Reflection

Chapter in a nutshell: Reflection flips a point/figure across a mirror line (an axis, the origin, or a line $x=a$ / $y=b$). Each reflection has a simple coordinate rule. Points that map onto themselves are invariant points.

1. Key Ideas

• A reflection is a mirror image; the mirror line is the axis/line of reflection.
• The image is the same size (reflection is an isometry); the figure is laterally inverted.
• Invariant point: a point that stays in place (lies on the mirror line).

2. Formulas (quick reference — all reflection rules)

• In the x-axis: $M_x(x,\,y)=(x,\,-y)$
• In the y-axis: $M_y(x,\,y)=(-x,\,y)$
• In the origin: $(x,\,y)\to(-x,\,-y)$
• In the line $x=a$ (vertical): $(x,\,y)\to(2a-x,\,y)$
• In the line $y=b$ (horizontal): $(x,\,y)\to(x,\,2b-y)$
• In the line $y=x$: $(x,\,y)\to(y,\,x)$
• In the line $y=-x$: $(x,\,y)\to(-y,\,-x)$
• Invariant points: on the x-axis $y=0$; on the y-axis $x=0$; on $x=a$ the points with that x; etc.

3. Prerequisite Ideas

• Cartesian plane: plotting $(x, y)$; the four quadrants and the sign of coordinates.

4. Worked Example (one, for the method)

5. Common Mistakes to Avoid

• Swapping the rules: x-axis changes the sign of y; y-axis changes the sign of x.
• For $x=a$, using $a-x$ instead of $2a-x$.
• Forgetting that reflection keeps the figure the same size (only flips it).
• Missing that points on the mirror line are invariant.

6. Likely Exam Questions (with crisp answers)

1. Rule for reflection in the x-axis? → $(x,y)\to(x,-y)$.
2. Rule for reflection in the y-axis? → $(x,y)\to(-x,y)$.
3. Rule for reflection in the origin? → $(x,y)\to(-x,-y)$.
4. Rule for the line $x=a$? → $(x,y)\to(2a-x,y)$.
5. Rule for the line $y=b$? → $(x,y)\to(x,2b-y)$.
6. What is an invariant point? → A point that maps onto itself (lies on the mirror line).
7. Reflect (2, −3) in the y-axis. → (−2, −3).
8. Which points are invariant under reflection in the x-axis? → All points with $y=0$ (on the x-axis).
9. Does reflection change the size of a figure? → No.
10. Rule for reflection in $y=x$? → $(x,y)\to(y,x)$.

Extended & Prerequisite Formulas (full)

• x-axis (x,y)→(x,−y); y-axis →(−x,y); origin →(−x,−y); x=a →(2a−x,y); y=b →(x,2b−y); y=x →(y,x); y=−x →(−y,−x).
• Invariant: on x-axis y=0; on y-axis x=0; on x=a all points with that x.
• Prereq: plotting (x,y), quadrant sign rules (+,+),(−,+),(−,−),(+,−); distance √[(x₂−x₁)²+(y₂−y₁)²].