Problem 51 - College Algebra

Question

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

Step-by-Step Solution

Answer

Enter data, set up the scatter plot, adjust window settings, and view the graph.

Step 1: Enter Data into Calculator

Begin by turning on your graphing calculator. Navigate to the 'STAT' menu and choose 'EDIT'. In the L1 column, enter the x-values: 0, 2, 4, 5, 7, 8, 10, 11, 15, and 17. In the L2 column, enter the corresponding \( f(x) \) values: 12, 28.6, 52.8, 70.3, 99.9, 112.5, 125.8, 127.9, 135.1, and 135.9.

Step 2: Set Up Scatter Plot

Press the '2nd' button followed by 'Y=' to access the 'STAT PLOT' menu. Select 'Plot1' and choose 'ON'. Select the scatter plot icon (the first option), then set 'Xlist' to L1 and 'Ylist' to L2. This tells the calculator to use L1 as x-values and L2 as y-values.

Step 3: Adjust Window Settings

Press the 'WINDOW' button. Set an appropriate window for your data: Xmin = -1, Xmax = 20, Ymin = 0, and Ymax = 150. Adjusting the window ensures that all data points will be visible in the graph.

Step 4: Plot the Data

Press the 'GRAPH' button to view the scatter plot. The plotted points will appear on the screen, visually representing the relationship between \( x \) and \( f(x) \). Ensure all points display correctly in the chosen window range.

Key Concepts

Using a Graphing Calculator

A graphing calculator is a powerful tool used to visualize mathematical data quickly and easily. It's particularly useful for subjects like algebra and calculus. When you use a graphing calculator for a scatter plot, you can see how data points relate to each other visually. This is helpful when trying to understand patterns among variables. To begin, you should first enter all of your data into the calculator. In this case, it involves inputting the given set of \( x \) coordinates and their corresponding \( f(x) \) values into the calculator's lists, typically labeled L1 for \( x \) and L2 for \( f(x) \).

• Turn on your calculator and find the 'STAT' button to enter the edit mode.
• Enter the x-values into L1 and y-values into L2.

Switch to the 'STAT PLOT' menu to activate a plot, selecting a scatter plot type. This process ties your data entries to the visualization function of the graphing calculator.

Visualizing Data with Scatter Plots

Data visualization is a method of representing numerical data in a visual context, such as a scatter plot. Scatter plots are particularly beneficial when examining relationships and trends within data sets. They provide immediate visual insight, allowing us to detect correlation, clusters, or outliers. In a scatter plot:

• Each pair of \( (x, f(x)) \) values is plotted as a point on a Cartesian plane.
• The positioning reveals any apparent patterns or trends in the data.

This technique turns raw numbers into something more accessible, enabling a deeper understanding of what the numbers signify. In the calculator, each point gets plotted as guided by the L1 and L2 list entries, providing a clear picture of the data's distribution as you press the 'GRAPH' button. Adjusting the view settings ensures that each point is clearly visible and correctly positioned on the graph.

Understanding Function Values

Function values are the outputs you get when you input specific numbers into a function. They exemplify how the function behaves over different inputs. In this example, the \( f(x) \) values reflect the output for each \( x \) value entered. This output is crucial because:

• It determines the position of the points on the scatter plot.
• It tells us how the function progresses and changes.

Analyzing these values can offer insights into the rate and direction of the progression of a function. For instance, by observing how the function values increase as the x-values increase, one can infer the nature of the function's growth. The scatter plot helps visualize this behavior by showing how concentrated or dispersed the points are, hinting at the possible mathematical relationship governing the function.