Problem 53 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Answer: A horizontal shift to the right by 4 units is needed.

Step 1: Identify the original function and the transformed function

We have the original function g(x) as: $$g(x) = \ln x$$ And the transformed function h(x) as: $$h(x) = \ln (x-4)$$

Step 2: Compare the functions and identify the transformation

We can see that the only difference between the two functions is the argument inside the natural logarithm. In the transformed function, instead of having \(x\), we have \((x-4)\): $$h(x) = \ln (x-4)$$ This indicates a horizontal shift.

Step 3: Determine the direction and magnitude of the horizontal shift

The horizontal shift can be determined by looking at the value inside the parentheses. In this case, we have \((x-4)\) in the transformed function, which means the graph will shift to the right by 4 units.

Step 4: List the transformations

To change the graph of \(g(x) = \ln x\) into the graph of \(h(x) = \ln (x-4)\), the following transformation is needed: 1. A horizontal shift to the right by 4 units.

Key Concepts

Horizontal Shift

Understanding the concept of a horizontal shift is crucial when studying transformations of functions. A horizontal shift moves every point on a graph to the left or right by a certain amount. This transformation doesn't change the shape of the graph; it simply translates it along the x-axis.

Let's decode the horizontal shift in the context of the exercise. Given the original function, \( g(x) = \ln x \), and the transformed function, \( h(x) = \ln (x-4) \), we observe a modification in the argument of the logarithm from \( x \) to \( (x-4) \). This minus four inside the logarithm indicates that every point on the graph of \( g(x) \) should be shifted horizontally to the right by 4 units to obtain the graph of \( h(x) \).

In practical terms, if you were to draw the original function on a graph paper, shifting it horizontally would involve taking a 'ruler' and shifting each point on the curve to the right side by the distance of four 'units' on the graph paper, where a unit corresponds to the distance between two consecutive lines. This type of manipulation is key in mathematics, as it allows us to visualize how functions can be transformed and how their graphs relate to each other.

Transformations of Functions

The term 'transformations of functions' refers to the various operations that can be applied to a function's equation to alter its graph. These changes include shifts, stretches, compressions, and reflections. Transformations can be applied to any part of the function's formula, resulting in a corresponding change in its graph.

Transformations can be horizontal or vertical shifts, moving the graph along the axes; stretches or compressions, which elongate or shorten the graph in the vertical or horizontal direction; or reflections, flipping the graph over an axis. Understanding how these transformations affect the shape and position of a graph is invaluable in graphing new functions based on familiar ones and plays a central role in calculus and algebra.

When applying a transformation, like the horizontal shift displayed in the exercise, it is important to analyze the algebraic changes in the function's formula. For our example with logarithmic functions, the horizontal shift is conveyed through an alteration inside the argument of the natural logarithm. Grasping the nature of these changes enhances a student's ability to quickly sketch graphs and predict the effects of varying different parameters in the functions' equations.

Natural Logarithm Graph

The natural logarithm graph represents the function \( f(x) = \ln x \), which is the inverse of the natural exponential function \( e^x \). It's important to note that the natural logarithm is only defined for positive real numbers, and its graph is characterized by a distinctive shape that approaches the y-axis (asymptote) but never touches it, while steadily increasing without bound as x becomes larger.

The graph of the natural logarithm passes through the point (1,0) since \( \ln(1) \) equals zero. It also exhibits certain properties, such as being continuously increasing (monotonically) and having a domain of all positive real numbers (\( x > 0 \)).

Understanding the graph of the natural logarithm function helps us better grasp various transformations, such as those presented in the exercise. When the argument of the natural logarithm is altered, as seen with \( h(x) = \ln (x-4) \), it shifts the graph horizontally without changing its overall shape. Hence, recognizing the base graph of the natural logarithm is an essential step before analyzing how transformations like shifts will affect the function's visual representation.