Problem 80 - 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. $$ [-4.7,4.7,1] \text { by }[-3.1,3.1,1] $$

Step-by-Step Solution

Answer

4 tick marks on the positive x-axis and 3 on the positive y-axis.

Step 1: Understand the Viewing Window

The viewing rectangle given is \([-4.7, 4.7, 1]\) by \([-3.1, 3.1, 1]\). This means the \(x\)-axis spans from \(-4.7\) to \(4.7\) with tick marks every \(1\) unit, and the \(y\)-axis spans from \(-3.1\) to \(3.1\) with tick marks every \(1\) unit.

Step 2: Calculate Tick Marks on the Positive x-axis

To find the number of tick marks on the positive \(x\)-axis, count the number of units from \(0\) to \(4.7\). Since the tick marks are placed every \(1\) unit, the tick marks will be at \(1, 2, 3, 4\). Therefore, there are 4 tick marks on the positive \(x\)-axis.

Step 3: Calculate Tick Marks on the Positive y-axis

To find the number of tick marks on the positive \(y\)-axis, count the number of units from \(0\) to \(3.1\). Placing tick marks every \(1\) unit means there are tick marks at \(1, 2, 3\). Thus, there are 3 tick marks on the positive \(y\)-axis.

Key Concepts

Viewing Window

When using a graphing calculator, the concept of a viewing window is crucial for understanding how the graph is displayed. The viewing window refers to the range of values that will be displayed on the calculator's screen for both the x-axis and y-axis. In our exercise, the viewing window is set as \([-4.7, 4.7, 1]\) by \([-3.1, 3.1, 1]\). This notation means:

• The x-axis spans from -4.7 to 4.7.
• The y-axis spans from -3.1 to 3.1.
• Each tick mark is 1 unit apart on both axes.

A carefully chosen viewing window allows you to see the important features of the graph, such as intercepts, maxima, and minima. This customization helps in focusing on specific parts of a graph or observing entire patterns. When setting up a viewing window, consider what you want to analyze and adjust the window accordingly. By choosing specific minimum and maximum limits for axes, you can make sure that the graph fits well within the screen and provides the information you need.

Tick Marks

Tick marks are the small lines that appear at regular intervals along the axes of a graph. They help in measuring and understanding distances on the graph, making it easier to interpret data. In our specific exercise, the axes are set with tick marks every 1 unit, which means:

• On the x-axis, tick marks will appear at 1, 2, 3, and 4, moving positively from zero.
• On the y-axis, tick marks will appear at 1, 2, and 3 in the positive direction from zero.

Tick marks play an essential role in accurately depicting the scale of the graph. The distance between them (here, it's 1 unit) determines how much of the graph will be visible and the level of detail that can be observed. In setting up tick marks, it's crucial to pick an interval that reflects the nature of the data or function you're observing. Using appropriate tick mark intervals ensures clarity, thereby making your graph easy to read and understand.

Coordinate Axes

Coordinate axes are the fundamental part of any graph, providing a framework upon which graphs are plotted. In the context of our exercise, the coordinate axes refer to both the x-axis and y-axis. These axes intersect at the origin point, \((0, 0)\), and extend infinitely in both directions.The x-axis is the horizontal line, while the y-axis is the vertical line. Each point on the graph is identified by a pair of values: \((x, y)\). The coordinates tell us the position of that point relative to the axes.

• The x-coordinate shows the horizontal distance from the y-axis.
• The y-coordinate indicates the vertical distance from the x-axis.

In our example, the viewing window and tick marks together define the visible portion of these axes on a graphing calculator. The setup decides which values of \(x\) and \(y\) are displayed, framed by the chosen limits. Understanding how coordinate axes work allows for accurate placement and interpretation of data points, as they hold the reference system for all plotting activities.