Problem 65 - Algebra and Trigonometry

Question

Explaining the Concepts What does a \([-20,2,1]\) by \([-4,5,0.5]\) viewing rectangle mean?

Step-by-Step Solution

Answer

A \([-20,2,1]\) by \([-4,5,0.5]\) viewing rectangle means that the graph is drawn for x-values ranging from -20 to 2 (with increments of 1) and y-values ranging from -4 to 5 (with increments of 0.5). This defines the size and the scale of the viewing area on the graph.

Step 1: Understand the Definition of 'Viewing Rectangle'

A 'viewing rectangle' on a graph is determined by four values: the minimum and maximum values of x-axis (x-min and x-max) and the minimum and maximum values of y-axis (y-min and y-max). Additionally, we often also have the scales for both axes: x-scale and y-scale. These refer to the increments on each axis.

Step 2: Interpret x Values

The \([-20,2,1]\) setting means the x-axis starts from -20 (x-min) and ends at 2 (x-max), taking steps at increments of 1 (x-scale). In other words, the graph is drawn for the x values from -20 to 2, each unit on the x-axis representing 1.

Step 3: Interpret y Values

The \([-4,5,0.5]\) setting tells us the y-axis starts from -4 (y-min) and goes until 5 (y-max), with the increments being 0.5 (y-scale). This means the graph is drawn for the y values from -4 to 5, each unit on the y-axis representing 0.5.

Key Concepts

Understanding the X-Axis

In the world of graphs, the x-axis is your horizontal guide. It stretches from left to right across the graph. Every point on this axis is defined by a numerical value which helps determine the position of data points in a horizontal direction. In our example of

• [-20, 2, 1]

this means:

• **X-min:** -20, the smallest value on the x-axis.
• **X-max:** 2, the largest value on the x-axis.
• **X-scale:** 1, the increments or steps between each value on the x-axis.

In simple terms, the x-axis starts at -20 and ends at 2, moving in steps of 1. This provides the basic framework for evaluating data horizontally across the graph. Knowing this helps visualize how far left or right data points will lie.

Examining the Y-Axis

The y-axis stands tall, stretching vertically from bottom to top on a graph. It provides a vertical framework for data representation. Similar to the x-axis, the y-axis gives numbers to guide where points are placed vertically. In the setting

• [-4, 5, 0.5]

let’s break it down:

• **Y-min:** -4, the lowest value shown on the y-axis.
• **Y-max:** 5, the highest value on the y-axis.
• **Y-scale:** 0.5, the increments or steps between each vertical value.

This configuration means the y-axis starts at -4 and stretches upwards to 5, moving up in steps of 0.5. Understanding these increments is crucial because it dictates how spaced out or closely packed the data points will appear vertically. It essentially sets the stage for how detailed the data will look from a top-down perspective.

Visualizing the Graph

The graph combines both the x-axis and y-axis to create a visual representation of data. Picture it as a grid where each intersection between the horizontal and vertical lines represents a possible data point. The term "viewing rectangle" is thus used to describe the visible area of this grid. In our example,

• **X values** range from -20 to 2.
• **Y values** range from -4 to 5.

This rectangle structure is framed by the axes you just learned about, creating a window for viewing data. Through this window, the "scale" values...

• **1** increment for x-axis
• **0.5** increment for y-axis

...show how tight or loose the gridlines are within this area. Essentially, these settings help decide how detailed or zoomed-in the graph appears. If someone wanted more detail, they might decrease the scale to see more increments. Conversely, increasing the scale might give a broader, less detailed view, akin to zooming out.