Problem 51 - A Graphical Approach to College Algebra

Question

Set the viewing window of your calculator to the given specifications. Make a sketch of your window. $$\begin{aligned} &[-100,100] \text { by }[-50,50]\\\ &\mathrm{Xscl}=20 \quad \mathrm{Yscl}=25 \end{aligned}$$

Step-by-Step Solution

Answer

Set window: X from -100 to 100, Y from -50 to 50, Xscl 20, Yscl 25; sketch with tick marks accordingly.

Step 1: Understanding the Viewing Window

The viewing window is the range of values displayed on your calculator screen for both the x-axis and the y-axis. Here, you need to set the x-axis from -100 to 100 and the y-axis from -50 to 50.

Step 2: Setting the X-axis Range

On your calculator, enter the 'WINDOW' settings. Set the Xmin to -100 and the Xmax to 100. This defines the horizontal view of your graph between these two numbers.

Step 3: Setting the Y-axis Range

While still in the 'WINDOW' settings, set the Ymin to -50 and the Ymax to 50. This defines the vertical view of your graph between these two values.

Step 4: Configuring the X-Scale (Xscl)

In the 'WINDOW' settings, set the Xscl to 20. This means that the tick marks on the x-axis will appear every 20 units.

Step 5: Configuring the Y-Scale (Yscl)

Similarly, set the Yscl to 25. This sets the spacing between tick marks on the y-axis to 25 units, which provides a consistent measure for your graph.

Step 6: Sketching the Viewing Window

Draw a rectangle to represent the calculator screen. Label the x-axis from -100 to 100 and the y-axis from -50 to 50. Mark the x-axis tick marks every 20 units, and the y-axis tick marks every 25 units.

Key Concepts

Viewing Window

The viewing window on a graphing calculator is like a frame through which you see your graph. Imagine it as the screen where your graph appears. Setting the right viewing window ensures that you see all the important parts of the graph. You adjust the viewing window by setting limits for both the x-axis and the y-axis.

• The **x-axis range** is the horizontal span, which is specified as Xmin and Xmax.
• The **y-axis range** is the vertical span, given by Ymin and Ymax.
• Within the window, logical increments for marks are set by the X-scale and Y-scale.

By adjusting these ranges, you can zoom in to capture details or zoom out to view the broader trend. You can think of it like adjusting a zoom lens on a camera, capturing everything from close-up details to wide, sweeping landscapes.

X-axis and Y-axis Range

The x-axis and y-axis ranges determine how wide and tall your graph will appear on the screen. Setting these ranges correctly is crucial for displaying the right section of the graph you want to analyze.

- **X-axis Range:** When you set the Xmin to -100 and Xmax to 100, you are defining the boundaries of your graph. The x-axis will display values starting from -100 and going up to 100. This wide range lets you see both the negative and positive portions of the graph.
- **Y-axis Range:** Similarly, setting the Ymin to -50 and Ymax to 50 establishes the top and bottom edges of your graph. This range shows values from -50 to 50 on the y-axis. Having these values in mind helps in displaying a balanced view of the graph.

These ranges are key to centering your graph and making sure all relevant data points are visible. Always consider what parts of the graph you need before choosing these values.

X-Scale and Y-Scale

The X-scale and Y-scale settings are like rulers for each axis, marking equal intervals on the x and y axes. These scales help you interpret the graph more easily by giving consistent tick marks.

- **X-Scale (Xscl):** With an Xscl of 20, tick marks will appear every 20 units on the x-axis. This means that space between each tick mark represents 20 units, simplifying the reading of graph data points along the horizontal line.
- **Y-Scale (Yscl):** When the Yscl is set to 25, it indicates each tick mark on the y-axis is 25 units apart. This regular spacing on the vertical line helps in quickly understanding the changes in values.

Choosing the right scale depends on what you want to analyze in the graph. The scale should match the level of detail you need: finer scales for detailed analyses or broader ones for general patterns. Adjust it to suit your study topic, ensuring that every significant point is easy to discern without cluttering the graph.