Problem 456 - Precalculus

Question

For the following exercises, refer to Table 4.27. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {555} & {383} & {307} & {210} & {158} & {122} \\\ \hline\end{array}$$ Use a graphing calculator to create a scatter diagram of the data.

Step-by-Step Solution

Answer

Enter the data into a graphing calculator, activate a scatter plot, adjust the viewing window, and graph.

Step 1: Understand the Data Table

The table consists of two rows. The first row represents the variable \( x \) with values \([1, 2, 3, 4, 5, 6]\). The second row, \( f(x) \), contains corresponding function values \([555, 383, 307, 210, 158, 122]\). This data will be plotted on a scatter plot.

Step 2: Set Up the Graphing Calculator

Turn on your graphing calculator. Go to the 'Stat' section and select 'Edit' to enter the data points. Input the \( x \)-values into \( L_1 \) and the \( f(x) \)-values into \( L_2 \).

Step 3: Input Data into Lists

In \( L_1 \), input \( 1, 2, 3, 4, 5, 6 \). In \( L_2 \), input \( 555, 383, 307, 210, 158, 122 \). This organizes the data for plotting.

Step 4: Configure Plot Settings

Press '2nd' and 'Y=' (Stat Plot menu) to access plot settings. Choose 'Plot 1' and turn it on. Set it to a scatter plot, with \( L_1 \) for \( Xlist \) and \( L_2 \) for \( Ylist \).

Step 5: Adjust the Viewing Window

Press 'Window' to adjust the axes so all points are visible. For \( x \), set the range a bit wider than the data points, like \( [0, 7] \). For \( f(x) \), set \( [100, 600] \) to capture all output values.

Step 6: Display the Scatter Plot

Press 'Graph' to display the scatter diagram. The points should appear on the screen, showing the relationship between \( x \) and \( f(x) \).

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that allows you to visualize mathematical equations and data sets. Unlike basic calculators, graphing calculators can display graphs and are essential for plotting more complex functions and datasets like the one in the exercise. To begin, make sure your graphing calculator is properly functioning and know how to access its 'Stat' section, which is crucial for entering and manipulating data. You'll be making use of several features within the calculator to graph a scatter plot, so familiarizing yourself with its different keys and options, found typically across models from brands like TI or Casio, will greatly streamline this process.

Data Table

The data table in this exercise is the foundation of your scatter plot. It organizes your data into two rows:

• The first row represents the values of your independent variable, in this case, the variable \( x \), with values \( [1, 2, 3, 4, 5, 6] \).
• The second row contains corresponding values of your dependent function, \( f(x) \), which are \( [555, 383, 307, 210, 158, 122] \).

To create a scatter plot, these pairs of values will be plotted as individual points, with \( x \) values indicating horizontal positions and \( f(x) \) values determining vertical positions. This visualization helps to easily identify patterns or relationships in the data.

Plot Settings

Configuring plot settings correctly is essential for a successful scatter plot. After entering your data into the graphing calculator:

• Navigate to the 'Stat Plot' menu, usually found by pressing '2nd' and then 'Y=' on most calculators.
• Select a plot (typically 'Plot 1') and ensure it's turned on.
• Set the plot type to scatter plot, an option that allows you to see individual points without connecting lines. This is suitable when examining discrete data sets from a data table.
• Assign your Lists; typically, \( L_1 \) is used for \( Xlist \) (where your \( x \)-values are entered) and \( L_2 \) for \( Ylist \) (where your \( f(x) \)-values are entered).

This setup ensures that each pair of \( x \) and \( f(x) \) values are properly aligned and ready to be displayed on the graph.

Viewing Window Adjustment

Ensuring the proper viewing window on your graphing calculator is key to seeing all of your plotted points clearly.

• Press 'Window' on your calculator, which lets you adjust the display settings for your graph's axes.
• For this exercise, given the \( x \)-values range from 1 to 6, set your \( x \min \) to slightly below your smallest value, such as 0, and \( x \max \) slightly above your largest value, such as 7.
• Similarly, use \( f(x) \)-values ranging from 122 to 555 to guide your setting of \( y \min \) at 100 and \( y \max \) at 600.

These adjustments create a buffer around your data points, ensuring they aren't placed too close to the edge of your screen, thus preserving clarity. Once set, pressing 'Graph' will display your scatter plot within these bounds, making sure that all relevant points are visible for analysis.