Problem 31 - Essential Calculus

Question

Sketch the graphs of the function \(g(x)=f(x)+C\) for \(C=-2, C=0,\) and \(C=3\) on the same set of coordinate axes. $$ f(x)=\ln x $$

Step-by-Step Solution

Answer

The graphs of function g(x) for \(C=-2\), \(C=0\), and \(C=3\) are vertical translations of the base function \(f(x) = \ln x\). For \(C=-2\), the graph moves 2 units downwards, for \(C= 0\), there is no change, and for \(C=3\), the graph moves 3 units upwards.

Step 1: Graphing the Original Function

First, sketch the graph of the base function \(f(x) = \ln x\). Recall that the graph of \(f(x) = \ln x\) has a vertical asymptote at \(x=1\), crosses the x-axis at \(y=0\), and increases without bound as \(x\) approaches infinity.

Step 2: Graphing for C=-2

Next, sketch the graph of the function \(g(x) = f(x) - 2\). As \(f(x) = \ln x\), the function \(g(x) = f(x) - 2\) corresponds to moving the graph of the base function down by 2 units.

Step 3: Graphing for C=0

Sketch the graph of the function \(g(x) = f(x) + 0\). As \(f(x) = \ln x\), the function \(g(x) = f(x) + 0\) is the same as the base function \(f(x) = \ln x\), so there is no change.

Step 4: Graphing for C=3

Lastly, sketch the graph of the function \(g(x) = f(x) + 3\). As \(f(x) = \ln x\), the function \(g(x) = f(x) + 3\) corresponds to moving the graph of the base function up by 3 units.

Key Concepts

Transformation of Functions

Understanding the transformation of functions is essential in graphing logarithmic functions as well as other types of functions. The process involves altering the appearance of a graph through various modifications, such as shifting, stretching, compressing, or reflecting. In our exercise, the function \( f(x) = \ln x \) undergoes a simple transformation known as a vertical shift. This alteration doesn't change the shape of the graph but rather moves it up or down on the coordinate plane.

When we add a constant C to the function, we apply a vertical shift in the direction dependent on the sign of C. If C is positive, the entire graph shifts upwards by C units, while a negative value of C will shift it downward by the absolute value of C units. This principle holds true for any parent function, not just logarithmic ones, making it a versatile and widely applicable concept in the world of algebraic and transcendental functions.

Vertical Shifts

A vertical shift is a specific type of transformation that moves a graph up or down along the y-axis without changing its shape or orientation. In our example, when we set C to -2, 0, or 3, we are applying vertical shifts to the function \( f(x) = \ln x \).

To effectively graph \( g(x) = f(x) + C \), we keep the x-values the same and add C to each corresponding y-value of the function. The graph of \( f(x) \) moves along the y-axis in accordance with C:

• For C = -2, each y-value is decreased by 2 units.
• For C = 0, the y-values remain unchanged, and so does the graph.
• For C = 3, each y-value is increased by 3 units.

The understanding of vertical shifts is key to quickly and successfully manipulating the position of graphs that represent mathematical functions.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a fundamental function in mathematics, particularly in calculus. It serves as the inverse to the exponential function \( e^x \), where e is the mathematical constant approximately equal to 2.71828. The graph of the natural logarithm has distinctive characteristics:

• It passes through the point (1, 0), meaning \( \ln 1 = 0 \).
• It has a vertical asymptote at x=0, indicating the values approach negative infinity as x gets closer to zero from the right.
• The function increases without bound as x increases, displaying a gradual, concave upward shape.

Understanding the behavior of the natural logarithm is crucial for various fields such as natural sciences, economics, and engineering, as it helps in modeling growth processes and calculating time decay, among other applications.

Asymptotes

An asymptote is a line that a graph approaches but never actually reaches. Asymptotes can be horizontal, vertical, or oblique. In the context of logarithmic functions, we predominantly deal with vertical asymptotes. For instance, the function \( f(x) = \ln x \) has a vertical asymptote at x=0.

This means that as x approaches zero from the right, the y-values of the function decrease without bound, making the graph get infinitesimally close to the y-axis without intersecting it. Asymptotes are important because they help us understand the behavior of functions at extreme values and are indicative of the limitations of certain functions in real-world contexts, like scenarios where a certain threshold cannot be crossed or a certain limit can never be reached.