Problem 68 - Algebra for College Students

Question

Graph \(\ln x\). Now predict the graphs for \(f(x)=\ln (x-2)\), \(f(x)=\ln (x-6)\), and \(f(x)=\ln (x+4)\). Graph the three functions on the same set of axes with \(f(x)=\ln x\).

Step-by-Step Solution

Answer

The graphs of \( \ln(x-2) \), \( \ln(x-6) \), and \( \ln(x+4) \) are horizontal shifts of \( \ln x \) by 2 right, 6 right, and 4 left, with vertical asymptotes at \( x = 2, 6, -4 \) respectively.

Step 1: Understand the Base Function

The graph of the base function, \( f(x) = \ln x \), is defined only for \( x > 0 \). It passes through the point (1,0), has a vertical asymptote at \( x = 0 \), and is increasing and concave down for all \( x > 0 \).

Step 2: Transformations for \( f(x) = \ln(x-2) \)

The function \( f(x) = \ln(x-2) \) represents a horizontal shift of the graph \( f(x) = \ln x \) to the right by 2 units. Its domain is \( x > 2 \) and the vertical asymptote moves to \( x = 2 \).

Step 3: Transformations for \( f(x) = \ln(x-6) \)

The function \( f(x) = \ln(x-6) \) indicates a horizontal shift of the graph \( f(x) = \ln x \) to the right by 6 units. The domain is \( x > 6 \), and the vertical asymptote is at \( x = 6 \).

Step 4: Transformations for \( f(x) = \ln(x+4) \)

The function \( f(x) = \ln(x+4) \) involves a horizontal shift of the graph \( f(x) = \ln x \) to the left by 4 units. The domain changes to \( x > -4 \), with the vertical asymptote at \( x = -4 \).

Step 5: Plotting the Graphs

On the graph, plot the base function \( f(x) = \ln x \) with its asymptote at \( x = 0 \). Then, sequentially plot \( f(x) = \ln(x-2) \), \( f(x) = \ln(x-6) \), and \( f(x) = \ln(x+4) \), ensuring to depict each function's unique horizontal shift and respective asymptote.

Key Concepts

Transformations of Functions

Graphing functions often involves understanding their transformations. A transformation changes the function's graph, altering its position or shape. In the case of logarithmic functions like (f(x) = \ln x)\, transformations usually involve shifts, stretches, or reflections. Horizontal shifts are among the most common transformations.

A horizontal shift occurs when you add or subtract a constant from the input variable \(x\). For instance:

• In \(f(x) = \ln(x-2)\), the graph shifts to the right by 2 units.
• In \(f(x) = \ln(x-6)\), the shift is 6 units to the right.
• For \(f(x) = \ln(x+4)\), the shift is 4 units to the left.

When graphing these transformations, start with the base function and move the entire graph in the direction indicated by the shift. This way, you maintain the shape and the overall behavior of the function but change its horizontal position.

Vertical Asymptote

A vertical asymptote is a line that a graph approaches but never actually touches or crosses. For logarithmic functions, understanding the asymptote is crucial for graphing, as it signifies where the function is undefined.

The base logarithmic function, \(f(x) = \ln x\), has a vertical asymptote at \(x = 0\). This is because the logarithm is undefined for non-positive values. When we apply a horizontal shift to our logarithmic function, the location of the vertical asymptote shifts as well:

• For \(f(x) = \ln(x-2)\), the asymptote moves to \(x = 2\).
• For \(f(x) = \ln(x-6)\), it shifts to \(x = 6\).
• In \(f(x) = \ln(x+4)\), the asymptote is found at \(x = -4\).

Remember, the position of the vertical asymptote directly relates to the domain of your function, dictating for which \(x\) values the function is defined.

Domain of a Function

The domain of a function indicates all the permissible \(x\) values for which the function exists. For logarithms, this means \(x > 0\) in their base form, as the logarithm of a non-positive number is undefined.

When dealing with transformations, particularly horizontal shifts, you must adjust the domain accordingly. Consider these scenarios:

• The domain for \(f(x) = \ln(x-2)\) is \(x > 2\).
• For \(f(x) = \ln(x-6)\), the domain is \(x > 6\).
• In \(f(x) = \ln(x+4)\), the domain becomes \(x > -4\).

Always remember, the domain is altered by horizontal shifts, identifying areas where the functions are defined and rejecting those where they are not. Recognizing this helps ensure accurate and effective graphing of logarithmic functions.