8.1: The Cartesian Coordinate System and the Quadrants
- Page ID
- 176594
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A Cartesian coordinate system in the plane consists of two perpendicular number lines, called the coordinate axes, that meet at a point called the origin. The horizontal axis is the \(x\)-axis and the vertical axis is the \(y\)-axis, and both axes are marked off using the same unit length. Every point in the plane is specified uniquely by an ordered pair of real numbers.
The coordinates of a point are the ordered pair \((x,y)\) that locate it, where \(x\) (the abscissa) is the signed distance from the point to the \(y\)-axis and \(y\) (the ordinate) is the signed distance from the point to the \(x\)-axis. A positive first coordinate lies to the right of the \(y\)-axis and a negative one to the left; a positive second coordinate lies above the \(x\)-axis and a negative one below. The origin has coordinates \((0,0)\).
The coordinates \((x,y)\) form an ordered pair: the first entry always measures horizontal position and the second always measures vertical position. Consequently, when \(x \neq y\), the point \((x,y)\) is different from the point \((y,x)\).
In the Cartesian coordinate system, the section of the plane where
- \( x \) and \( y \) are both positive is called Quadrant I (QI),
- \( x < 0 \) and \( y > 0 \) is called Quadrant II (QII),
- \( x \) and \( y \) are both negative is called Quadrant III (QIII), and
- \( x > 0 \) and \( y < 0 \) is called Quadrant IV (QIV).
A point with a zero coordinate lies on an axis and therefore in no quadrant. A point of the form \((x,0)\) lies on the \(x\)-axis, and a point of the form \((0,y)\) lies on the \(y\)-axis. The origin \((0,0)\) lies on both axes.
Examples
State the quadrant in which each point lies.
- \((3,-7)\)
- \((-2,-5)\)
- \((6,1)\)
- \((-4,9)\)
- Answers
-
- The first coordinate is positive and the second is negative, so \((3,-7)\) lies in Quadrant \(\text{IV}\).
- Both coordinates are negative, so \((-2,-5)\) lies in Quadrant \(\text{III}\).
- Both coordinates are positive, so \((6,1)\) lies in Quadrant \(\text{I}\).
- The first coordinate is negative and the second is positive, so \((-4,9)\) lies in Quadrant \(\text{II}\).
State the quadrant in which each point lies, or state that the point lies on an axis.
- \((0,-4)\)
- \((5,0)\)
- \((0,0)\)
- Answers
-
- The first coordinate is \(0\), so \((0,-4)\) lies on the \(y\)-axis (below the origin) and in no quadrant.
- The second coordinate is \(0\), so \((5,0)\) lies on the \(x\)-axis (right of the origin) and in no quadrant.
- Both coordinates are \(0\), so \((0,0)\) is the origin, which lies on both axes and in no quadrant.
Describe how to plot the point \(P(-3,4)\), and state the quadrant in which it lies.
- Answer
-
Begin at the origin. The first coordinate is \(-3\), so move \(3\) units to the left along the \(x\)-axis. The second coordinate is \(4\), so move \(4\) units up. Mark the point at that location. Because the first coordinate is negative and the second is positive, \(P\) lies in Quadrant \(\text{II}\).
A point \(Q\) lies in Quadrant \(\text{II}\). What can you conclude about the signs of its coordinates? Give one point that could be \(Q\).
- Answer
-
By the Quadrant Classification theorem, a point in Quadrant \(\text{II}\) has a negative first coordinate and a positive second coordinate; that is, \(x<0\) and \(y>0\). Any point meeting both conditions is a valid choice, for example \(Q(-5,2)\).
Suppose \(a>0\) and \(b>0\). State the quadrant in which each point lies.
- \((-a,b)\)
- \((-a,-b)\)
- \((a,-b)\)
- Answers
-
Because \(a>0\) and \(b>0\), the expressions \(-a\) and \(-b\) are negative while \(a\) and \(b\) are positive. Applying the Quadrant Classification theorem:
- The first coordinate \(-a\) is negative and the second coordinate \(b\) is positive, so \((-a,b)\) lies in Quadrant \(\text{II}\).
- Both coordinates are negative, so \((-a,-b)\) lies in Quadrant \(\text{III}\).
- The first coordinate \(a\) is positive and the second coordinate \(-b\) is negative, so \((a,-b)\) lies in Quadrant \(\text{IV}\).
Explain why \((2,7)\) and \((7,2)\) represent different points, and state the quadrant of each.
- Answer
-
The coordinates form an ordered pair, so the first entry gives horizontal position and the second gives vertical position. For \((2,7)\), move \(2\) units right and \(7\) units up; for \((7,2)\), move \(7\) units right and \(2\) units up. These are different locations, so the points are different. Both have two positive coordinates, so each lies in Quadrant \(\text{I}\).
Sources
Several parts of this text use modifications from the following sources:
- Wikipedia article: "Cartesian coordinate system"
- Wikipedia article: "Quadrant (plane geometry)"
All of these sources are released under the Creative Commons Attribution-Share-Alike License 4.0.