Problem 12 - Algebra and Trigonometry

Question

Sketch the region given by the set. \(\\{(x, y) | 0 \leq y \leq 4\\}\)

Step-by-Step Solution

Answer

It's a horizontal strip between \(y = 0\) and \(y = 4\), covering all \(x\).

Step 1: Understand the Set

The set given is \(\{(x, y) | 0 \leq y \leq 4\}\). This means we are considering all points \((x, y)\) where the value of \(y\) is between 0 and 4, inclusive. In other words, \(y\) can be any number from 0 to 4, and there is no restriction on the value of \(x\).

Step 2: Interpret the Region on the Coordinate Plane

Since there are no restrictions on \(x\), \(x\) can take any value. Therefore, every horizontal line defined by a specific \(y\)-value between 0 and 4 is part of the region. This forms a strip that extends infinitely in both directions along the \(x\)-axis.

Step 3: Sketch the Region

Draw a set of axes on a graph with the \(x\)-axis across the bottom and the \(y\)-axis vertically. Draw horizontal boundary lines at \(y = 0\) and \(y = 4\). Shade the entire region between these two lines to indicate that all values of \(y\) between 0 and 4 are included, while \(x\) can be any number. This shaded strip covers all \(x\) from -∞ to ∞ and \(y\) from 0 to 4.

Key Concepts

Graphing Inequalities

When grappling with inequalities in mathematics, a common task involves graphing them on the coordinate plane. This process helps us visualize the set of solutions that satisfy the inequality. Instead of just plotting points, we shade regions that represent all possible solutions.
For example, consider the inequality \(0 \leq y \leq 4\). This inequality describes a range of \(y\)-values from 0 to 4. \(y\) can be any number within this range, including the boundaries 0 and 4.
To showcase this on a graph:

• Draw dashed or solid lines for boundary conditions (solid when equal to, dashed when not).
• Shade the region where the inequality is true. In this case, shade between the lines \(y=0\) and \(y=4\).

The absence of conditions on \(x\) indicates that it can take any value. Knowing how to graph inequalities turns abstract conditions into concrete geometric regions on a plane, making it easier to analyze solutions.

Coordinate System

The coordinate system is a foundational tool used to locate points and plot graphs in mathematics. It consists of two number lines that intersect at a point called the origin. The horizontal line is known as the x-axis, and the vertical one is the y-axis.
These axes divide the plane into four quadrants, allowing us to categorize points based on their signs:

• Quadrant I (+x, +y)
• Quadrant II (-x, +y)
• Quadrant III (-x, -y)
• Quadrant IV (+x, -y)

A point on this plane is represented as \( (x, y) \), indicating its position relative to the origin. In graphing inequalities, it's crucial to understand this system thoroughly, as it helps identify which parts of the plane should be shaded.
With no restrictions on \(x\), any value can be used. Thus, the coordinate system allows you a versatile scope of analysis, essential for visualizing broader regions, such as the one defined between \(y=0\) and \(y=4\) in this exercise.

Two-Dimensional Region

In mathematics, a two-dimensional region is a part of the plane identified or characterized by specific criteria, like inequalities. These regions encompass all points that meet the set conditions. In our exercise, we focus on the region defined by \(0 \leq y \leq 4\) and all values of \(x\).
This forms a strip-like region extending horizontally across the entire coordinate plane, with its boundaries stretching from negative to positive infinity along the x-axis.
To clarify:

• The top boundary is the line \(y=4\).
• The bottom boundary is the line \(y=0\).
• The region includes all those vertical locations between these y-values.

Understanding and interpreting these two-dimensional regions on the coordinate plane provide a more tangible grasp of how inequalities control and define spaces, which can be vital in various applications such as area calculations, optimization problems, and more.