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E1.3 · Plot four-quadrant coordinates and describe translations

Learn to plot four-quadrant coordinates and describe translations through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Spatial Sense

Locate points in all four quadrants and describe how they slide

Imagine a map with streets running east-west and north-south. The streets meet at one starting place. A coordinate grid works in a similar way: two number lines cross, and their meeting point is the starting place. A coordinate pair tells where a point is. A translation is a slide from one place to another. In this lesson, you will use grid directions and unit steps to plot points and describe slides.

What you will learn

  • Read and plot coordinate pairs in all four quadrants.
  • Use positive and negative coordinates to describe a point's position.
  • Describe a translation by its direction and number of grid units.
  • Find the new position of a point after a stated translation.

1. Meet the coordinate grid

A number line helps you understand the grid. On a horizontal number line, numbers get greater as you move right and smaller as you move left. On a vertical number line, numbers get greater as you move up and smaller as you move down. A negative number is less than zero. For example, moving two units left from zero reaches −2-2.
The horizontal number line is called the x-axis. The vertical number line is called the y-axis. They cross at the origin, which is the point (0,0)(0,0). Each space between neighbouring grid lines is one grid unit, unless the grid says otherwise.
The two axes divide the grid into four regions called quadrants. The upper-right quadrant is where points are right of the y-axis and above the x-axis. The upper-left quadrant is left and above. The lower-left quadrant is left and below. The lower-right quadrant is right and below. A point on an axis is not inside a quadrant.
A coordinate pair gives two positions in a set order. The first number tells how far left or right the point is. The second tells how far up or down it is. For example, (3,−2)(3,-2) means three units right and two units down from the origin.
The signs help you check where a point belongs. A positive first coordinate means right; a negative first coordinate means left. A positive second coordinate means up; a negative second coordinate means down. A coordinate of zero means the point is on one of the axes.
  • Read the horizontal position first and the vertical position second.
  • Right and up use positive numbers; left and down use negative numbers.
  • A point with a zero coordinate lies on an axis.

2. Plot a point and name its quadrant

To plot a point, start at the origin. First follow the horizontal direction given by the first coordinate. Then move up or down according to the second coordinate. Place a dot where you stop and write the point's label beside it.
For example, to plot (−4,1)(-4,1), move four units left from the origin. Then move one unit up. The point is left of the y-axis and above the x-axis, so it is in the upper-left quadrant.
You can also name a point that is already plotted. Count its horizontal distance from the y-axis and decide whether it is left or right. Then count its vertical distance from the x-axis and decide whether it is above or below. Write the horizontal value first.
A quick check is to compare the signs with the point's location. A point left and below must have two negative coordinates. A point right and below must have a positive first coordinate and a negative second coordinate.
  • Begin at the origin when plotting.
  • Move horizontally first, then vertically.
  • Check that the signs match the point's location.

3. Describe a translation

A translation is a slide. It moves a point or shape without turning or flipping it. Every point of a translated shape moves the same number of grid units in the same direction.
Describe a translation with clear directions and counts. For example, “move three units left and two units up” tells both how far to move and which way to go. The count is measured in grid units.
When a point moves right or left, its horizontal position changes while its vertical position stays the same. When it moves up or down, its vertical position changes while its horizontal position stays the same. You can track the two moves separately using the grid.
For a shape, match each original corner with its new corner. A corner is also called a vertex. If each matching corner has moved the same way and the same distance, you can describe the slide for the whole shape.
  • A translation is a slide, not a turn or flip.
  • State the direction and number of grid units.
  • All points in a translated shape make the same move.

Worked example

Plot a point in the lower-left quadrant

Plot point A(−3,−2)A(-3,-2) and name its quadrant.
Point A in the lower-left quadrant
Point A in the lower-left quadrantThe point is three units left and two units down from the origin.xyA(-3,-2)OriginHypothetical example

The point is three units left and two units down from the origin.

  1. Move horizontally
    The first coordinate is −3-3. From the origin, move three grid units left because the coordinate is negative.
    −3-3
  2. Move vertically
    The second coordinate is −2-2. From that place, move two grid units down.
    −2-2
  3. Check the quadrant
    The point is left of the y-axis and below the x-axis. Those directions place it in the lower-left quadrant.
    A(−3,−2)A(-3,-2)
Answer: Plot AA three units left and two units down from the origin. It is in the lower-left quadrant.
Check: Both coordinates are negative. That matches a point left of the y-axis and below the x-axis.

Worked example

Name a point from its location

Point BB is two units right of the y-axis and four units below the x-axis. Write its coordinates and name its quadrant.
Point B in the lower-right quadrant
Point B in the lower-right quadrantThe point is two units right and four units down from the origin.xyB(2,-4)OriginHypothetical example

The point is two units right and four units down from the origin.

  1. Read the horizontal position
    The point is right of the y-axis, so its first coordinate is positive. Its distance from that axis is two grid units.
    22
  2. Read the vertical position
    The point is below the x-axis, so its second coordinate is negative. Its distance from that axis is four grid units.
    −4-4
  3. Write the pair and name the quadrant
    Write the horizontal value first and the vertical value second. A point right and below is in the lower-right quadrant.
    B(2,−4)B(2,-4)
Answer: The coordinates are B(2,−4)B(2,-4). Point BB is in the lower-right quadrant.
Check: The first coordinate is positive because the point is right of the y-axis. The second is negative because it is below the x-axis.

Worked example

Translate a point

Point C(−2,1)C(-2,1) moves four units right and three units down. Describe the translation and find the new coordinates.
Point C before and after a translation
Point C before and after a translationThe stated slide moves C four units right and three units down to C′.xyC(-2,1)C′(2,-2)Hypothetical example

The stated slide moves C four units right and three units down to C′.

  1. Follow the horizontal move
    Point CC starts two units left of the y-axis. Moving four units right takes it to two units right of the y-axis.
    −2+4=2-2+4=2
  2. Follow the vertical move
    The point starts one unit above the x-axis. Moving three units down takes it to two units below the x-axis.
    1−3=−21-3=-2
  3. State the result
    The stated slide is four units right and three units down. The new point is right of the y-axis and below the x-axis.
    C′(2,−2)C'(2,-2)
Answer: The translation is four units right and three units down. The new coordinates are C′(2,−2)C'(2,-2).
Check: The horizontal position changes from −2-2 to 22. The vertical position changes from 11 to −2-2, matching the stated right and down moves.

Common mistakes and how to avoid them

Reading the vertical coordinate first.
Correction: Write and read the horizontal position first, then the vertical position.
Thinking every negative coordinate means down.
Correction: A negative first coordinate means left. A negative second coordinate means down.
Changing the vertical position during a left or right move.
Correction: Left and right change only the horizontal position. Up and down change only the vertical position.
Calling a point on an axis part of a quadrant.
Correction: The axes mark the edges of the quadrants. A point on an axis is not inside one.

Lesson summary

  • Coordinates name a horizontal position first and a vertical position second.
  • Use right and up for positive directions; use left and down for negative directions.
  • A point's signs and position show which quadrant it is in.
  • Describe a translation by naming its direction and number of grid units.

Check your understanding

Question 1

Which coordinates describe a point four units left and three units up from the origin?
  1. (4,3)(4,3)
  2. (−4,3)(-4,3)
  3. (−3,4)(-3,4)
  4. (4,−3)(4,-3)
Show answer and explanation
(−4,3)(-4,3)
Left gives a negative first coordinate, and up gives a positive second coordinate. The pair is (−4,3)(-4,3).

Question 2

Point D(1,−2)D(1,-2) moves two units left and one unit up. What are its new coordinates?
  1. (3,−3)(3,-3)
  2. (−1,−3)(-1,-3)
  3. (−1,−1)(-1,-1)
  4. (1,−1)(1,-1)
Show answer and explanation
(−1,−1)(-1,-1)
Moving two units left takes the horizontal position from 11 to −1-1. Moving one unit up takes the vertical position from −2-2 to −1-1. The new coordinates are (−1,−1)(-1,-1).

Key terms

Coordinate grid
A grid made by a horizontal number line and a vertical number line.
Origin
The point where the two axes meet, written as (0,0)(0,0).
Quadrant
One of the four regions made by the axes.
Coordinate pair
Two numbers that give a point's horizontal and vertical positions, in that order.
Translation
A slide that moves a point or shape the same distance in the same direction.
Vertex
A corner of a shape.

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