts do not affect the x-values of the graph; they simply raise or lower the entire graph. Therefore, the new function's graph will maintain the shape and horizontal placement of the original function while being repositioned vertically.

Graph Transformation

Graph transformation involves changing the position or size of a graph using operations such as shifts, stretches, and reflections. In our case, we are dealing with a vertical shift. Graph transformations are integral in understanding how different algebraic manipulations to a function's formula will visually impact its graph. These transformations give us the tools to adapt functions to better model real-world situations without changing the fundamental nature of their relationships. In addition to vertical shifts, other common transformations include horizontal shifts, vertical stretching and shrinking, and horizontal stretching and shrinking. Recognizing these transformations just by looking at the equation of a function, as in identifying the vertical shift in our exercise, is a valuable skill.

Base of Logarithm

The base of a logarithm in a logarithmic function like \( \text{log}_b x \) is the number \( b \) that is raised to a power to obtain \( x \). This is a crucial concept because the base determines the logarithm's rate of increase or decrease. For instance, the common logarithm uses base 10, and the natural logarithm uses the constant \( e \). In our function, the base 2 means that the logarithm is part of the binary logarithm family, where every increase by 1 in \( y \) means that \( x \) is doubled. The choice of base has a significant influence on the shape of the logarithmic graph. Understanding the impact of different bases is essential, especially considering that logarithms of different bases are related by a constant factor and can thus be transformed into one another.