Problem 96 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln x+4$$

Step-by-Step Solution

Answer

The function \(g(x) = \ln x + 4\) is a result of shifting the function \(f(x) = \ln x\) 4 units upwards.

Step 1: Understand the original function graph

The log function \(f(x) = \ln x\) is defined for \(x > 0\), and as \(x\) approaches infinity, the y-values of the graph also gradually increase. As \(x\) approaches 0 from the positive side, the values of \(y\) heads towards negative infinity. The graph crosses the x-axis at \(x=1\).

Step 2: Analyze the transformation

The function \(g(x) = \ln x + 4\) is obtained from \(f(x) = \ln x\) by adding 4 to every y-coordinate of the graph of \(f(x)\). This is known as a vertical shift or vertical translation, and the positive value indicates that it's an upward shift.

Step 3: Describe the transformed graph

In the function \(g(x) = \ln x + 4\), the graph will be identical to that of \(f(x) = \ln x\) but every point will be moved 4 units upwards. Therefore, the graph still gradually increases as \(x\) approaches infinity and heads towards negative infinity as \(x\) approaches 0 from the positive side. However, instead of crossing the x-axis at \(x=1\), it will cross at the point (1,4).

Key Concepts

Vertical Translation

A vertical translation causes the entire graph of a function to move up or down in the y-direction without affecting its shape or orientation. In the context of our exercise, when we consider the transformation from the function \(f(x) = \ln x \) to \(g(x) = \ln x + 4 \), a vertical translation occurs. Here, each point on the graph of \(f(x)\) is shifted upward by 4 units.

The concept of vertical translation is quite straightforward because it simply involves adjusting the output values of the function:

• For positive upward shifts, you add a constant to the function values.
• For downward shifts, you subtract a constant from the function values.

Vertical translations do not change any roots or x-intercepts of the function, but they alter the point where the graph crosses the y-axis. This straightforward method of translating functions is essential in transforming graphs efficiently.

Graph of Logarithmic Function

The graph of the natural logarithm function, \(f(x) = \ln x\), serves as the base for understanding transformations. This logarithmic function is defined for all positive values of \(x\). Unlike polynomial functions, which might oscillate or have several roots, the natural logarithm graph has several distinct characteristics:

• It increases gradually as \(x\) becomes larger.
• As \(x\) approaches zero from the positive side, the function's y-values rapidly approach negative infinity.
• The graph passes through the point (1, 0) because \(\ln 1 = 0\).

This logarithmic curve is asymptotic to the y-axis, meaning it gets closer and closer to the y-axis but never actually touches or crosses it.

Understanding this basic structure of a logarithmic graph helps to easily visualize any transformations applied, such as vertical shifts or other operations.

Properties of Logarithmic Functions

Logarithmic functions possess unique properties that stand out from other types of functions. Being the inverse of exponential functions, they often appear in situations where growth or decay needs to be measured, especially over extensive ranges. Some characteristics of logarithmic functions include:

• Domain and Range: The natural logarithmic function \(f(x) = \ln x\) has a domain of \(x > 0\) and a range of all real numbers.
• Asymptotic Nature: As a critical property, the graph of \(\ln x\) is asymptotic to the y-axis, meaning it will keep getting closer without ever touching it.
• Base Point: At \(x = 1\), the value is always zero, since \(\ln 1 = 0\). This is the fundamental point where the graph crosses the x-axis.

Because of these properties, logarithmic functions are pivotal in modeling real-world phenomena where growth stabilizes over time, or when interpreting scales like the Richter scale or decibels. Familiarity with these properties not only aids in function transformations but also in applying logarithms in various scientific and mathematical contexts.