Question

Describe the transformation of the graph of \(f\) represented by the graph of \(g\). Then give an equation of the asymptote. \(f(x)=\ln x, g(x)=\ln (x+6)\)

Short answer

The function \(g(x) = \ln (x+6)\) is obtained from \(f(x) = \ln x\) by shifting it 6 units to the left. The equation of the asymptote is \(x=-6\).

Step-by-step solution

- Identify the transformation

The transformation from \(f(x) = \ln x\) to \(g(x) = \ln (x+6)\), is a horizontal shift. This is caused by the addition of a scalar inside the function.

- Describe the direction and distance of the shift

The +6 inside the bracket causes a shift in the horizontal direction. The graph of \(f(x)\) is moved 6 units to the left to form the graph of \(g(x)\). Thus the effect of modifying \(x\) to \(x+6\) is a horizontal shift to the left by 6 units.

- Determination of the asymptote

The asymptote is the line which the function approaches but never reaches. For the function \(f(x) = \ln x\), the asymptote is \(x=0\). As the function is shifted 6 units to the left, the new asymptote of \(g(x)\) is \(x=-6\)

Key concepts

Logarithmic Functions

Logarithmic functions are a type of mathematical function often denoted as \( f(x) = \ln x \) where "\( \ln \)" represents the natural logarithm. These functions are the inverse of exponential functions. They have specific characteristics, such as their shape and asymptotic behavior, that distinguish them from other types of functions.
In the context of graph transformations, understanding logarithmic functions is crucial because each transformation, like shifting or stretching, impacts the graph's shape and the equation's behavior. For \( f(x) = \ln x \), the function only takes positive values of \( x \) because the logarithm of zero or a negative number is undefined. This is due to the properties of logarithms, which are based on exponentiation. Here are some key points about logarithmic functions:

• They pass through the point \((1, 0)\) because \( \ln(1) = 0 \).
• The basic shape is a curve that rises slowly, never crossing the vertical line \( x = 0 \) (asymptote).
• They are continuous and differentiable wherever they are defined.

Understanding \( f(x) = \ln x \) will give you the foundation needed to comprehend how graph transformations, like horizontal shifts, affect these functions.

Horizontal Shifts

Horizontal shifts in graph transformations occur when a function is translated along the x-axis. For instance, transforming \( f(x) = \ln x \) to \( g(x) = \ln(x + 6) \) results in a horizontal shift. Specifically, adding a constant inside the logarithm function shifts the graph horizontally.

When we add a positive number inside the function as \( x+6 \), the graph of \( f(x) \) shifts to the left by that constant amount. This might seem counterintuitive, but it's a typical behavior in horizontal transformations:

• For \( g(x) = \ln(x + 6) \), the graph moves 6 units to the left.
• If we were to take \( g(x) = \ln(x - 6) \), on the other hand, the graph would shift 6 units to the right.

When performing horizontal shifts, the asymptote of the original function also shifts by the same amount. Understanding this concept is essential for accurately representing the new position of the graph after transformation.

Asymptotes

An asymptote is a line that a graph approaches but never actually touches or crosses. In the graph of the function \( f(x) = \ln x \), the vertical asymptote is at \( x = 0 \). This means the graph approaches the y-axis as \( x \) gets close to zero but doesn't actually touch the axis.
The concept of asymptotes is particularly important when dealing with transformations like horizontal shifts. For the transformed function \( g(x) = \ln(x + 6) \), the asymptote shifts alongside the graph movement.
Since the entire graph of \( g(x) \) is shifted 6 units to the left, the asymptote also shifts. Therefore, the new asymptote for \( g(x) \) is at \( x = -6 \). Recognizing the shift in the asymptote is critical for understanding how the transformed function behaves and where it is undefined or tends towards infinity. Asymptotes help define the boundary and the behavior of a function as it approaches infinity or certain undefined points.