Problem 388 - 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. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1.25} & {2.25} & { 3.56} & {4.2} & {5.65} & {6.75} & {7.25} & {8.6} & {9.25} & {10.5}\\\ \hline f(x) & {5.75} & {8.75} & {12.68} & {14.6} & {18.95} & {22.25} & {23.75} & {27.8} & {29.75} & {33.5}\\\ \hline\end{array}$$

Step-by-Step Solution

Answer

The data likely represents a linear function.

Step 1: Enter Data into Calculator

Begin by entering the data points into a graphing calculator. Use the list feature to input the pairs \((x, f(x))\) from the table: \((1.25, 5.75)\), \((2.25, 8.75)\), \((3.56, 12.68)\), etc., until you enter all points to \((10.5, 33.5)\). Ensure that each pair is correctly paired with the respective x and f(x) values.

Step 2: Create Scatter Plot

Once the data has been entered, utilize the graphing calculator’s function to create a scatter plot of the data. This will give a visual representation of the data points plotted on a coordinate plane.

Step 3: Analyze Scatter Plot for Pattern

Examine the scatter plot to identify the pattern of the points. Determine if the points exhibit a straight-line pattern (linear), an exponential curve (exponential), or a logarithmic curve (logarithmic). Observe if the points seem to increase by constant addition, multiplication, or by varying rates.

Step 4: Determine Function Type

Given that the data points are evenly spread and show a linear increase with a relatively constant rate of change, the points closely form a straight line suggesting that the data could represent a linear function rather than exponential or logarithmic functions.

Key Concepts

Linear Functions

Linear functions are an essential concept in mathematics, often used to model relationships where there is a constant rate of change between variables. In a linear function, every increase in one variable results in a consistent increase (or decrease) in the other variable.
When graphing a linear function, the result is a straight line. The general form of a linear equation is \( y = mx + b \), where:

• \( m \) is the slope, representing the rate of change
• \( b \) is the y-intercept, where the line crosses the y-axis

The slope \( m \) is calculated as the change in the y-values divided by the change in the x-values, known as \( \Delta y / \Delta x \).
Linear functions provide a simple and clear model for data that changes at a steady rate. This makes them useful in prediction and estimation in various fields.

Graphing Calculator

A graphing calculator is a powerful tool for visually representing mathematical data. It can graph functions, plot data points, and perform a vast array of computations, making it invaluable for data analysis.
To use a graphing calculator for scatter plotting, follow these steps:

• Enter data pairs (x, f(x)) into the calculator using list functions.
• Access the graphing feature and select the option to create a scatter plot.
• Adjust the settings for optimal viewing, such as zooming in on specific sections.

Graphing calculators can also be used to fit lines to data sets, helping determine the type of function that best describes the data. They provide a visual method to verify the linearity, exponential nature, or other functional forms of the data.

Data Analysis

Data analysis involves examining data sets to discern patterns and draw conclusions. It is a critical step in the mathematical process, especially when predicting future outcomes or understanding past trends.
The data analysis process for a scatter plot includes:

• Visual inspection to identify overall patterns or trends.
• Finding consistent rates of change to suggest a linear function.
• Noting any increasing, decreasing, or constant behaviors in the data.

In the context of the given exercise, data analysis focused on determining whether the data points aligned to a particular function type. Upon analyzing the scatter plot, the points suggested a linear relationship, as they spread out evenly and form a nearly straight line. By confirming a linear pattern, one can confidently use linear functions for further analysis and predictions.