Problem 55 - Contemporary Precalculus

Question

List the transformations that will change the graph of \(g(x)=\ln x\) into the graph of the given function. $$h(x)=\ln (x+3)-4$$

Step-by-Step Solution

Answer

Answer: The transformations are a horizontal shift of 3 units to the left and a vertical shift of 4 units down.

Step 1: Identify Horizontal Shift

Observe that in the function \(h(x) = \ln (x+3) - 4\), the argument of the natural logarithm is \((x+3)\). This represents a horizontal shift of the graph to the left by 3 units. So, the first transformation is a horizontal shift of 3 units to the left.

Step 2: Identify Vertical Shift

In the function \(h(x) = \ln (x+3) - 4\), the \(-4\) outside the natural logarithm represents a vertical shift of the graph. In this case, it's a vertical shift down by 4 units. So, the second transformation is a vertical shift of 4 units down.

Step 3: Check for Stretches or Compressions

There is no constant multiplying the natural logarithm function, which means there are no stretches or compressions. Therefore, we do not need to consider any other transformations.

Step 4: List the Transformations

We have identified two transformations that change the graph of \(g(x) = \ln x\) into the graph of \(h(x) = \ln (x+3) - 4\). They are: 1. A horizontal shift of 3 units to the left. 2. A vertical shift of 4 units down.

Key Concepts

Horizontal Shift

Understanding horizontal shifts in graph transformations is essential for analyzing the behavior of functions. Imagine sliding a function left or right along the x-axis without altering its shape—this is what a horizontal shift does. In our context, the function given by \(h(x)=\ln (x+3)-4\) exhibits this phenomenon.

To identify a horizontal shift, focus on the part of the function inside the brackets. If the x-variable is replaced with \(x-c\), the graph shifts to the right by c units; conversely, if it is replaced with \(x+c\), the graph shifts to the left by c units. In the solution provided, we witness a horizontal shift to the left by 3 units, indicated by the transformation \(x\to x+3\). This effectively moves every point on the graph of the natural logarithm 3 units to the left.

Vertical Shift

In addition to horizontal movement, functions can also experience vertical shifts, which translate the graph up or down in relation to the y-axis. In the function \(h(x)=\ln (x+3)-4\), the number subtracted from the logarithm function indicates a vertical displacement. A positive number after the logarithm suggests an upward shift, while a negative number, as seen with the \(-4\), signifies a downward shift.

To quantify this transformation in our exercise, we note that the vertical shift is 4 units down. No other variable aside from x is affected, maintaining the original shape of the graph. Think of this as moving the function's graph along an elevator—up (+) or down (−)—without changing its curvature or orientation.

Natural Logarithm Graph

The natural logarithm graph, represented as \(y=\ln(x)\), is a fundamental concept in precalculus and higher mathematics. It possesses a distinctive shape: it passes through the point (1,0), asymptotically approaches the y-axis (but never touches it), and continues to rise slowly as x increases. This growth is unbounded as x tends towards infinity, though it does so at a decreasing rate.

The key features to identify on a natural logarithm graph include its asymptote at x=0, intercept at (1,0), and the overall increasing behavior. When transformations are applied to \(\ln(x)\), these features shift accordingly, but the fundamental shape of the graph remains unhindered.

Function Transformations

When we speak of function transformations, we refer to operations that alter the appearance of a graph. This could involve translations (shifts), reflections, and dilations (stretches or compressions). These transformations are tools that help in visualizing changes without re-plotting the whole function.

Key transformations include:

• Horizontal and vertical shifts, moving the graph along the axes without rotation.
• Reflections over the x-axis or y-axis, creating a mirror image of the original graph.
• Stretches or compressions that either elongate or squish the graph horizontally or vertically.

Understanding these modifications is invaluable for predicting the result of complicated function manipulations. It also greatly simplifies the process of graphing functions without utilizing a point-by-point plotting method.