Geometry Reference
Coordinate Transformations
Transformations move, flip, turn, or resize a figure while preserving the geometric information that the rule allows.
Solved Examples
Jump directly to a worked example by transformation type.
Fact Table
| Transformation | Coordinate Rule | What It Preserves |
|---|---|---|
| Translation | $(x,y)\to(x+a,y+b)$ | Lengths, angles, and orientation. |
| Reflection across y-axis | $(x,y)\to(-x,y)$ | Lengths and angles, but reverses orientation. |
| $90^\circ$ counterclockwise rotation | $(x,y)\to(-y,x)$ | Lengths, angles, and orientation. |
| Dilation about origin | $(x,y)\to(kx,ky)$ | Angles and shape; lengths scale by $|k|$. |
Content Formulas
Reflection Across x-axis
$$(x,y)\to(x,-y)$$
$180^\circ$ Rotation
$$(x,y)\to(-x,-y)$$
Area Under Dilation
$$A' = k^2A$$
A rotation rule is ordered: $90^\circ$ counterclockwise is not the same as $90^\circ$ clockwise. State both the angle and direction.
Classic Examples
Translate a Point
Translate $P(2,-1)$ right $3$ units and down $2$ units.
Solution Steps
- Add $3$ to the x-coordinate.
- Subtract $2$ from the y-coordinate.
- State the translated point.
$$x'=2+3=5$$$$y'=-1-2=-3$$$$\boxed{P'(5,-3)}$$
Reflect Across the y-axis
Reflect $Q(-4,3)$ across the y-axis.
Solution Steps
- Use the y-axis reflection rule.
- Change only the sign of the x-coordinate.
- State the reflected point.
$$(-4,3)\to(-(-4),3)$$$$x'=4,\qquad y'=3$$$$\boxed{Q'(4,3)}$$
Rotate Counterclockwise
Rotate $R(2,-1)$ by $90^\circ$ counterclockwise about the origin.
Solution Steps
- Use $(x,y)\to(-y,x)$.
- Substitute the point coordinates in that order.
- State the rotated point.
$$(x,y)\to(-y,x)$$$$(2,-1)\to(-(-1),2)$$$$\boxed{R'(1,2)}$$
Dilate About the Origin
Dilate $S(-2,3)$ by scale factor $2$ about the origin.
Solution Steps
- Multiply the x-coordinate by the scale factor.
- Multiply the y-coordinate by the scale factor.
- State the image point.
$$x'=2(-2)=-4$$$$y'=2(3)=6$$$$\boxed{S'(-4,6)}$$