The Point (3, 4) and the Point (4, 3) Are Not the Same Place
Learn to plot an ordered pair and name its quadrant, read coordinates off a graph, use the sign pattern to reflect a point in either axis, and turn a street map into coordinates.
Why is (3, 4) a different point from (4, 3)?
Because the pair is ordered — the first number always means the horizontal step and the second always means the vertical one.
The point is reached by going ** units right and then units up**. The point is reached by going ** units right and then units up. Those are two different places on the page, and both lie in the first quadrant.
The only thing separating them is the order of the two numbers, and that is exactly why the notation is called an ordered pair. A pair in which order did not matter could not name a unique point at all.
So the very first rule of the whole chapter is: read and write first, always**. Everything else on this page — quadrants, signs, reflections, map coordinates — depends on that convention being followed without exception.
This page covers the first part of the CBSE Class 9 Mathematics chapter on the use of coordinates.
The point is reached by going ** units right and then units up**. The point is reached by going ** units right and then units up. Those are two different places on the page, and both lie in the first quadrant.
The only thing separating them is the order of the two numbers, and that is exactly why the notation is called an ordered pair. A pair in which order did not matter could not name a unique point at all.
So the very first rule of the whole chapter is: read and write first, always**. Everything else on this page — quadrants, signs, reflections, map coordinates — depends on that convention being followed without exception.
This page covers the first part of the CBSE Class 9 Mathematics chapter on the use of coordinates.
How do you plot a point and say which quadrant it is in?
Count across from the origin first, then up or down, and the signs tell you the quadrant.
The Cartesian plane is made of two perpendicular number lines: the **-axis running horizontally and the -axis running vertically, crossing at the origin** .
For a point :
- is the abscissa — the distance from the -axis, positive to the right and negative to the left
- is the ordinate — the distance from the -axis, positive upwards and negative downwards
The two axes divide the plane into four quadrants, numbered anticlockwise starting from the top right: I, II, III, IV.
Worked examples.
- — right and up, so Quadrant I
- — left and up, so Quadrant II
- — left and down, so Quadrant III
- — right and down, so Quadrant IV
- — no vertical step at all, so it lies **on the -axis**
- — no horizontal step, so it lies **on the -axis**
- — the origin itself
A point on an axis is in no quadrant. The axes are boundaries, not parts of the regions they separate. So and have no quadrant, and answering Quadrant I for is wrong even though the is positive.
The test is simple: if either coordinate is zero, the point is on an axis.
- puts the point on the **-axis**
- puts it on the **-axis
- Both zero gives the origin
Mark the scale on both axes before plotting.** If one square stands for unit on the -axis, say so, and use the same or a stated scale on the -axis. A graph without a scale cannot be read, and the scale carries marks of its own in a plotting question.
The Cartesian plane is made of two perpendicular number lines: the **-axis running horizontally and the -axis running vertically, crossing at the origin** .
For a point :
- is the abscissa — the distance from the -axis, positive to the right and negative to the left
- is the ordinate — the distance from the -axis, positive upwards and negative downwards
The two axes divide the plane into four quadrants, numbered anticlockwise starting from the top right: I, II, III, IV.
Worked examples.
- — right and up, so Quadrant I
- — left and up, so Quadrant II
- — left and down, so Quadrant III
- — right and down, so Quadrant IV
- — no vertical step at all, so it lies **on the -axis**
- — no horizontal step, so it lies **on the -axis**
- — the origin itself
A point on an axis is in no quadrant. The axes are boundaries, not parts of the regions they separate. So and have no quadrant, and answering Quadrant I for is wrong even though the is positive.
The test is simple: if either coordinate is zero, the point is on an axis.
- puts the point on the **-axis**
- puts it on the **-axis
- Both zero gives the origin
Mark the scale on both axes before plotting.** If one square stands for unit on the -axis, say so, and use the same or a stated scale on the -axis. A graph without a scale cannot be read, and the scale carries marks of its own in a plotting question.
How do you read the coordinates of a marked point off a graph?
Drop a perpendicular to each axis and read where it lands — the -axis first.
The method:
- From the marked point, draw a **perpendicular to the -axis and read the value there. That is the -coordinate
- From the same point, draw a perpendicular to the -axis and read the value there. That is the -coordinate**
- Write them as an ordered pair, first
Worked example 1. A point lies units to the right of the origin and units below it. Its perpendiculars meet the axes at and , so the point is
and it lies in Quadrant IV.
Worked example 2. A point lies units to the left and units up. The point is , in Quadrant II.
Worked example 3 — with a scale. On a graph where ** cm represents units**, a point is marked cm to the right of the origin and cm above it.
so the point is . Reading the squares instead of the units would have given — a completely different point, and the standard error when the scale is not .
Worked example 4 — a point on an axis. A point sits on the -axis, units below the origin. Its -coordinate is , so the point is .
**Read first, every time.** A point described as * up and right* is , not — the description gave the vertical step first and the notation still wants the horizontal one first. The notation's order never changes to match the wording of the question, and questions are sometimes phrased in the reverse order deliberately.
A quick check on any reading. Look at the signs you have written and see whether they match the quadrant the point is visibly in. A point plainly up and to the left must read , and if your answer has two positives you have made an error before doing any arithmetic. That sign pattern is the subject of the next section.
The method:
- From the marked point, draw a **perpendicular to the -axis and read the value there. That is the -coordinate
- From the same point, draw a perpendicular to the -axis and read the value there. That is the -coordinate**
- Write them as an ordered pair, first
Worked example 1. A point lies units to the right of the origin and units below it. Its perpendiculars meet the axes at and , so the point is
and it lies in Quadrant IV.
Worked example 2. A point lies units to the left and units up. The point is , in Quadrant II.
Worked example 3 — with a scale. On a graph where ** cm represents units**, a point is marked cm to the right of the origin and cm above it.
so the point is . Reading the squares instead of the units would have given — a completely different point, and the standard error when the scale is not .
Worked example 4 — a point on an axis. A point sits on the -axis, units below the origin. Its -coordinate is , so the point is .
**Read first, every time.** A point described as * up and right* is , not — the description gave the vertical step first and the notation still wants the horizontal one first. The notation's order never changes to match the wording of the question, and questions are sometimes phrased in the reverse order deliberately.
A quick check on any reading. Look at the signs you have written and see whether they match the quadrant the point is visibly in. A point plainly up and to the left must read , and if your answer has two positives you have made an error before doing any arithmetic. That sign pattern is the subject of the next section.
Formula
What are the signs in each quadrant, and how do you reflect a point?
Each quadrant has its own sign pattern, and a reflection changes exactly one sign.
The pattern is easy to reconstruct rather than memorise: is positive on the right half of the plane and negative on the left; is positive in the upper half and negative in the lower.
The reflection rules:
Which coordinate flips, and why. Reflecting in the **-axis moves the point up or down across a horizontal mirror, so the **-coordinate changes sign and is untouched. Reflecting in the **-axis moves it left or right, so changes sign. The coordinate that flips is the one measured perpendicular to the mirror.
Worked example 1 — a first-quadrant point.** Take .
- In the -axis: — the point moves from Quadrant I to Quadrant IV
- In the -axis: — from Quadrant I to Quadrant II
Worked example 2. Take , in Quadrant II.
- In the -axis: — II to III
- In the -axis: — II to I
Notice that the -axis image is , with a positive . Changing the sign of gives ; mechanically writing a minus in front would give , which is the point you started with.
Worked example 3. Take , in Quadrant III.
- In the -axis: — III to II
- In the -axis: — III to IV
Worked example 4 — a point that does not move. Reflect in the -axis:
The point is unchanged, because it already lies on the mirror. **Every point of the -axis is unchanged by reflection in the -axis**, and the same holds for the -axis.
Reflecting in both axes, one after the other, gives :
Both signs have changed, which moves the point from Quadrant I to Quadrant III — diagonally opposite. Two mirror reflections have produced a half turn about the origin, which is worth noticing because it means the two operations are connected rather than separate.
Change the sign; do not simply add a minus. That is the one rule to hold on to here, and it is the only thing that goes wrong when the starting point already has a negative coordinate.
The pattern is easy to reconstruct rather than memorise: is positive on the right half of the plane and negative on the left; is positive in the upper half and negative in the lower.
The reflection rules:
Which coordinate flips, and why. Reflecting in the **-axis moves the point up or down across a horizontal mirror, so the **-coordinate changes sign and is untouched. Reflecting in the **-axis moves it left or right, so changes sign. The coordinate that flips is the one measured perpendicular to the mirror.
Worked example 1 — a first-quadrant point.** Take .
- In the -axis: — the point moves from Quadrant I to Quadrant IV
- In the -axis: — from Quadrant I to Quadrant II
Worked example 2. Take , in Quadrant II.
- In the -axis: — II to III
- In the -axis: — II to I
Notice that the -axis image is , with a positive . Changing the sign of gives ; mechanically writing a minus in front would give , which is the point you started with.
Worked example 3. Take , in Quadrant III.
- In the -axis: — III to II
- In the -axis: — III to IV
Worked example 4 — a point that does not move. Reflect in the -axis:
The point is unchanged, because it already lies on the mirror. **Every point of the -axis is unchanged by reflection in the -axis**, and the same holds for the -axis.
Reflecting in both axes, one after the other, gives :
Both signs have changed, which moves the point from Quadrant I to Quadrant III — diagonally opposite. Two mirror reflections have produced a half turn about the origin, which is worth noticing because it means the two operations are connected rather than separate.
Change the sign; do not simply add a minus. That is the one rule to hold on to here, and it is the only thing that goes wrong when the starting point already has a negative coordinate.
How do you turn a street map into coordinates?
Choose an origin, fix a positive direction for each axis, and state a scale. After that every position is an ordered pair.
All three choices are yours to make, and the coordinates depend entirely on them — so they must be written down before any position is recorded.
Worked example — a town grid. Take the main crossing as the origin, east as positive , north as positive , and let ** unit represent m.
- A school** m east and m north of the crossing is at
- A hospital m west and m north is at
- A market m east and m south is at
- A bus stand m west and m south is at
- A post office m due east, on the main road itself, is at — **on the -axis
Reading a distance off the grid.** The school is at and the market at . A person walking along the streets must travel
going one unit west and five units south. That is the distance along the streets, not the straight-line distance — which needs the formula in the next part of this chapter.
Worked example — the reverse question. *Where is the point ?* Two units west and four units north of the crossing, so m west and m north.
Change the origin and every coordinate changes. Move the origin to the school and the school becomes , while the market becomes . What does not change is the relative position — the market is still one unit west and five units south of the school, whichever point is called the origin.
That is the same relativity the motion chapter of this course established: a position is always relative to a chosen reference point, and stating the reference is part of stating the position.
Real addresses are coordinates already. Third house on the fourth cross street is an ordered pair with the streets as axes, and a seat described as *row , seat * is another. The Cartesian plane is not a new idea imposed on the world; it is the tidying up of something already in use — and that is why fixing the origin, the directions and the scale is the whole of the work.
All three choices are yours to make, and the coordinates depend entirely on them — so they must be written down before any position is recorded.
Worked example — a town grid. Take the main crossing as the origin, east as positive , north as positive , and let ** unit represent m.
- A school** m east and m north of the crossing is at
- A hospital m west and m north is at
- A market m east and m south is at
- A bus stand m west and m south is at
- A post office m due east, on the main road itself, is at — **on the -axis
Reading a distance off the grid.** The school is at and the market at . A person walking along the streets must travel
going one unit west and five units south. That is the distance along the streets, not the straight-line distance — which needs the formula in the next part of this chapter.
Worked example — the reverse question. *Where is the point ?* Two units west and four units north of the crossing, so m west and m north.
Change the origin and every coordinate changes. Move the origin to the school and the school becomes , while the market becomes . What does not change is the relative position — the market is still one unit west and five units south of the school, whichever point is called the origin.
That is the same relativity the motion chapter of this course established: a position is always relative to a chosen reference point, and stating the reference is part of stating the position.
Real addresses are coordinates already. Third house on the fourth cross street is an ordered pair with the streets as axes, and a seat described as *row , seat * is another. The Cartesian plane is not a new idea imposed on the world; it is the tidying up of something already in use — and that is why fixing the origin, the directions and the scale is the whole of the work.
Exam tip
Exam tip: write x first and check the sign against the picture
**Always write and read first**, whatever order the question describes the steps in. Four up and three right is .
is the abscissa (from the -axis), the ordinate (from the -axis).
Quadrants run anticlockwise from the top right: I , II , III , IV .
A point on an axis is in no quadrant. If either coordinate is , name the axis: means the **-axis**, means the **-axis, both zero means the origin.
Mark the scale** on both axes, and if cm represents units, multiply — a point cm right is at , not .
For a reflection, flip the coordinate perpendicular to the mirror: in the -axis ; in the -axis .
Change the sign; do not write a minus in front. The -axis image of is .
Say which quadrant the image lands in — reflection in the -axis swaps I with IV and II with III.
Points on the mirror do not move: reflected in the -axis is still .
And check every answer's signs against the picture — a point visibly up and to the left must read .
is the abscissa (from the -axis), the ordinate (from the -axis).
Quadrants run anticlockwise from the top right: I , II , III , IV .
A point on an axis is in no quadrant. If either coordinate is , name the axis: means the **-axis**, means the **-axis, both zero means the origin.
Mark the scale** on both axes, and if cm represents units, multiply — a point cm right is at , not .
For a reflection, flip the coordinate perpendicular to the mirror: in the -axis ; in the -axis .
Change the sign; do not write a minus in front. The -axis image of is .
Say which quadrant the image lands in — reflection in the -axis swaps I with IV and II with III.
Points on the mirror do not move: reflected in the -axis is still .
And check every answer's signs against the picture — a point visibly up and to the left must read .
Did you know
Why a seat number is already a coordinate
Look at a cinema ticket, a train reservation or an examination hall seating plan and you find a pair of numbers or a letter and a number: **row , seat . That is an ordered pair, and the hall is a Cartesian plane with the rows as one axis and the seats as the other.
The conventions are all there too. There is an origin — the corner where counting begins. There are positive directions — away from the screen, and left to right as you face it. And the order matters**: row seat is nowhere near row seat .
The same structure turns up wherever a position has to be recorded unambiguously. A spreadsheet cell is column C, row 5. A chessboard square is e4. A pixel on a screen is a pair of numbers counted from one corner. A place on the Earth's surface is a pair of angles, latitude and longitude — which is exactly the system the earth-systems chapter of this course used to explain why the poles are cold.
What the Cartesian plane adds to all of these is negative numbers. A cinema has no row , because counting starts at one end of a finite hall. Extending both axes in both directions produces four quadrants instead of one and lets a single system describe positions on every side of the origin — which is what makes the reflections in this page possible at all.
So coordinates are not a new invention to be learned for mathematics. They are the tidying-up of a habit everybody already has, with the addition of an origin in the middle and directions that run both ways.
The conventions are all there too. There is an origin — the corner where counting begins. There are positive directions — away from the screen, and left to right as you face it. And the order matters**: row seat is nowhere near row seat .
The same structure turns up wherever a position has to be recorded unambiguously. A spreadsheet cell is column C, row 5. A chessboard square is e4. A pixel on a screen is a pair of numbers counted from one corner. A place on the Earth's surface is a pair of angles, latitude and longitude — which is exactly the system the earth-systems chapter of this course used to explain why the poles are cold.
What the Cartesian plane adds to all of these is negative numbers. A cinema has no row , because counting starts at one end of a finite hall. Extending both axes in both directions produces four quadrants instead of one and lets a single system describe positions on every side of the origin — which is what makes the reflections in this page possible at all.
So coordinates are not a new invention to be learned for mathematics. They are the tidying-up of a habit everybody already has, with the addition of an origin in the middle and directions that run both ways.
Exam relevance
How does coordinate geometry in Class 9 feed into JEE Main?
Because coordinate geometry is one of the largest topics in JEE Main Mathematics, and this page is where its notation and conventions are fixed.
This is the foundation for the Class 11 Mathematics chapter Straight Lines and the chapters on Conic Sections — circle, parabola, ellipse and hyperbola — all examined in JEE Main. Every one of them is written in the coordinate notation established here, and every formula in them assumes the quadrant conventions and the sign pattern without restating them.
Where the reflection rules are reused. The transformations on this page become the formal study of shifting of origin and of transformation of axes in Class 11, and reflection reappears in Straight Lines when the image of a point in a line is required — a standard JEE Main question. The Class 9 insight that the coordinate flipping is the one perpendicular to the mirror is what generalises to reflection in an arbitrary line.
The quadrant sign pattern is used constantly in Class 11 Trigonometric Functions, where the sign of each trigonometric ratio in each quadrant follows directly from the signs of and there. Students who know the four sign pairs find the trigonometric sign rules obvious; those who do not have to memorise a second table.
The map-to-coordinates work leads into Class 11 Introduction to Three Dimensional Geometry, where a third axis is added and the plane becomes space with eight octants instead of four quadrants. The reasoning is identical, with one more coordinate.
What the questions look like. For board work, plotting questions and name the quadrant questions are common and quick. For JEE Main, this material almost never appears alone — it appears as the first line of a longer problem, where a point's coordinates or its reflection have to be written down before the real work begins. An error there makes every subsequent step wrong for a reason that has nothing to do with the topic being tested.
How board and competitive emphasis differ. A board paper asks you to plot given points on a graph sheet, name their quadrants, and write the reflection of a point — with marks for the axes, the scale and the neatness of the plotting. A competitive paper assumes all of that and asks for something that follows from it. So the board paper rewards the drawing and the competitive paper rewards the speed and accuracy of writing coordinates down.
The single trap that costs the most marks. Reversing the order of the coordinates. A question describing a position as * units above and units to the right* wants , and the wording deliberately puts the vertical step first. The notation's order never adapts to the question's order, and this is worth checking on every single answer because the error is invisible once written.
A second trap worth naming. Assigning a quadrant to a point on an axis. The point lies **on the -axis** and is in no quadrant, even though is positive. If either coordinate is zero, name the axis — and a question offering such a point is testing exactly that distinction.
This is the foundation for the Class 11 Mathematics chapter Straight Lines and the chapters on Conic Sections — circle, parabola, ellipse and hyperbola — all examined in JEE Main. Every one of them is written in the coordinate notation established here, and every formula in them assumes the quadrant conventions and the sign pattern without restating them.
Where the reflection rules are reused. The transformations on this page become the formal study of shifting of origin and of transformation of axes in Class 11, and reflection reappears in Straight Lines when the image of a point in a line is required — a standard JEE Main question. The Class 9 insight that the coordinate flipping is the one perpendicular to the mirror is what generalises to reflection in an arbitrary line.
The quadrant sign pattern is used constantly in Class 11 Trigonometric Functions, where the sign of each trigonometric ratio in each quadrant follows directly from the signs of and there. Students who know the four sign pairs find the trigonometric sign rules obvious; those who do not have to memorise a second table.
The map-to-coordinates work leads into Class 11 Introduction to Three Dimensional Geometry, where a third axis is added and the plane becomes space with eight octants instead of four quadrants. The reasoning is identical, with one more coordinate.
What the questions look like. For board work, plotting questions and name the quadrant questions are common and quick. For JEE Main, this material almost never appears alone — it appears as the first line of a longer problem, where a point's coordinates or its reflection have to be written down before the real work begins. An error there makes every subsequent step wrong for a reason that has nothing to do with the topic being tested.
How board and competitive emphasis differ. A board paper asks you to plot given points on a graph sheet, name their quadrants, and write the reflection of a point — with marks for the axes, the scale and the neatness of the plotting. A competitive paper assumes all of that and asks for something that follows from it. So the board paper rewards the drawing and the competitive paper rewards the speed and accuracy of writing coordinates down.
The single trap that costs the most marks. Reversing the order of the coordinates. A question describing a position as * units above and units to the right* wants , and the wording deliberately puts the vertical step first. The notation's order never adapts to the question's order, and this is worth checking on every single answer because the error is invisible once written.
A second trap worth naming. Assigning a quadrant to a point on an axis. The point lies **on the -axis** and is in no quadrant, even though is positive. If either coordinate is zero, name the axis — and a question offering such a point is testing exactly that distinction.
Key takeaways
The Cartesian plane, quadrants and reflections: quick revision
- An ordered pair means horizontally first, then vertically. and are different points.
- The **-axis is horizontal, the -axis vertical, and they meet at the origin** .
- is the abscissa (distance from the -axis); is the ordinate (distance from the -axis).
- Quadrants run anticlockwise from the top right: is I, is II, is III, is IV.
- A point on an axis is in no quadrant. is on the **-axis**, on the **-axis**, and is the origin.
- If either coordinate is zero, name the axis rather than a quadrant.
- To read a point off a graph, drop a perpendicular to each axis and write first. A point right and down is .
- Use the scale: if cm represents units, a point cm right and cm up is , not .
- Signs by quadrant: I , II , III , IV . is positive on the right, positive above.
- **Reflection in the -axis**: . **In the -axis**: . The coordinate that flips is the one perpendicular to the mirror.
- gives and ; gives and ; gives and .
- Change the sign, do not add a minus — the -axis image of is .
- Reflection in the -axis swaps I with IV and II with III; points on the mirror do not move, so stays .
- Reflecting in both axes gives , which is a half turn about the origin — becomes .
- For a map, fix the origin, the positive directions and the scale first. With east and north positive and unit m, a school m east and m north is .
- The street distance from to is units m — not the straight-line distance.
- Change the origin and every coordinate changes, while the relative positions do not.
Plot five points including two with negative coordinates, name each quadrant, then write both reflections of each — the ones that start negative are the ones that catch people out.
- The **-axis is horizontal, the -axis vertical, and they meet at the origin** .
- is the abscissa (distance from the -axis); is the ordinate (distance from the -axis).
- Quadrants run anticlockwise from the top right: is I, is II, is III, is IV.
- A point on an axis is in no quadrant. is on the **-axis**, on the **-axis**, and is the origin.
- If either coordinate is zero, name the axis rather than a quadrant.
- To read a point off a graph, drop a perpendicular to each axis and write first. A point right and down is .
- Use the scale: if cm represents units, a point cm right and cm up is , not .
- Signs by quadrant: I , II , III , IV . is positive on the right, positive above.
- **Reflection in the -axis**: . **In the -axis**: . The coordinate that flips is the one perpendicular to the mirror.
- gives and ; gives and ; gives and .
- Change the sign, do not add a minus — the -axis image of is .
- Reflection in the -axis swaps I with IV and II with III; points on the mirror do not move, so stays .
- Reflecting in both axes gives , which is a half turn about the origin — becomes .
- For a map, fix the origin, the positive directions and the scale first. With east and north positive and unit m, a school m east and m north is .
- The street distance from to is units m — not the straight-line distance.
- Change the origin and every coordinate changes, while the relative positions do not.
Plot five points including two with negative coordinates, name each quadrant, then write both reflections of each — the ones that start negative are the ones that catch people out.