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The Quadrants of the Cartesian Plane (page 3 of 3)

Sections: Introduction to the plane, Plotting points, The four quadrants

he two axes divide the plane into four sections called "quadrants". The quadrants are labelled with Roman numerals (not Arabic numerals), starting at the positive x-axis and going around anti-clockwise:

When you get to trigonometry, this method of numbering the quadrants will make perfect sense and will be very useful. For now, just memorize the information. You'll probably have only a few questions on quadrants, and then you'll hardly see the topic again until trigonometry. Typical questions are generally similar to the following:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

• In which quadrant is the point (–2, –3)?

The simplest way to answer this is to plot the point:

As you can see, the point (–2, –3) is in Quadrant III.

• In which quadrant is the point (4, y)?

Since they don't tell you what "y" is, then, in this case, y can be anything. That is, (4, y) is not just one point! Since y can be –5, then (4, –5) is a valid answer. So is (4, –3), (4, 0), (4, 2), and (4, 4). So is any point that has x = 4:

In fact, the "point" (4, y) is actually the line "x = 4", which passes through two quadrants! So the answer is:

Quadrants I and IV.

By the way, please use standard notation. You can maybe abbreviate "Quadrant" as "Q", but don't completely omit it; use Roman "I, II, III, IV", not Arabic "1, 2, 3, 4"; and use "&" or "and", not "plus" or "+".

• In which quadrant is (x, y), xy < 0?

This is asking for the points (x, y), where x and y have values such that, when you multiply x and y together, you get a negative number. To figure this one out, it's probably simplest to just pick a sample point from each quadrant. For instance, in Quadrant I, pick, say, (2, 3). Then the product is 2 × 3 = 6, which is positive. Any product in Quadrant I will be positive, so Quadrant I is not part of the answer.

In Quadrant II, pick, say, (–4, 5). Then the product is (–4) × 5 = –20, which is negative. Any product in Quadrant II will be negative, so Quadrant II is part of the answer.

In Quadrant III, pick, say, (–2, –1). Then the product is (–2) × (–1) = 2, which is positive. Any other product in Quadrant III will be positive also, so Quadrant III is not part of the answer.

In Quadrant IV, pick, say, (3, –4). Then the product is 3 × (–4) = –12, which is negative. Any other product in Quadrant IV will be negative also, so Quadrant IV is part of the answer.

The points (x, y), with xy < 0, lie in Quadrants II and IV.

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