The Lumenary · Grades 6–8

The coordinate plane

Name: ______________________

Date: ______________________

You will learn to

Plot and read points $(x, y)$ with integer and fraction coordinates in all four quadrants
Explain why every point corresponds to exactly one ordered pair, and why the order of the coordinates matters
Find horizontal and vertical distances between points using absolute value, and distances along grid lines
Reflect points in the axes, describe the effect on their coordinates and quadrants, and prove that reflecting twice returns every point

1.
In which quadrant is the point $(-3, 5)$ ?
- A.Quadrant I
- B.Quadrant II
- C.Quadrant III
- D.Quadrant IV
2.
Reflect the point $(4, -7)$ in the $y$ -axis. Where does it land?
- A. $(-4, -7)$
- B. $(4, 7)$
- C. $(-4, 7)$
- D. $(-7, 4)$
3.
What is the distance between $(-6, 3)$ and $(5, 3)$ ?
4.
A cyclist in a city of square blocks rides from $(-2, -3)$ to $(4, 5)$ along the streets, never doubling back. How many blocks does she ride?
5.
A point $P$ is reflected in the $x$ -axis and then the result is reflected in the $y$ -axis. It ends up at $(-2, 5)$ . Where was $P$ ?
- A. $(2, -5)$
- B. $(-2, 5)$
- C. $(2, 5)$
- D. $(-2, -5)$
6.
Kofi was asked to plot $(3, 2)$ . He started at the origin, went up 3 and then right 2. Which point did he actually plot?
- A. $(2, 3)$
- B. $(3, 2)$
- C. $(-3, 2)$
- D. $(3, 3)$
7.
Which point is exactly halfway between $(-3, 2)$ and $(4, 2)$ ?
- A. $(\tfrac{1}{2}, 2)$
- B. $(1, 2)$
- C. $(3\tfrac{1}{2}, 2)$
- D. $(\tfrac{1}{2}, 0)$
8.
Which statements are true for every point $(a, b)$ ? (Choose all that apply.)
- A.Reflecting in the $x$ -axis and then the $y$ -axis gives the same point as the other order.
- B.Reflection in the $x$ -axis keeps the $x$ -coordinate.
- C.Reflection in the $x$ -axis keeps the point in the same quadrant.
- D.The distance from $(a, b)$ to $(-a, b)$ is $2|a|$ .
- E.The distance from $(a, b)$ to $(-a, b)$ is $2a$ .
9.
In the Lean file, reflectX P is the reflection of the point P in the $x$ -axis. Which statement says that reflecting twice in the $x$ -axis brings every point back?
- A.∀ P : Point, reflectX (reflectX P) = P
- B.∀ P : Point, reflectX P = P
- C.∀ P : Point, reflectX (reflectY P) = P
- D.∃ P : Point, reflectX (reflectX P) = P
10.
Write a function quadrant(x, y) that returns 1, 2, 3 or 4 for a point in quadrant I, II, III or IV, and 0 for a point on an axis. It should work for fractions too.
Computational — Practice

Answer key — The coordinate plane

1.
$x = -3 \lt 0$ , so the point is left of the $y$ -axis; $y = 5 > 0$ , so it is above the $x$ -axis. Left and up is quadrant II.
2.
Reflection in the $y$ -axis sends $(a, b)$ to $(-a, b)$ : the point moves straight across the vertical mirror to the same distance on the other side. So $(4, -7) \mapsto (-4, -7)$ .
3.
The points lie on the horizontal line $y = 3$ , so the distance is $|-6 - 5| = |-11| = 11$ : 6 units from $x = -6$ to $0$ , and 5 more to $x = 5$ .
4.
She rides $|-2 - 4| = 6$ blocks east and $|-3 - 5| = 8$ blocks north: $6 + 8 = 14$ blocks, whatever route she takes.
5.
Each reflection undoes itself, so work backwards: reflect $(-2, 5)$ in the $y$ -axis to get $(2, 5)$ , then in the $x$ -axis to get $(2, -5)$ . So $P = (2, -5)$ . Equivalently, the two reflections together are the half turn $(a, b) \mapsto (-a, -b)$ .
6.
Right 2 and up 3 is $(2, 3)$ . To plot $(3, 2)$ he should go 3 right (the $x$ -coordinate comes first) and 2 up. The two points are mirror images of each other in the line $y = x$ .
7.
The points are $7$ apart on the line $y = 2$ . Halfway means $3.5$ from each: $-3 + 3.5 = \tfrac12$ and $4 - 3.5 = \tfrac12$ . The midpoint is $(\tfrac12, 2)$ — a point with a fraction coordinate. In general the midpoint of $(a, b)$ and $(c, b)$ is $(\tfrac{a + c}{2}, b)$ .
8.
(a), (b) and (d) are always true. (c) fails for every point off the axes: reflection in the $x$ -axis moves it to the quadrant on the other side of the $x$ -axis. (e) fails whenever $a \lt 0$ ; the absolute value in (d) is exactly what fixes it.
9.
The statement is ∀ P : Point, reflectX (reflectX P) = P. Option (b) is false in general, (c) describes a different composite (the half turn), and (d) uses ∃ (“there exists”) instead of ∀ (“for all”).

10.

def quadrant(x, y):
    if x == 0 or y == 0:
        return 0
    if x > 0 and y > 0:
        return 1
    if x < 0 and y > 0:
        return 2
    if x < 0 and y < 0:
        return 3
    return 4

After the axis test, both coordinates are non-zero, so the only remaining case after the first three is $x > 0$ , $y \lt 0$ : quadrant IV.