Problem 386 - Precalculus

Question

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10}\\\ \hline f(x) & {2.4} & {2.88} & {3.456} & {4.147} & {4.977} & {5.972} & {7.166} & {8.6} & {10.32} & { 12.383}\\\ \hline\end{array}$$

Step-by-Step Solution

Answer

The data follows an exponential pattern.

Step 1: Plot the points

Enter the given pairs of \(x\) and \(f(x)\) values into a graphing calculator. Plot these points on a scatter plot to visually inspect the nature of the data.

Step 2: Analyze the data trend

Upon graphing, observe the shape and pattern of the scatter plot. Look for patterns or curves that could indicate a particular type of relationship, such as a linear (straight line), exponential (rapidly increasing or decreasing curve), or logarithmic (curve that rises quickly and then levels off) fit.

Step 3: Check for linearity

Calculate the differences between successive \(f(x)\) values to determine if the sequence is arithmetic, hinting at a linear relationship: \(\Delta f(x) = f(x+1) - f(x)\). If the differences are constant, the data is linear.

Step 4: Analyze exponential relationship

Compute the ratio \(\frac{f(x+1)}{f(x)}\) for successive terms to see if this ratio is constant or approximately constant, which would suggest an exponential function. For example: the first ratio is \(\frac{2.88}{2.4} \approx 1.2\). If ratios are roughly constant, the relationship might be exponential.

Step 5: Verify against a logarithmic model

Testing for a logarithmic pattern typically involves seeing whether \(f(x)\) values increase rapidly but then stabilize. In this sequence, check whether the growth rate decreases with increasing \(x\). However, if an exponential pattern is already indicated, this step is likely unnecessary.

Step 6: Conclusion

The calculated ratios of successive \(f(x)\) values are roughly constant, suggesting an exponential function fits the data best. The ratios were close to \(1.2\), reinforcing this conclusion.

Key Concepts

graphing calculator

A graphing calculator is a powerful tool that can help represent data visually and analyze mathematical functions. When dealing with tables like the one given in the exercise, a graphing calculator allows you to input the values of \( x \) and \( f(x) \) and plot them on a graph. This visualization, called a scatter plot, helps in looking for patterns in the data.

Using a graphing calculator involves a few simple steps:

• Enter the data points from the table into the calculator’s data entry interface.
• Choose the plot option to graph the points on a grid.
• Use the graphing feature to fit different models, such as linear or exponential, by selecting the appropriate function type.

Graphing calculators make analyzing complex data easier, providing a clear visual indication of trends, which is fundamental in identifying the nature of relationships in data sets. They also offer tools to fit curves, calculate ratios, and differentiate among types of functions.

scatter plot

A scatter plot is a type of data visualization that displays values for typically two variables for a set of data. In this context, it shows the relationship between \( x \) values and their associated \( f(x) \) values. The plotted points reveal the overall trend the data follows, aiding in the understanding of the function type.

The scatter plot gives a quick visual summary and can indicate whether the trend is:

• Linear - forming a straight line, suggesting a linear function
• Exponential - showing rapid increases or decreases, characteristic of exponential functions
• Logarithmic - where points swiftly rise then level out

After plotting, the exercise indicates analyzing these patterns to hypothesize what kind of mathematical model the data might represent. The visual pattern helps determine whether further mathematical calculations are necessary to solidify hypotheses about the type of function at play.

arithmetic sequences

Arithmetic sequences are sequences of numbers with a common difference between consecutive terms. When checking for arithmetic sequences in functions, you analyze whether differences between consecutive \( f(x) \) values are constant.

In the given exercise:

• Calculate the difference \( \Delta f(x) = f(x+1) - f(x) \).
• If these differences remain constant across the data set, the sequence is arithmetic and the pattern suggests a linear relationship.

In our specific exercise, checking these differences helps determine if the data is linear. If all \( \Delta f(x) \) are the same, then the sequence confirms a linear function. However, when differences aren't constant, further analyses, like checking ratios, are needed to determine if another function type fits better.

analyzing data trends

Analyzing data trends involves looking at how data points relate to one another and determining the nature of their relationship. From a data-driven perspective, this exercise involves examining whether the data conforms to a recognizable mathematical model.

Key components include:

• Visual inspection - using a scatter plot for a preliminary understanding.
• Mathematical calculations - compute sequential differences and ratios.
• Comparisons - between calculated values to look for constants or patterns.

The choice to pursue a linear, exponential, or logarithmic model arises from these steps. For instance, in this exercise, the constant ratio between successive \( f(x) \) values, observed to be around \( 1.2 \), points to an exponential relationship. Recognizing such a ratio suggests the data increases multiplicatively, which is a hallmark of exponential functions. Analyzing data trends thus plays a crucial role in successfully identifying the most fitting mathematical model.