Problem 88 - Precalculus with Calculus Previews

Question

In Problems \(85-88\), discuss how the graph of the given function can be obtained from the graph of \(f(x)\) \(=\ln x\) by means of a rigid transformation (a shift or a reflection)? $$ y=\ln (-x) $$

Step-by-Step Solution

Answer

The graph of \( y = \ln (-x) \) is the reflection of \( y = \ln x \) over the y-axis.

Step 1: Identify the Base Function

The base function given is \( f(x) = \ln x \). This is the natural logarithm function, which is defined only for positive values of \( x \). Its graph passes through the point (1,0) and is asymptotic to the y-axis.

Step 2: Analyze the Transformation

The given function is \( y = \ln (-x) \). This function introduces a negative sign inside the argument of the logarithm function, which represents a transformation of the base function \( f(x) \).

Step 3: Determine the Type of Transformation

The function \( y = \ln (-x) \) can be viewed as \( y = \ln(x') \) where \( x' = -x \). This is a reflection over the y-axis since each \( x \) value is replaced with \(-x\).

Step 4: Describe the Graph Changes

Reflecting \( f(x) = \ln x \) over the y-axis results in a graph that is now defined in the negative x-domain. The original point that passes through (1,0) would now be transformed to pass through (-1,0). Thus, the graph of \( y = \ln (-x) \) is a mirror image of \( \ln(x) \) flipped over the y-axis.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. The most common base for a logarithm is the natural base, represented as \( e \) (approximately 2.718). A natural logarithm function is denoted as \( f(x) = \ln x \). This function is only defined for positive values of \( x \) because you can only take the logarithm of positive numbers. The graph of \( \ln x \) has a few critical features:

• It passes through the point \((1,0)\), as \( \ln 1 = 0 \).
• The graph approaches the y-axis but never touches it, which is known as being asymptotic to the y-axis.
• As \( x \) increases, \( \ln x \) increases slowly, reflecting the logarithmic growth.

Understanding these features is crucial, as transformations such as reflections or shifts will manipulate the graph, altering these characteristics based on the transformation type applied.

Graph Transformations

Graph transformations allow us to manipulate functions systematically. They can include translations (shifts), reflections, and stretches or compressions. When dealing with these transformations, knowing the base graph—such as \( \ln x \)—helps in predicting the resulting graph:- **Horizontal Shifts:** Occur when adding or subtracting a value directly to \( x \). For example, \( y = \ln(x - 2) \) would shift the graph of \( \ln x \) two units to the right.- **Vertical Shifts:** Result from adding or subtracting a number to the entire function. Such as \( y = \ln x + 3 \), which shifts it three units up.Reflections are another form of transformation and involve flipping the graph over a specified axis, which directly changes the domain or range.

Reflection Over Axes

Reflection is a type of transformation that flips the graph of a function over a specific axis. For logarithmic functions like \( \ln x \), reflections can significantly alter the domain:- **Reflection Over the y-axis:** This reflection is achieved by replacing \( x \) with \(-x\). In the function \( y = \ln(-x) \), this transformation flips the graph of \( \ln x \) over the y-axis. Initially defined for positive \( x \), the new function is now defined for negative \( x \) values. This results in the original point \( (1,0) \) becoming \( (-1,0) \), making the transformed graph a mirror image across the y-axis.- **Reflection Over the x-axis:** Works by taking the negative of the entire function, but is less common in the context of logarithms due to their specific domain requirements.By understanding which values to replace or change, one can accurately predict how the graph will be adjusted. This foundational knowledge is helpful when sketching graphs quickly or analyzing transformation effects in complex functions.