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Points on the Coordinate Plane

Plot, read, move and reflect points and name their quadrant from the signs of the coordinates.

Practice this skill 18 min Points in the coordinate plane Leer en español
Key idea

What you'll learn

In this lesson you will learn to read and plot points on the coordinate plane — the grid that almost every GED graph question is built on.

You will be able to:

  • name the parts of the coordinate plane (axes, origin, quadrants);
  • plot any point (x,\ y), including negatives, zeros, fractions and decimals;
  • read the coordinates of a point from a graph, even when the grid is scaled;
  • tell which quadrant a point is in from the signs alone;
  • move, reflect and connect points to answer geometry and map questions.

Why it matters on the GED. The test uses this skill directly in "click on the graph" (hot-spot) questions, where you must click the exact location of a point. It also hides inside many other questions: finding slope, graphing a line, reading a chart, finding a missing corner of a rectangle, or measuring distance on a map. If you can place points quickly and correctly, all of those questions get easier.

Real life. Maps, seat charts, spreadsheets, GPS locations and video-game screens all use the same idea: one number for "across" and one number for "up or down".

Words to know

  • coordinate plane — a flat grid made by two number lines that cross at right angles (the grid on graph paper).
  • x-axis — the horizontal number line. Right is positive, left is negative.
  • y-axis — the vertical number line. Up is positive, down is negative.
  • origin — the point where the axes cross, (0,\ 0).
  • ordered pair — two numbers in parentheses, (x,\ y), that name one point. Order matters: (2, 5) and (5, 2) are different points.
  • x-coordinate — the first number; how far left or right of the origin (e.g. in (-4, 1) it is -4, 4 to the left).
  • y-coordinate — the second number; how far up or down (in (-4, 1) it is 1, 1 up).
  • quadrant — one of the four regions the axes cut the plane into, numbered I, II, III, IV counter-clockwise starting at the upper right.
  • plot — to mark a point at its location on the grid.
  • scale — how many units each grid square stands for (e.g. 1, 2, 5 or 10).
  • reflection — a "mirror flip" of a point across an axis (e.g. (3, 2) reflected across the x-axis is (3, -2)).

All Math words in the glossary

The grid and the ordered pair

Coordinate plane from −5 to 5 on both axes with the four quadrants labeled: Quadrant I (+, +) upper right, Quadrant II (−, +) upper left, Quadrant III (−, −) lower left, Quadrant IV (+, −) lower right. Point P(3, 2) is plotted with a dashed path from the origin 3 units right then 2 units up. Point Q(−4, −1) is plotted in Quadrant III.−5−4−3−2−112345−5−4−3−2−1123450xyQuadrant I(+, +)Quadrant II(−, +)Quadrant III(−, −)Quadrant IV(+, −)P(3, 2)Q(−4, −1)

The coordinate plane is two number lines glued together at zero.

  • The x-axis runs left–right. Numbers to the right of zero are positive; to the left they are negative.
  • The y-axis runs up–down. Numbers above zero are positive; below are negative.
  • They cross at the origin, (0, 0).

Every point has an address written as an ordered pair (x,\ y):

  1. The first number, x, says how far to move left or right from the origin.
  2. The second number, y, says how far to move up or down.

A good memory hook: "Walk to the elevator, then ride it." You walk across the floor (x) first, then go up or down (y).

Example. Plot (3,\ 2): start at the origin, walk 3 right, then go 2 up.

Example. Plot (-4,\ -1): start at the origin, walk 4 left, then go 1 down.

Coordinate plane from −6 to 6 on the x-axis and −5 to 5 on the y-axis. To plot (3, 2), a blue arrow walks 3 units right along the x-axis from the origin, then a second arrow rides 2 units up to the point (3, 2). To plot (−4, −1), a red arrow walks 4 units left, then rides 1 unit down to the point (−4, −1).−6−5−3−2−112456−5−4−3−2−1123450xy3 right2 up4 left1 down(3, 2)(−4, −1)
Sign of the numberx (first) meansy (second) means
positivemove rightmove up
negativemove leftmove down
zerodon't move left/rightdon't move up/down

Points on an axis. If a coordinate is 0, you skip that move.

  • (0,\ -3): no left/right move, 3 down → the point is on the y-axis.
  • (5,\ 0): 5 right, no up/down move → the point is on the x-axis.

Fractions and decimals. (1.5,\ -2.5) is halfway between the grid lines: 1.5 right and 2.5 down. \left(-\tfrac{1}{2},\ 3\right) is half a square left and 3 up.

Coordinate plane from −6 to 6 by −5 to 5. Four plotted points: (0, −3) lies on the y-axis; (5, 0) lies on the x-axis; (1.5, −2.5) sits halfway between grid lines, 1.5 right and 2.5 down; (−1/2, 3) is half a square left of the y-axis and 3 up.−6−5−4−3−2−112346−5−4−3−2−112340xy(0, −3) on the y-axis(5, 0)on the x-axis(1.5, −2.5)(−½, 3)

Reading a point from a graph is the same process backwards: drop straight down (or up) to the x-axis to read x, then go straight across to the y-axis to read y.

The four quadrants and their signs

The two axes split the plane into four quadrants. They are numbered with Roman numerals, starting in the upper right and going counter-clockwise (the opposite way a clock turns):

QuadrantLocationSign of xSign of yExample
Iupper right++(3,\ 2)
IIupper left-+(-4,\ 2)
IIIlower left--(-4,\ -1)
IVlower right+-(5,\ -1)
Coordinate plane from −6 to 6 on both axes, divided into four quadrants numbered counter-clockwise. Quadrant I (upper right) has signs (+, +) and contains (3, 2). Quadrant II (upper left) has (−, +) and contains (−4, 2). Quadrant III (lower left) has (−, −) and contains (−4, −1). Quadrant IV (lower right) has (+, −) and contains (5, −1). The points (0, 4) and (−6, 0) lie on the axes and are in no quadrant.−6−5−4−3−2−1123456−6−5−4−3−2−11234560xyI (+, +)II (−, +)III (−, −)IV (+, −)(3, 2)(−4, 2)(−4, −1)(5, −1)(0, 4)(−6, 0)Green points lie on an axis: no quadrant.

So you can name the quadrant without drawing anything — just look at the signs.

  • (-7,\ 0.5): (-,\ +) → Quadrant II.
  • (2,\ -9): (+,\ -) → Quadrant IV.

Points on the axes are in no quadrant. (0,\ 4) lies on the y-axis and (-6,\ 0) lies on the x-axis. They sit on the boundary lines between quadrants. The origin is on both axes.

Working backwards. If a question says "(a,\ b) is in Quadrant II," you know a < 0 and b > 0 — even without numbers. You can also reason: "(a, b) is in Quadrant II, so (-a, b) is in Quadrant I," because changing the sign of a makes it positive.

Moving, flipping and measuring points

Moving (translating) a point.

  • Right k units: add k to x. Left k: subtract k from x.
  • Up k units: add k to y. Down k: subtract k from y.

Example: start at (-2,\ 1), move 5 right and 3 down → (-2 + 5,\ 1 - 3) = (3,\ -2).

Coordinate plane from −5 to 5. The point (−2, 1) moves 5 units right (blue arrow) to (3, 1), then 3 units down (red arrow) to its new position (3, −2). The x-coordinate goes −2 + 5 = 3 and the y-coordinate goes 1 − 3 = −2.−5−4−3−2−11245−4−3−2−12340xy5 right3 downstart (−2, 1)end (3, −2)

Reflecting (flipping) a point.

  • Across the x-axis: keep x, change the sign of y. (5,\ -1) \to (5,\ 1).
  • Across the y-axis: keep y, change the sign of x. (5,\ -1) \to (-5,\ -1).
Coordinate plane from −6 to 6 by −3 to 3. The point (5, −1) is reflected across the x-axis to (5, 1): same x, opposite y. It is reflected across the y-axis to (−5, −1): same y, opposite x. Dashed lines show each mirror flip crossing the axis at a right angle.−6−4−3−2−112346−3−2230xy(5, −1)(5, 1)(−5, −1)across x-axisacross y-axis

Distance along a grid line. When two points share an x- or a y-coordinate, the segment between them is vertical or horizontal. Its length is the difference of the other coordinates (bigger minus smaller).

  • (-3,\ 2) to (4,\ 2): same y, length 4 - (-3) = 7.
  • (4,\ 2) to (4,\ -3): same x, length 2 - (-3) = 5.

When the segment crosses an axis, you are really adding two pieces: 3 units on one side of zero plus 4 on the other is 7.

Coordinate plane from −5 to 6 by −5 to 4. A horizontal segment from (−3, 2) to (4, 2) has length 7: 3 units left of the y-axis plus 4 units right of it. A vertical segment from (4, 2) to (4, −3) has length 5: 2 units above the x-axis plus 3 units below it.−5−4−2−112356−5−4−3−2−11340xy34length 723length 5(−3, 2)(4, 2)(4, −3)

Scaled grids. Not every grid counts by 1. Before counting squares, read the axis labels. If they read 2, 4, 6, \dots then each square is 2 units; if they read 10, 20, 30, \dots each square is 10. A point 3 squares left and 2 squares up on a "by-2" grid is (-6,\ 4), not (-3,\ 2).

Coordinate plane whose grid squares are 2 units each: the x-axis is labelled −10, −8, …, 8, 10 and the y-axis −8, …, 8. A point is 3 squares left and 2 squares up from the origin. Because each square is 2 units, the point is (−6, 4), not (−3, 2).−10−8−6−4−2246810−8−6−424680xy3 squares × 2 = 6 left2 squares × 2 = 4 up(−6, 4)1 square = 2 units
Formula

Rules to memorize (not on the formula sheet)

There is no single formula for plotting, but these rules are used constantly. They are not printed on the GED formula sheet, so memorize them:

\text{point} = (x,\ y) = (\text{left/right},\ \text{up/down}) \text{Quadrants: } \ \text{I}\,(+,+) \quad \text{II}\,(-,+) \quad \text{III}\,(-,-) \quad \text{IV}\,(+,-) \text{Reflect across } x\text{-axis: } (x,\ y) \to (x,\ -y) \qquad \text{Reflect across } y\text{-axis: } (x,\ y) \to (-x,\ y) \text{Horizontal distance} = \lvert x_2 - x_1 \rvert \qquad \text{Vertical distance} = \lvert y_2 - y_1 \rvert
  • x = horizontal position, y = vertical position.
  • \lvert \ \rvert means absolute value: take the difference and drop any minus sign.

(The general distance formula for slanted segments comes from the Pythagorean theorem, which is on the formula sheet — that is covered in a later lesson.)

Worked example

Example 1 — Plot a point with a negative

Problem. Plot the point (2,\ -4). In which quadrant is it?

  1. Start at the origin (0, 0).
  2. x = 2 is positive → move 2 units right.
  3. y = -4 is negative → move 4 units down.
  4. Mark the point. It is right of the y-axis and below the x-axis.
Coordinate plane from −5 to 5 by −6 to 3. Starting at the origin, an arrow moves 2 units right, then a second arrow moves 4 units down to the plotted point (2, −4). The lower-right region, Quadrant IV with signs (+, −), is shaded.−5−4−3−2−11345−6−5−4−3−2−11230xyQuadrant IV (+, −)2 right4 down(2, −4)

Answer: the point is in Quadrant IV.

Check: Sign pattern (+,\ -) matches Quadrant IV ✓

Worked example

Example 2 — A map problem (real world)

Problem. On a city map grid, each unit is 1 block. Maria's apartment is at (-5,\ -1). The grocery store is at (-5,\ 6). She walks straight north along the street. How many blocks does she walk?

  1. Both places have x = -5, so the path is vertical.
  2. Vertical distance = difference of the y-coordinates: 6 - (-1) = 6 + 1 = 7
City map grid, each unit 1 block, x from −7 to 3 and y from −3 to 7. Maria's apartment is at (−5, −1) and the grocery store is at (−5, 6). She walks straight north: 1 block from y = −1 up to the x-axis, then 6 more blocks up to y = 6, for 7 blocks in all.−7−6−3−2−1123−3−2−112345670xy161 + 6 = 7 blockshome (−5, −1)store (−5, 6)

Answer: 7 blocks.

Check: From y = -1 up to 0 is 1 block, and from 0 up to 6 is 6 more: 1 + 6 = 7 ✓

Worked example

Example 3 — Work backwards: the missing corner

Problem. Three corners of a rectangle are J(-3,\ 2), K(4,\ 2) and L(4,\ -3). Find the fourth corner M, then the rectangle's perimeter in units.

  1. J and K share y = 2: top side. K and L share x = 4: right side.
  2. M is the bottom-left corner: same x as J (-3) and same y as L (-3). So M = (-3,\ -3).
  3. Side lengths: JK = 4 - (-3) = 7 and KL = 2 - (-3) = 5.
  4. Perimeter = 2\ell + 2w = 2(7) + 2(5) = 14 + 10 = 24.
Coordinate plane from −5 to 6 by −5 to 4 showing rectangle JKLM. J(−3, 2), K(4, 2) and L(4, −3) are given; the missing corner is M(−3, −3). The top side JK is 7 units long and the right side KL is 5 units long, so the perimeter is 7 + 5 + 7 + 5 = 24 units. Each corner is a right angle.−5−4−2−112356−5−4−2−11340xyJ(−3, 2)K(4, 2)L(4, −3)M(−3, −3)7575

Answer: M = (-3,\ -3); perimeter 24 units.

Check: ML = 4 - (-3) = 7 matches JK, and JM = 2 - (-3) = 5 matches KL ✓

Worked example

Example 4 — GED-hard: move, then reflect

Problem. Point A(-1,\ 3) is moved 4 units right and 6 units down to point B. Then B is reflected across the y-axis to point C. What are the coordinates of C, and which quadrant is it in?

  1. Move right 4: x = -1 + 4 = 3. Move down 6: y = 3 - 6 = -3. So B = (3,\ -3).
  2. Reflect across the y-axis: keep y, change the sign of x. C = (-3,\ -3).
  3. Signs (-,\ -) → Quadrant III.
Coordinate plane from −5 to 5. Point A(−1, 3) moves 4 units right (blue arrow) to (3, 3) and then 6 units down (blue arrow) to B(3, −3). B is then reflected across the y-axis (red dashed line) to C(−3, −3) in Quadrant III. B and C are each 3 units from the y-axis at the same height.−5−4−3−2−11245−5−4−2−112450xy4 right6 down33reflect across y-axisA(−1, 3)B(3, −3)C(−3, −3)

Answer: C = (-3,\ -3), in Quadrant III.

Check: B(3, -3) and C(-3, -3) are both 3 units from the y-axis on opposite sides, at the same height ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

Plotting and reading points is a pencil-and-eyes skill — you do not need the calculator, and reaching for it wastes time.

Use the TI-30XS only for the arithmetic around a point when the numbers get messy:

  • To enter a negative coordinate, use the (−) key (the negative key, bottom row), not the subtraction key. Example: 4 - (-3) is typed 4 − (−) 3 enter → 7.
  • For fractional coordinates, the n/d key enters a fraction, and the toggle key switches the answer between fraction and decimal, so you can see that \tfrac{5}{2} is 2.5 — halfway between 2 and 3 on the grid.

When NOT to use it: deciding quadrants, reading coordinates off a graph, and simple moves like "3 left, 2 up." Do those in your head or on the erasable note board.

Common mistake

Common mistakes

  • Swapping x and y. Plotting (2, -4) at "2 down, 4 right" puts it at (4, -2). ✗ Always go across first (x), then up/down (y). ✓
  • Dropping the negative sign. Plotting (-4, 2) at (4, 2) puts the point in the wrong quadrant. ✗ Negative x = left; negative y = down. ✓
  • Numbering quadrants clockwise. Calling the lower-right region "Quadrant II." ✗ Quadrants go counter-clockwise from the upper right: I, II, III, IV. ✓
  • Putting axis points in a quadrant. Saying (0, 4) is in Quadrant I or II. ✗ A point with a zero coordinate lies on an axis and is in no quadrant. ✓
  • Ignoring the scale. Counting 3 squares left as x = -3 when each square is 2 units. ✗ Read the axis labels first: 3 squares × 2 = 6, so x = -6. ✓
  • Subtracting across zero wrongly. Saying the distance from -3 to 4 is 4 - 3 = 1. ✗ It is 4 - (-3) = 7. ✓
Test strategy

GED test strategy

  • Hot-spot questions: say the move out loud in your head ("3 right, 2 down") and point with the mouse before you click. Click the center of the grid intersection, not near it.
  • Read the question twice. "Reflect across the x-axis" and "reflect across the y-axis" give different answers; "right" vs "left" and "up" vs "down" are easy to misread under time pressure.
  • Check the scale on every graph before counting squares.
  • Eliminate by signs. For "which quadrant" or "which point" questions, cross out every choice with the wrong sign pattern first — often only one choice survives.
  • Plug back in. After finding a missing vertex or a moved point, reverse the move and make sure you return to where you started.
  • Budget time: a plotting question should take under a minute. If you're stuck, mark it, move on, and come back.

Check yourself

Points in the coordinate planeHot spot

Click the point (-4,\ 2) on the coordinate plane.

Click on the figure to mark your answer (or focus it and use the arrow keys).

Coordinate plane from -5 to 5 on both axes-5-5-4-4-3-3-2-2-1-11122334455xy
Hint

Read the ordered pair in order: the first number tells you left/right, the second tells you up/down.

Check yourself

Points in the coordinate planeNo calculator

In which quadrant is the point (-3,\ -5) located?

Answer choices
Hint

Look at the sign of each coordinate, then picture where left/right and up/down send you.

Summary

  • A point is written (x,\ y): across first (right +, left -), then up/down (up +, down -).
  • The origin is (0, 0). Points with a 0 coordinate lie on an axis and are in no quadrant.
  • Quadrants run counter-clockwise: I (+,+), II (-,+), III (-,-), IV (+,-).
  • Move a point by adding/subtracting from x (left/right) or y (up/down). Reflect across the x-axis by changing the sign of y; across the y-axis by changing the sign of x.
  • Horizontal/vertical distance = difference of coordinates; across zero, 4 - (-3) = 7.
  • Always read the scale before counting grid squares.
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