Problem 18 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, sketch the graph of the function. $$ y=\frac{1}{4} \ln x $$

Step-by-Step Solution

Answer

The graph of the function \( y=\frac{1}{4} \ln x \) passes through the points (1,0), (e,1/4), and (e^4,1). It rises more slowly than the graph of \( y = \ln x \) due to the factor of \( \frac{1}{4} \). The curve approaches the y-axis but never intersects it, reflecting the fact that the function is undefined for \( x \leq 0 \).

Step 1: Identify the Domain

The domain of the function \( y=\frac{1}{4} \ln x \) is defined as all real numbers \( x \) such that \( x>0 \), because the natural logarithm function is undefined for \( x \leq 0 \).

Step 2: Identify points for plotting

Choose a few points within the domain of x to plot. Consider values such as x=1, x=e, and x=e^4, as they produce nicely rounded values when plugged into the natural logarithm function. For instance, when x=1, \( y=\frac{1}{4} \ln 1 = 0 \).When x=e, \( y=\frac{1}{4} \ln e = \frac{1}{4} \), and when x=e^4, \( y=\frac{1}{4} \ln e^4 = 1 \). These points can be mapped on the graph.

Step 3: Draw the graph

Plot the selected points on the graph, and draw the curve of the function which passes through these points. Since the coefficient of \( \ln x \) is \( \frac{1}{4} \), the graph will rise more slowly than the graph of \( y = \ln x \). The curve will approach the y-axis but never intersect it, as the function is undefined for \( x \leq 0 \).

Key Concepts

Natural Logarithm

The natural logarithm is a mathematical function denoted by \( \ln x \). It is the inverse of the exponential function \( e^x \), where \( e \) is approximately equal to 2.71828. The natural logarithm has distinct properties:

• \( \ln 1 = 0 \)
• \( \ln e = 1 \)
• \( \ln(ab) = \ln a + \ln b \)
• \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• \( \ln(a^b) = b\ln a \)

These properties are useful when solving equations and when graphing the function. Since \( \ln x \) is undefined for \( x \leq 0 \), its domain is limited to positive real numbers. In the given function, \( y = \frac{1}{4}\ln x \), the graph has a vertical stretch/compression based on the coefficient \( \frac{1}{4} \). This means the natural logarithm's rate of increase is reduced by a factor of four.

Domain and Range

The domain of a function is the complete set of possible input values (x-values) for the function, while the range is the complete set of possible output values (y-values).
For the function \( y = \frac{1}{4} \ln x \), its domain is all real numbers greater than zero. This is because logarithmic functions, especially the natural logarithm, cannot have zero or negative numbers as inputs.

• Domain: \( (0, \infty) \)

The range of \( y = \frac{1}{4} \ln x \) includes all real numbers. As \( x \) increases from above zero, \( y \) can take any value from negative infinity to positive infinity. The graph will rise but at a decreasing rate due to the \( \frac{1}{4} \) factor, indicating a slower growth than the standard logarithm function.

Function Graphing

Graphing a function involves plotting points and drawing a curve through those points to represent the function visually. For \( y = \frac{1}{4} \ln x \), we start by selecting specific x-values that are easy to compute with the logarithm.

• When \( x = 1 \), \( y = \frac{1}{4} \ln 1 = 0 \).
• When \( x = e \), \( y = \frac{1}{4} \ln e = \frac{1}{4} \).
• When \( x = e^4 \), \( y = \frac{1}{4} \ln e^4 = 1 \).

These calculated points are plotted on a graph, ensuring they reflect the function correctly. The smooth curve connecting these points approaches the y-axis and increases slowly due to the factor \( \frac{1}{4} \), indicating stretched growth compared to \( \ln x \). By understanding these points and the curve, we can better visualize how the natural logarithm behaves with different modifications.

Mathematical Plotting

Mathematical plotting refers to the process of drawing graphs using points that reflect the behavior of a mathematical function. For \( y = \frac{1}{4} \ln x \), plotting helps in visually understanding how variations in the logarithmic function change its shape and growth.
In plotting:

• First, choose relevant x-values that are easy to compute.
• Second, calculate the corresponding y-values using the function rule.
• Third, plot these (x, y) coordinates on a graph.
• Finally, draw a smooth curve through the points.

The plotted graph of \( y = \frac{1}{4} \ln x \) reveals how the curve approaches the y-axis asymptotically and stretches horizontally. Visual plotting is crucial as it provides intuition about the function's behavior, domain, and range.