What Are All the Quadrants on a Graph? Quadrants I to IV

So, what are all the quadrants on a graph? A standard coordinate graph has four quadrants, numbered I, II, III and IV, created when the horizontal x-axis and vertical y-axis cross at the origin. Quadrant I is top right, II is top left, III is bottom left and IV is bottom right, and each has its own pattern of positive and negative coordinates.

What are all the quadrants on a graph? The four at a glance

The system is called the Cartesian plane, named after the mathematician René Descartes, and it is described on Wikipedia’s Cartesian coordinate system page. Two number lines cross at right angles. The horizontal line is the x-axis and the vertical one is the y-axis. Where they meet is the origin, written (0, 0).

Quadrant	Position	Sign of x	Sign of y	Example point
I	Top right	Positive	Positive	(3, 4)
II	Top left	Negative	Positive	(-3, 4)
III	Bottom left	Negative	Negative	(-3, -4)
IV	Bottom right	Positive	Negative	(3, -4)

Notice also how the quadrants relate to each other through reflection. If you take a point in Quadrant I, such as (3, 4), reflect it across the y-axis and you land in Quadrant II at (-3, 4). Reflect that across the x-axis and you arrive in Quadrant III at (-3, -4), and one more reflection across the y-axis lands you in Quadrant IV at (3, -4). Seeing this chain of four points makes the sign patterns much easier to remember, because only the signs change and the size of the numbers stays the same.

The same idea works for distances. Every one of those four points is the same distance from the origin, which you can check with the Pythagorean theorem: the square root of 3 squared plus 4 squared is 5. Quadrants therefore tell you the direction, while the numbers tell you how far.

How to remember the order

Quadrants are numbered counterclockwise, which means in the opposite direction to a clock’s hands. Begin in the top right, then sweep up and to the left, down, and finally round to the bottom right. A handy trick is to draw a large letter C or a backwards arrow around the origin, starting at the top right.

Another trick is to remember the signs as “all, sine, tangent, cosine” for trigonometry later on, but for now it is enough to remember: top right is all positive, and the sign changes as you go round the circle.

Working out the quadrant of a point

1. Look at the x-coordinate, the first number. Is it positive (right) or negative (left)?
2. Look at the y-coordinate, the second number. Is it positive (up) or negative (down)?
3. Match the pair of signs to the table above.

For example, the point (-5, 2) has a negative x and a positive y, so it lies in Quadrant II. The point (6, -1) has positive x and negative y, so it is in Quadrant IV.

Points on the axes

Points like (4, 0) or (0, -3) sit exactly on an axis. They do not lie in any quadrant, because the quadrants are the open regions between the axes. The origin (0, 0) is also not in a quadrant.

How to plot a point

1. Start at the origin.
2. Move left or right along the x-axis by the first number.
3. From there, move up or down by the second number.
4. Mark the point and label it with its coordinates.

Remember: “run before you jump”, meaning go along the x-axis before you go up or down the y-axis.

Why quadrants matter

Quadrants help you quickly describe where a shape, line or solution sits.

• Graphing lines: a line with a positive slope passes through Quadrants I and III if it goes through the origin.
• Reflections: reflecting a point across the x-axis changes the sign of y, while reflecting across the y-axis changes the sign of x.
• Real data: a graph of temperatures, profits and losses, or positions on a map may use all four quadrants to show positive and negative values.
• Geometry and trigonometry: angles are measured from the positive x-axis, and the quadrant tells you whether sine and cosine are positive or negative.

If you are tackling shapes next, see our notes on the formulas for a semicircle and the page on the four quadrants on a graph, which approaches the same idea with extra practice points. If you are brushing up on terminology, also read what is meant by the quotient of two numbers.

Common mistakes

• Swapping x and y, so the point (2, 5) is plotted at (5, 2).
• Numbering quadrants clockwise rather than counterclockwise.
• Putting points on an axis into a quadrant.
• Forgetting that a negative number moves left or down.
• Using an uneven scale on the axes.

Quick practice

Try to say which quadrant each point is in: (7, 2), (-1, -8), (-4, 6), (0, 5), (3, -3). The answers are I, III, II, none (on the y-axis) and IV. If you got them all right, you have got the idea.

Frequently asked questions

Conclusion

Once you can answer what are all the quadrants on a graph, plotting and describing points gets much easier. Remember the counterclockwise order, match the signs of x and y, and keep points on the axes separate from the four regions.