Problem 46 - College Algebra

Question

For the following exercises, refer to Table 11. $$\begin{array}{ccccccccccc}{x} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ {f(x)} & {8.7} & {12.3} & {15.4} & {18.5} & {20.7} & {22.5} & {23.3} & {24} & {24.6} & {24.8}\end{array}$$ Use a graphing calculator to create a scatter diagram of the data.

Step-by-Step Solution

Answer

Use a graphing calculator's 'Stat Plot' feature to plot \( x \) and \( f(x) \) data as a scatter diagram with settings adjusted for window viewing.

Step 1: Understand the Data Table

The table presents two variables: \( x \) and \( f(x) \). The \( x \) values represent the independent variable ranging from 1 to 10, while \( f(x) \) represents the dependent values corresponding to each \( x \).

Step 2: Initialize Graphing Calculator

Turn on your graphing calculator and navigate to the 'Stat Plot' feature, which allows you to plot scatter diagrams.

Step 3: Enter Data into Lists

Enter the \( x \) values \((1, 2, 3, \, \ldots, \, 10)\) into List 1 (often labeled as L1) and the corresponding \( f(x) \) values \((8.7, 12.3, 15.4, \, \ldots, \, 24.8)\) into List 2 (L2) of the calculator.

Step 4: Set Up the Scatter Plot

Go to 'Stat Plot' and select 'Plot 1'. Ensure that it is turned on and set the type to 'Scatterplot'. Set L1 as the Xlist and L2 as the Ylist. Choose a mark or style for your scatter points.

Step 5: Adjust Window Settings

Modify the 'Window' settings to ensure all your data points fit into the view. Set the Xmin to 1, Xmax to 10, and adjust Ymin and Ymax slightly below your smallest value, 8.7, and above your largest value, 24.8.

Step 6: Graph and Observe

Press the 'Graph' button to display the scatter plot. Observe the pattern or trend of the plotted points to analyze how \( f(x) \) changes as \( x \) increases.

Key Concepts

Independent Variable

In mathematics and data analysis, an independent variable is a variable that stands alone and isn't changed by the other variables you are trying to measure. In the context of a scatter plot, the independent variable is typically plotted along the x-axis. For the table provided in the exercise, the independent variable is represented by the variable \(x\), which ranges from 1 to 10.

The independent variable is crucial because it is often the variable you control or adjust in experiments or data collection. By altering the independent variable, you can observe how other variables react or change—these are called dependent variables. Understanding the role of the independent variable helps you predict and assess patterns within a set of data.

Dependent Variable

The dependent variable is what you measure in the experiment and is affected during the experiment when you change the independent variable. In the scatter plot context, it’s typically plotted along the y-axis.

In the exercise provided, \(f(x)\) acts as the dependent variable. Its values, which range from 8.7 to 24.8, depend on the respective values of \(x\). As \(x\) increases from 1 to 10, you can observe changes in \(f(x)\). This setup allows you to explore the relationship between the two variables, helping to answer questions like "How does \(f(x)\) change when \(x\) changes?"

• Helps identify trends.
• Shows cause-and-effect relationships.

The dependent variable lets you see the effect of altering the independent variable in the context of real-world data analysis.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing data and mathematical functions. It provides a visual representation of data points, helping students and researchers to make sense of complex data sets easily.

To create a scatter plot of the data given in the exercise, follow these steps:

• Enter independent variable values \(x\) (1 through 10) into List 1.
• Enter dependent variable values \(f(x)\) (8.7 through 24.8) into List 2.
• Use the 'Stat Plot' feature to generate a scatter plot where List 1 is used as the x-axis data (\(x\)), and List 2 is used as the y-axis data (\(f(x)\)).
• Adjust viewing settings to ensure the graph covers all data points adequately.

Graphing calculators can plot functions, calculate statistics, and perform other complex computations, making them an essential tool for students working on data-related tasks.

Data Analysis

Data analysis involves examining and processing data to uncover useful information, draw conclusions, and support decision-making. Scatter plots, like the one described in the exercise, are graphical representations that help in visual data analysis by displaying relationships between two variables.

By analyzing the scatter plot created from the data sets of \(x\) and \(f(x)\), you can identify:

• The overall trend of the data points (e.g., linear, non-linear).
• Correlations between variables.
• Outliers or unusual observations.

Understanding these patterns is essential in fields like economics, biology, and engineering, where interpreting data sets can lead to groundbreaking insights and solutions. The scatter plot allows you to visually assess these data characteristics quickly and effectively.