Reflection

A reflection is a rigid flip: the mirror line acts like a hinge, and every point swaps sides while staying the same perpendicular distance from the line. For a vertical mirror the x-coordinate is the one measured perpendicular to the line, so it flips; for a horizontal mirror the y-coordinate flips. Everything below follows from that one idea. Notice too that a reflection preserves shape and size exactly — if you reflect every vertex of a triangle, the image is congruent to the original triangle, just flipped, which is why reflections belong to the family of isometries (distance-preserving transformations) alongside translations and rotations. It helps to separate the five prescribed reflections into two families. The first three — reflection in the y-axis, the x-axis, and the origin — are all reflections in lines (or a point) that pass through the origin, so their rules involve only a sign change, nothing more. The second two — reflection in $x=a$ and $y=a$ — are reflections in lines that may sit anywhere on the plane, so their rules involve the constant $a$ as well as a sign change. Once you see the origin cases as the special case $a=0$ of the general vertical or horizontal mirror, all five rules collapse into just two patterns to remember.

Reflection in the y-axis, $x=0$ (a vertical line): only the x-coordinate flips sign.

Reflection in the x-axis, $y=0$ (a horizontal line): only the y-coordinate flips sign.

Reflection in the origin: both coordinates flip sign — the same result as reflecting in $x=0$ and then in $y=0$.

Reflection in the vertical line $x=a$: the y-coordinate is untouched; the x-coordinate reflects about $a$ rather than about $0$.

where: $a$ is the fixed x-value of the mirror line, and $2a-x$ is the point on the opposite side of $a$ that is the same distance from $a$ as $x$ is.

Reflection in the horizontal line $y=a$: the x-coordinate is untouched; the y-coordinate reflects about $a$.

An invariant point is one whose image is itself, $P'=P$. Every point on the mirror line is invariant under reflection in that line — a point on $x=0$ (the y-axis) stays put when reflected in $x=0$; a point on $y=a$ stays put when reflected in $y=a$. Only the origin is invariant under reflection in the origin, since $(-x,-y)=(x,y)$ forces $x=0$ and $y=0$ together. This is worth stating as a general principle: a point is invariant under a given reflection precisely when it already satisfies the equation of the mirror line, because a point on the mirror is its own perpendicular foot, so there is zero distance to reflect across. For $x=a$, that means any point with x-coordinate exactly $a$ is invariant, no matter what its y-coordinate is — the mirror line $x=a$ contains infinitely many such points, one for every value of $y$.

Question: The point $Q(4,7)$ is reflected in the line $x=4$ to get $Q'$, and $Q'$ is then reflected in the line $y=2$ to get $Q''$. Find $Q'$ and $Q''$, and state whether $Q$ is invariant under the first reflection.

Step 1. Reflect $Q(4,7)$ in $x=4$ using $(x,y)\to(2a-x,\ y)$ with $a=4$: $Q'\big(2(4)-4,\ 7\big) = Q'(4,7)$.

Step 2. Compare $Q'$ with $Q$. Since $Q'=(4,7)$ is identical to $Q=(4,7)$, the point $Q$ is invariant under reflection in $x=4$ — which makes sense, because $Q$'s x-coordinate is already $4$, so $Q$ lies exactly on the mirror line.

Step 3. Reflect $Q'(4,7)$ in $y=2$ using $(x,y)\to(x,\ 2a-y)$ with $a=2$: $Q''\big(4,\ 2(2)-7\big) = Q''(4,-3)$.

Final answer: $Q'=(4,7)$ and $Q''=(4,-3)$; $Q$ is an invariant point under the first reflection (in $x=4$) but not under the second (in $y=2$), since $Q''\ne Q'$.

Competitive stretch

If instead $Q(4,7)$ were reflected in $y=7$ first, would it be invariant? Check: $y=7$ passes through $Q$'s y-coordinate exactly, so $(x,\,2(7)-7)=(4,7)=Q$ — yes, invariant, by the same logic as Step 2.

• Mixing up which coordinate changes: reflection in $x=a$ (a vertical line) changes the x-coordinate; reflection in $y=a$ (a horizontal line) changes the y-coordinate. Match the axis letter in the line's equation to the coordinate that changes.
• Writing $(a-x, y)$ instead of the correct $(2a-x, y)$ for reflection in $x=a$ — a common arithmetic slip; always double the mirror's coordinate before subtracting.
• Assuming a point is invariant because one coordinate looks close to the mirror's value — invariance requires the coordinate to be exactly equal to $a$, not merely nearby.
• Confusing reflection in the origin (both coordinates flip) with reflection in a single axis (only one coordinate flips).

Ask the tutor: want to check a reflection image you calculated, or try chaining three reflections in a row to see where a point ends up?