Points on the Axes and Mirror Images

The two axes are the special "highways" of the coordinate plane: points on them have a coordinate equal to zero. Reflecting a point across an axis or the origin is one of the most common operations in coordinate geometry, and it boils down to flipping signs. This lesson sets up the three rules and gives you a feel for them.

Definitions

A point on the $x$-axis has $y$-coordinate $0$ and is of the form $(a, 0)$. A point on the $y$-axis has $x$-coordinate $0$ and is of the form $(0, b)$. The origin $(0, 0)$ is on both axes simultaneously.

A mirror image (or reflection) of a point $P$ in a line $\ell$ is the point $P'$ such that $\ell$ is the perpendicular bisector of the segment $PP'$. Geometrically, $P'$ is on the opposite side of $\ell$ at the same perpendicular distance.

Concept and the three sign-flip rules

Rule 1: Reflection across the $x$-axis sends $(x, y) \mapsto (x, -y)$. The horizontal coordinate is unchanged; the vertical coordinate flips sign.

Why. The $x$-axis is horizontal; reflecting across a horizontal line preserves the horizontal position and reverses the vertical position. The point $(3, 5)$ reflects to $(3, -5)$.

Rule 2: Reflection across the $y$-axis sends $(x, y) \mapsto (-x, y)$. The horizontal coordinate flips sign; the vertical coordinate is unchanged.

Why. The $y$-axis is vertical; reflection preserves vertical position. The point $(3, 5)$ reflects to $(-3, 5)$.

Rule 3: Reflection across the origin sends $(x, y) \mapsto (-x, -y)$. Both coordinates flip sign.

Why. Reflection through a point sends $P$ to the point on the line through $O$ and $P$, at the same distance, on the opposite side. This is the same as applying Rule 1 followed by Rule 2.

Compositions.

• Reflect in $x$-axis, then in $y$-axis: $(x, y) \mapsto (x, -y) \mapsto (-x, -y)$. Same as reflection through the origin.
• Reflect in $y$-axis, then in $x$-axis: same end result, $(-x, -y)$.
• Reflect across $y = x$ (an axis-bisector): swaps the coordinates, $(x, y) \mapsto (y, x)$. We will see this once we discuss the line $y = x$ in the next chapter.

Distances are preserved. Reflection is a rigid motion: distances and angles do not change. If $A, B$ are two points and $A', B'$ are their reflections, then $|AB| = |A'B'|$.

Quadrant changes. Each reflection moves a point from one quadrant to a specific other:

• $x$-axis reflection: I $\leftrightarrow$ IV, II $\leftrightarrow$ III.
• $y$-axis reflection: I $\leftrightarrow$ II, III $\leftrightarrow$ IV.
• Origin reflection: I $\leftrightarrow$ III, II $\leftrightarrow$ IV.

Identifying axis points.

• A point with $y$-coordinate zero is on the $x$-axis.
• A point with $x$-coordinate zero is on the $y$-axis.
• The intersection of the axes is the origin, with both coordinates zero.

Distance from a point to an axis. The perpendicular distance from $(x, y)$ to the $x$-axis is $|y|$. The perpendicular distance to the $y$-axis is $|x|$. The distance to the origin uses Pythagoras: $\sqrt{x^2 + y^2}$. (We will not prove the last formula here; it follows from the Pythagoras theorem.)

Worked examples

Example 1. Find the mirror image of $(4, -3)$ across the $x$-axis.

By Rule 1, $(x, y) \mapsto (x, -y)$: image is $(4, 3)$.

Example 2. Find the mirror image of $(-5, 2)$ across the $y$-axis.

By Rule 2, $(x, y) \mapsto (-x, y)$: image is $(5, 2)$.

Example 3. Find the mirror image of $(7, -8)$ across the origin.

By Rule 3, $(x, y) \mapsto (-x, -y)$: image is $(-7, 8)$.

Example 4. Find the perpendicular distance from $(-3, 6)$ to the $x$-axis and to the $y$-axis.

To the $x$-axis: $|6| = 6$. To the $y$-axis: $|-3| = 3$.

Example 5. A point lies on the $x$-axis $4$ units to the right of the origin. Find its mirror image across the $y$-axis.

Point: $(4, 0)$. Reflecting across the $y$-axis gives $(-4, 0)$, on the $x$-axis $4$ units to the left.

Try it yourself

1. Find the reflection of $(5, 6)$ across the $x$-axis.
2. Find the reflection of $(5, 6)$ across the $y$-axis.
3. Find the reflection of $(5, 6)$ across the origin.
4. Reflect $(-3, -7)$ across the $x$-axis. In which quadrant is the image?
5. The reflection of a point $P$ across the $x$-axis is $(2, -4)$. Find $P$.
6. State the perpendicular distance from $(-2, 9)$ to the $x$-axis.
7. State the perpendicular distance from $(-2, 9)$ to the $y$-axis.
8. A point lies on the $x$-axis. What is its reflection across the $x$-axis?
9. A point lies on the $y$-axis. What is its reflection across the $y$-axis?
10. Are the points $(3, 4)$ and $(-3, -4)$ mirror images? If yes, in which line?

Pitfalls / Insight

• Be careful about which coordinate flips. Across the $x$-axis flips the $y$-sign; across the $y$-axis flips the $x$-sign. The flipped coordinate is the one perpendicular to the mirror line.
• A point on the axis is its own mirror image in that axis. $(3, 0)$ reflected in the $x$-axis is still $(3, 0)$.
• Origin reflection is not the same as a single axis reflection. Both signs change.

Insight. Every reflection in this lesson is a sign change rule. Memorise which sign(s) change for each axis or for the origin, and reflection problems collapse to a single line of arithmetic. The same idea will return in Class X as the foundation for the formal "distance formula" and "midpoint formula".