Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

In which quadrants are the \(y\) -coordinates positive?

Step-by-Step Solution

Answer

The y-coordinates are positive in the first and second quadrants.

Step 1: Understanding the Cartesian Coordinate System

The Cartesian coordinate system divides the plane into four quadrants. Each quadrant represents a specific combination of positive and negative x and y coordinates. Starting from the right upper quadrant and going anti-clockwise, we have the first, second, third, and fourth quadrants.

Step 2: Identify the quadrants with positive y-coordinates

We know that in the first quadrant, both x and y are positive. In the second quadrant, x is negative, but y is positive. In the third quadrant, both x and y are negative, and in the fourth quadrant, x is positive, but y is negative. Therefore, the y-coordinates are positive in the first and second quadrants.

Key Concepts

Quadrants of the Cartesian Coordinate System

The Cartesian Coordinate System is a two-dimensional plane defined by two perpendicular lines, the x-axis and the y-axis. These axes divide the plane into four distinct regions known as quadrants. Each quadrant is defined by specific combinations of positive and negative x and y coordinates, which help in locating points in the plane.

Here's how the quadrants are generally arranged:

• First Quadrant: Both the x-coordinate and y-coordinate are positive.
• Second Quadrant: The x-coordinate is negative, while the y-coordinate is positive.
• Third Quadrant: Both the x-coordinate and y-coordinate are negative.
• Fourth Quadrant: The x-coordinate is positive, but the y-coordinate is negative.

Each quadrant plays a significant role in mathematical calculations and graphical interpretations. Observing the sign of coordinates in each quadrant helps quickly understand the nature and positioning of points in the plane.

Positive Coordinates

In the Cartesian Coordinate System, coordinates are defined as positive or negative based on their position in relation to the origin, which is the point of intersection of the x-axis and the y-axis. Positive coordinates can determine the location and nature of points, as well as their reflections and transformations across the axes.

Here is how positive coordinates are defined for each axis:

• Positive x-coordinates: Any value on the x-axis to the right of the origin is positive.
• Positive y-coordinates: Any value on the y-axis above the origin is considered positive.

Seeing how coordinates change sign when moving through different quadrants can help reinforce understanding of vector directions, transformations, and basic geometry principles.

Specifically, positive y-coordinates are evident in the first and second quadrants, where the y-value remains above the x-axis. Understanding these specific conditions is essential for mastering graph-related problems in geometry and algebra.

Graphical Representation

Graphical representation is a visual method of depicting mathematical concepts using graphs plotted on the Cartesian plane. It provides an intuitive way to understand and solve problems involving coordinates and to comprehend the spatial relationships between points.

Here are some benefits of graphical representation:

• It allows easy identification of the location of points and movements between points.
• Graphs help to visualize functions, making it easier to interpret their behavior.
• They can demonstrate the symmetry and geometry of figures, aiding in geometric and algebraic problem-solving.

Using a graph, you can see how the positive y-coordinates appear as points above the x-axis – precisely in the first and second quadrants. A graphical view can simplify the understanding of complex mathematical relationships, making concepts like slopes, intercepts, and symmetry more accessible.