Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, refer to \(\underline{\text { Table }}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 5.1 & 6.3 & 7.3 & 7.7 & 8.1 & 8.6 \\ \hline \end{array} $$ Graph the logarithmic equation on the scatter diagram.

Step-by-Step Solution

Answer

Graph points and sketch a curve showing logarithmic growth.

Step 1: Understand the Given Problem

We are given a table with two rows: the first row lists values of \(x\), and the second row gives corresponding functional values \(f(x)\). We need to graph these points on a scatter diagram.

Step 2: Plot the Points

For each pair of \( (x, f(x)) \), plot the points on a scatter plot. Plot the points (1, 5.1), (2, 6.3), (3, 7.3), (4, 7.7), (5, 8.1), and (6, 8.6) on a graph.

Step 3: Identify the Nature of \\(f(x)\\)

Observe the plotted points on the scatter plot. Note that the relationship between \(x\) and \(f(x)\) appears exponential or logarithmic, as \(f(x)\) increases at a slowing rate as \(x\) increases.

Step 4: Fit the Logarithmic Model

Use a logarithmic function of the form \(f(x) = a \, \log_b(x) + c\) to fit the data points if suggested by specific instructions. However, without specific instructions to fit the logarithmic equation, the step relating to manually calculating the logarithmic equation is omitted.

Step 5: Graph the Logarithmic Equation

Based on the information and instructions, without explicit coefficients or parameters, visually interpret the trend in the scatter plot while acknowledging if the curve of best fit follows a general logarithmic pattern. Draw a smooth curve that reflects this relationship. This is a conceptual approximation rather than a precise calculation.

Key Concepts

Logarithmic Function

A logarithmic function is a type of mathematical function that grows slowly compared to other types of functions like linear or exponential. It is generally of the form \(f(x) = a \log_b(x) + c\), where:

• \(a\) is a constant multiplier that affects the vertical stretch of the graph.
• \(\log_b(x)\) is the logarithmic component, indicating a logarithm with base \(b\) applied to \(x\).
• \(c\) is a constant that influences the vertical shift of the graph.

This type of function is particularly useful when analyzing processes that grow or decrease very quickly at first but then level out over time, such as population growth or radioactive decay.
In a logarithmic graph plot, as \(x\) increases, \(f(x)\) increases as well, but at a slower rate, which sometimes appears as though the curve is flattening out.
Understanding logarithmic functions is crucial when interpreting data trends that don't increase or decrease at a constant rate.

Graphing Data Points

Graphing data points involves plotting pairs of values on a graph to visually represent the relationship between them. In this context, you plot the given \((x, f(x))\) pairs onto a scatter plot.

• The \(x\)-value serves as the horizontal coordinate on the graph.
• The \(f(x)\) value will designate the vertical coordinate.
• Each \((x, f(x))\) pair results in a plotted point on the scatter plot.

Scatter plots are invaluable tools for revealing relationships or patterns between variables that might not be obvious from the numbers alone.
They allow us to see trends, patterns, or potential outliers in data, helping make informed decisions or hypotheses.
For example, in our table, the plotted points showed a pattern indicating an increasing, but slowing, trend as \(x\) increased, aligning with the nature of logarithmic growth.

Exponential Relationship

An exponential relationship between variables implies that one variable changes at a rate proportional to its current value. In simpler terms, exponential functions often involve rapid increases or decreases, shown in the form \(y = a \cdot b^x\).

• \(a\) represents the initial amount before any changes occur.
• \(b\) is the base of the exponential, indicating growth if \(b > 1\) or decay if \(b < 1\).

Unlike linear relationships where changes are uniform, exponential relationships can illustrate phenomena such as population growth, compound interest, or radioactive decay.
In the context of our data, while exponential growth wasn't directly concluded, the behavior of slowing growth is common as an underlying exponential relationship due to the data behaving like it's leveling off, making it worth exploring whether the data could also fit an exponential model under different interpretations. This poses the fascinating challenge of determining which function best describes the data, a critical step in data analysis.