Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-4,3)$$

Step-by-Step Solution

Answer

The point \((-4,3)\) is in the second quadrant.

Step 1: Understand the Coordinates

The given point is \((-4,3)\), which is represented as \((x,y)\). Here, \(x = -4\) and \(y = 3\).

Step 2: Determine the x-coordinate

Examine the \(x\)-coordinate of the point, \(-4\). Since it is less than zero, it lies to the left of the origin on the Cartesian plane.

Step 3: Determine the y-coordinate

Look at the \(y\)-coordinate of the point, \(3\). Since it is greater than zero, it lies above the origin on the Cartesian plane.

Step 4: Plot the Point on the Coordinate System

Using the coordinates, navigate to the position \((-4,3)\) on a rectangular coordinate system: move 4 units to the left from the origin, then move 3 units upwards.

Step 5: Identify the Quadrant

Recall that in a Cartesian plane, the second quadrant is where \(x\) is negative and \(y\) is positive. The point \((-4,3)\) is located in the second quadrant.

Key Concepts

Quadrants

The rectangular coordinate system, commonly known as the Cartesian plane, is divided into four regions called quadrants. Quadrants help us categorize the positions of points based on their coordinates.

The quadrants are numbered counter-clockwise starting from the upper-right section:

• First Quadrant (Quadrant I): Here, both the x-coordinate and y-coordinate are positive ( x > 0, y > 0 ).
• Second Quadrant (Quadrant II): In this quadrant, the x-coordinate is negative and the y-coordinate is positive ( x < 0, y > 0 ).
• Third Quadrant (Quadrant III): Both the x and y-coordinates are negative ( x < 0, y < 0 ).
• Fourth Quadrant (Quadrant IV): The x-coordinate is positive and the y-coordinate is negative ( x > 0, y < 0 ).

Understanding which quadrant a point lies in helps with graphing and analyzing data on a coordinate system.

Coordinates

Coordinates are a set of values that show an exact position on the Cartesian plane. Typically, a point is represented by two coordinates:

• The x-coordinate – This indicates how far left or right the point is.
• The y-coordinate – This tells us how far up or down the point is.

In the ordered pair ( x, y ) , the x-coordinate is the first number, and the y-coordinate is the second. For example, the point ( -4, 3 ) has an x-coordinate of -4 and a y-coordinate of 3. This means the point is 4 units left and 3 units up from the origin. The signs of these coordinates are essential, as they determine the point's position relative to the origin in the Cartesian plane.

Cartesian Plane

The Cartesian plane, an essential idea developed by René Descartes, lays out a flat, two-dimensional surface on which any point can be precisely located using a pair of numerical coordinates. This plane is formed by two perpendicular number lines that intersect at a point called the origin. The horizontal line is known as the x-axis, and the vertical line is the y-axis.

• The origin is at ( 0, 0 ) , the meeting point of the x-axis and y-axis.
• The x-axis moves left and right, while the y-axis stretches up and down.

The Cartesian plane allows for visualizing relationships between variables and is fundamental in geometry, graphing functions, and solving equations. Visually navigating the Cartesian plane involves understanding this two-axis system, where any point's location is determined by its coordinates. Being familiar with the plane's layout helps effortlessly plot points like ( -4, 3 ) and check their alignment and relationship with other points and lines on the graph.