Problem 81 - College Algebra with Modeling and Visualization

Question

Predict the number of tick marks on the positive \(x\) -axis and the positive y-axis. Then show the viewing rectangle on your graphing calculator. $$ [0,100,10] \text { by }[-50,50,10] $$

Step-by-Step Solution

Answer

11 tick marks on both x-axis and y-axis.

Step 1: Understand the Viewing Window

The notation \([0,100,10]\) by \([-50,50,10]\) represents the viewing window settings on a graphing calculator. \([0,100,10]\) indicates the range, starting value, and scale for the x-axis, while \([-50,50,10]\) indicates the same for the y-axis.

Step 2: Calculate Tick Marks on the X-Axis

For the x-axis, the viewing window is from 0 to 100 with a scale of 10. To find the number of tick marks, note the size of each tick, which is 10 units. The number of tick marks is calculated as \((100 - 0) / 10 + 1 = 11\). Therefore, there are 11 tick marks on the x-axis including both ends of the interval.

Step 3: Calculate Tick Marks on the Y-Axis

For the y-axis, the viewing window is from -50 to 50 with a scale of 10. The size of each tick is also 10 units. To find the number of tick marks on the y-axis, calculate \((50 - (-50)) / 10 + 1 = 11\). Therefore, there are 11 tick marks on the y-axis including both ends of the interval.

Step 4: Confirm the Viewing Rectangle

On your graphing calculator, set the window to \([0,100,10]\) for the x-axis and \([-50,50,10]\) for the y-axis. Ensure that every tick corresponds to an increment of 10 units from the start to the end of the specified range.

Key Concepts

Viewing Window

When using a graphing calculator, the viewing window is a crucial aspect that determines what portion of the graph you can see. It is essentially a rectangular region on the plane where the graph is displayed. The viewing window is defined by its x-axis and y-axis parameters, which include:- **Start value**: The lowest value of the axis that is visible.- **End value**: The highest value of the axis that is visible.- **Scale**: The increment between visible tick marks. In our example, the viewing window for the x-axis is \([0, 100, 10]\), meaning it starts at 0, ends at 100, and has tick marks every 10 units.Similarly, for the y-axis, it is \([-50, 50, 10]\), spanning from -50 to 50, with tick marks every 10 units.

Tick Marks

Tick marks play a vital role in assessing and interpreting graphs. They act as reference points, providing a consistent measure along the axes. When you set a viewing window, understanding tick marks to predict their number is essential.For the x-axis, each tick mark represents a jump of 10 units, starting at 0 and ending at 100. The calculation \((100 - 0) / 10 + 1 = 11\) confirms there are 11 tick marks in total, as it includes the starting point.On the y-axis, tick marks cover from -50 to 50. Every tick represents 10 units, and the calculation \((50 - (-50)) / 10 + 1 = 11\) shows there are 11 tick marks here as well. Each tick mark is crucial for visual interpretation of scale, helping you quickly gauge values on the graph.

X-Axis and Y-Axis Range

The x-axis and y-axis ranges define how much of the graph's plane is visible through the window. It's important to set these ranges correctly in your calculator to visualize the complete picture of a graph. For the x-axis: - **Range**: From 0 to 100 - **Increment**: Every 10 units - **Total Tick Marks**: 11, including both endpoints. For the y-axis: - **Range**: From -50 to 50 - **Increment**: Every 10 units - **Total Tick Marks**: 11, similarly covering from starting to ending points. Setting the ranges properly ensures that all critical areas of the graph are displayed, making it easier to analyze.

Calculator Settings

Adjusting the calculator settings to match the required viewing window is key for obtaining accurate results. It involves configuring the start, end, and scale values meticulously. Ensure the calculated increment of 10 units is correctly input for both x and y-axes. This indicates precise placement of tick marks and aids in understanding the behavior of plotted functions. More generally, verifying these settings eliminates visualization errors, showcasing an entire graph within desired limits. Always recheck your calculator's settings before solving problems to confirm all values align with the required graph display conditions. This proactive step prevents misinterpretation and maximizes the tool's efficiency and utility.