☰ خطة العام · الرياضيات (بالإنجليزية)

The coordinate plane and reflections

الأسبوع 24 · الأرباع الأربعة والأزواج المرتبة والانعكاس · 3 دروس
3

Reflecting a point across the x-axis or the y-axis

Discover Mathematics P6: Term 2, Unit 11, Lessons 1–3
GoalReflect a point across the x-axis or the y-axis and compare the coordinates.
New words
reflection · the mirror image of a point across a lineopposite · the number on the other side of zero at the same distance, such as 3 and −3error · a mistake that we can find and correct
Explanation

A reflection across an axis is a mirror image. The mirror is the axis. The point and its image are the same distance from the axis, on opposite sides. If you reflect across the x-axis, the point moves straight up or down, so x stays the same and y changes to its opposite: (3, 2) becomes (3, −2). If you reflect across the y-axis, the point moves straight left or right, so y stays the same and x changes to its opposite: (3, 2) becomes (−3, 2). A common error is to swap the two numbers, as in writing (2, −3) for the point (−3, 2). Swapping is not a reflection; it gives a different point.

Reflect acrossRule(3, 2) becomes
x-axis(x, y) → (x, −y)(3, −2)
y-axis(x, y) → (−x, y)(−3, 2)
Worked examples
xy−5−4−3−2−112345−5−4−3−2−1123450PP′
Reflect P(3, 2) across the x-axis. The x stays 3 and the y changes to −2. The image is P′(3, −2). It is 2 units below the axis, just as P is 2 units above it.
xy−5−4−3−2−112345−5−4−3−2−1123450PP″
Reflect P(3, 2) across the y-axis. The y stays 2 and the x changes to −3. The image is P″(−3, 2), which is in Quadrant II. P is in Quadrant I.
Find the error. Seif wants to plot (−3, 2). He goes 2 left and 3 up, so he reaches (−2, 3). He swapped the two numbers. The correct move is 3 left, then 2 up.
Class activity

Mirror game. Draw the x-axis on paper and put a small drawing close to it. Fold along the axis and copy it on the other side. Then place points on both halves and compare their coordinates.

Practice
1
Reflect (−4, 1) across the x-axis.
2
Reflect (−4, 1) across the y-axis.
3
Reflect (5, −2) across the x-axis.
4
Which reflection moves (3, 4) to (3, −4)?
5
Reflect (−2, −6) across the y-axis. In which quadrant is the image?