Problem 48 - A Graphical Approach to College Algebra

Question

Set the viewing window of your calculator to the given specifications. Make a sketch of your window. $$\begin{aligned} &[-40,40] \text { by }[-30,30]\\\ &\mathrm{Xscl}=5 \quad \mathrm{Yscl}=5 \end{aligned}$$

Step-by-Step Solution

Answer

The viewing window is set to x: -40 to 40, y: -30 to 30, with a scale of 5 for both axes.

Step 1: Identify Window Parameters

First, we need to identify the parameters given for the viewing window. From the exercise, the window for the x-axis, known as Xmin and Xmax, is from -40 to 40. Similarly, for the y-axis, Ymin and Ymax are from -30 to 30. The scaling factors for the x and y axes (Xscl and Yscl) are both 5.

Step 2: Set X-axis Window

Set the calculator's Xmin to -40 and Xmax to 40, matching the given window specifications for the x-axis. This sets the range of values that will be displayed on the x-axis of the graph.

Step 3: Set Y-axis Window

Set the calculator's Ymin to -30 and Ymax to 30 according to the given window specifications for the y-axis. This sets the range of values that will be displayed on the y-axis of the graph.

Step 4: Set Scale for Axes

Next, set the scale for both axes. This involves setting Xscl (the scale for the x-axis) to 5 and Yscl (the scale for the y-axis) to 5. This defines the intervals at which tick marks will be displayed along the axes.

Step 5: Sketch the Graph Window

Draw a sketch of the viewing window on graph paper or a drawing tool. The x-axis should range from -40 to 40 with tick marks every 5 units. The y-axis should range from -30 to 30 with tick marks every 5 units. Clearly label the axes and the tick marks to accurately represent the window settings.

Key Concepts

Viewing Window

A viewing window on a graphing calculator is the virtual frame through which you view your graph. It determines the visible portion of the graph by setting the range of values for both the x and y axes. Just like an actual window allows you to see only a specific part of the outside world, a viewing window on a calculator allows you to focus on a specific part of the graph.
Setting the viewing window correctly is crucial because it ensures that you capture all relevant features of the graph, like intercepts, peaks, and troughs. A well-set window prevents parts of the graph from getting cut off or, conversely, from seeing an excessively zoomed-out image where no details can be discerned.
In this example, the window is set to

• Xmin: -40 and Xmax: 40 for a total span of 80 units on the x-axis
• Ymin: -30 and Ymax: 30, spanning 60 units on the y-axis

These settings allow you to see a comprehensive part of the graph, facilitating better analysis and interpretation.

Scale Settings

Scale settings in a graphing calculator determine the intervals at which tick marks are placed along the axes. These settings help in reading and interpreting the graph by providing reference points.
In our example, both the x-axis and y-axis scales (denoted as Xscl and Yscl) are set at 5. This means every tick mark on both axes represents a gap of 5 units.

• On the x-axis, tick marks will appear at -40, -35, -30, ..., 35, and 40.
• On the y-axis, tick marks will show at -30, -25, -20, ..., 25, and 30.

The choice of scale can affect how the graph looks. A smaller scale shows more detail but may make the graph too cluttered, while a larger scale can simplify the graph but might skip over important details. Setting a consistent and appropriate scale is essential for clarity and easy understanding of the graph.

Graph Sketching

Graph sketching involves drawing a rough version of your graph based on the data and settings defined in your calculator's viewing window and scale settings. It is a visual representation of the function or data set within the designated window.
To start, draw horizontal and vertical lines to represent the x and y axes, respectively. Then, based on the viewing window settings

• draw the x-axis covering values from -40 to 40
• and the y-axis spanning from -30 to 30

Make sure to mark the tick marks every 5 units as per the scale settings.
Label these tick marks clearly, which aids in reading the graph accurately. The sketch doesn't have to be perfect or to scale but should correctly represent the position and scale of your graph.

Axis Settings

Axis settings involve configuring the limits and scale of the x and y axes on your graphing calculator. These settings play a foundational role in how your graph is displayed.

• The limits (Xmin, Xmax, Ymin, Ymax) define the window or the range of the graph visible on the screen.
• The scale (Xscl and Yscl) determines the spacing of tick marks along the axes.

Correctly setting the axis limits ensures the entire graph fits the viewing plane, preventing any part of it from being cut off. In our context, the axis settings let you visualize a comprehensive range. The right settings make it easier to analyze and interpret the graph, especially for understanding behaviors at various points within the given domain and range.