Problem 51 - College Algebra

Question

For the following exercises, refer to Table 12. $$\begin{array}{ccccccccccc}{x} & {0} & {2} & {4} & {5} & {7} & {8} & {10} & {11} & {15} & {17} \\ {f(x)} & {12} & {28.6} & {52.8} & {70.3} & {99.9} & {112.5} & {125.8} & {127.9} & {135.1} & {135.9}\end{array}$$ Use a graphing calculator to create a scatter diagram of the data.

Step-by-Step Solution

Answer

Use your calculator: Enter data, setup scatter plot, configure axes, then graph.

Step 1: Enter Data into the Calculator

First, access your graphing calculator and find the function or mode that allows you to enter data. This is typically found under the 'STAT' menu, followed by 'EDIT.' Enter the x-values (0, 2, 4, 5, 7, 8, 10, 11, 15, 17) into List 1, and the corresponding f(x) values (12, 28.6, 52.8, 70.3, 99.9, 112.5, 125.8, 127.9, 135.1, 135.9) into List 2.

Step 2: Set up the Scatter Plot

After entering the data, press the '2nd' key and then 'Y=' to access 'STAT PLOT.' Choose Plot 1 and press 'ENTER.' Turn the plot 'ON.' Select the scatter plot option, which is typically indicated by a small graph with unconnected points (dots).

Step 3: Configure the Axes

To ensure the scatter plot is correctly displayed, adjust the window settings. Press the 'WINDOW' button and set suitable minimum and maximum values for both x and y (for example, Xmin = 0, Xmax = 17, Ymin = 10, Ymax = 140), making sure all data points are visible.

Step 4: Graph the Data

Press the 'GRAPH' button to display the scatter plot. The calculator will plot the points based on the lists you entered. Check that all the points appear according to Table 12's data.

Key Concepts

Understanding Scatter Plots

A scatter plot is a powerful tool for analyzing relationships between two numerical variables. In this exercise, we're working with two sets of data: our x-values and corresponding function values, \(f(x)\). A scatter plot provides a visual representation of how one variable behaves in relation to another. Each pair of x and \(f(x)\) values is represented as a single point in a two-dimensional space.
This helps us see patterns, correlations, or even deviations within the data.

• The x-value is plotted along the horizontal axis.
• The \(f(x)\) value is plotted along the vertical axis.

It is especially useful for identifying linear relationships or clusters. By examining the shape and distribution of the data points, we can make inferences about the underlying trend or pattern between the variables.

Accurate Data Entry

Correctly entering data into the graphing calculator is crucial for creating a reliable scatter plot. This often involves using the 'STAT' menu on many calculators.

• Select 'EDIT' to open data lists for input.
• Place your x-values in the first column (List 1) and your \(f(x)\) values in the second column (List 2).

Ensure that the data entry is precise and double-check your lists for errors. Incorrect data entries can lead to inaccurate graphs, which will misrepresent the actual relationship between your variables. Attention to detail during this step ensures a solid foundation for the rest of the plotting process.

Adjusting Window Settings

Once the data is entered, it's essential to adjust your graphing calculator's window settings to view all data points clearly. The 'WINDOW' settings help define the range of both axes to accommodate all values.

• Set the x-axis minimum and maximum (Xmin and Xmax) based on your smallest and largest x-values.
• Set the y-axis minimum and maximum (Ymin and Ymax) based on your smallest and largest \(f(x)\) values.

For our table of values, settings might look like Xmin = 0, Xmax = 17, Ymin = 10, Ymax = 140. Correct window settings ensure that the entire scatter plot is visible and that data points are not crowded or overlapping. This step allows for a better visual analysis and understanding of the data.

Creating the Scatter Diagram

Creating a scatter diagram on a graphing calculator is the culmination of your data entry and window adjustments.

• Access the 'STAT PLOT' menu by pressing '2nd' followed by 'Y='.
• Select 'Plot 1', turn it 'ON', and choose the scatter plot type.
• Once everything is set, press 'GRAPH' to display the scatter plot.

As you view the plotted points, verify that they accurately represent the given data table. The scatter plot should illuminate any trends or patterns between the x-values and their corresponding \(f(x)\) values. This visual representation is key to interpreting the relationship within your data set.