Problem 4 - Technical Mathematics with Calculus

Question

If \(h\) and \(k\) are positive quantities, in which quadrants would the following points lie? $$(-h,-k)$$

Step-by-Step Solution

Answer

The point \((-h,-k)\) would lie in the third quadrant.

Step 1: Understanding the Coordinates

The point \((-h,-k)\) consists of a negative x-coordinate \((-h)\) and a negative y-coordinate \((-k)\). In a Cartesian coordinate system, negative x-values are located on the left side of the y-axis, and negative y-values are located below the x-axis.

Step 2: Identifying the Quadrant

Since both x and y coordinates are negative here, the point \((-h,-k)\) must lie in the quadrant where both x and y are negative. This is the third quadrant, which is in the bottom-left section of the Cartesian plane.

Key Concepts

Cartesian Coordinate System

Imagine a grid that lets you precisely pinpoint where you are on a map. This is the essence of the Cartesian coordinate system, named after René Descartes, which is designed to help you locate any point in a two-dimensional space using two numbers. These numbers are known as coordinates, represented as \(x, y\), where \(x\) corresponds to the horizontal position and \(y\) corresponds to the vertical position.

The system is visualized by two perpendicular lines that intersect at what is called the origin, denoted as \(0, 0\). These lines are called axes, with the horizontal one known as the x-axis and the vertical one as the y-axis. The point where they intersect, the origin, divides the plane into four sections: the quadrants.

• The first quadrant is where both x and y are positive.
• The second quadrant has a positive y but a negative x.
• The third quadrant is where both values are negative.
• The fourth quadrant has a positive x and a negative y.

When graphing points, knowing the sign of the coordinates swiftly tells you which quadrant the point belongs to, simplifying navigation across this system.

Negative Coordinates

In the coordinate plane, negative coordinates are just as important as positive ones. They correspond to directions that are opposite to the positive sides of the axes. If you're standing at the origin—the very center of the grid—moving left will take you through negative x-values, while moving down will take you through negative y-values.

Understanding Negative Coordinates

Whenever you see a negative sign in front of a coordinate, think of it as an instruction to move in a leftward or downward direction, depending on whether it's an x or y coordinate, respectively. A point with both negative x and y coordinates, such as \( -h, -k \), tells you to move left from the origin for the x-value and down for the y-value. Combining these two movements, you end up in the third quadrant. Think of it as a mirror world located below and to the left of the origin, where all points have that minus sign whispering, 'Go back and down.'

Graphing Points

Graphing points on the coordinate plane is like plotting treasure spots on a pirate's map. It requires attention to the order and sign of coordinates. Let’s learn how to graph a point step by step.

Step-by-Step Plotting

First, start with the x-coordinate. If it’s positive, move that number of units to the right of the origin; if it's negative, as in \( -h \), move left. Next, look at the y-coordinate. Move upwards for a positive and downwards for a negative, like \( -k \). The spot where you stop is where you’ll place your point. Mark it clearly.

In our example with a point \( -h, -k \), both steps take you towards the third quadrant. By practicing with different coordinates, you'll gain an instinct for how each point corresponds to a location on the plane, and you'll become an expert navigator of the Cartesian system. This skill is fundamental in various fields such as mathematics, physics, engineering, and computer graphics, revealing just how universal and crucial this concept is.