Geometric Transformations

Geometric Transformation

A geometric transformation moves a figure while keeping its shape and size the same. A translation adds one vector (a, b) to every point so the figure slides, yet the equation uses −a and −b instead. A reflection mirrors the figure across the x-axis, y-axis, origin, or line y = x, flipping a sign or swapping x and y. Slide a triangle here and switch the four reflections to see each rule yourself.

Translation — Same Shape, New Position

📦 Like Moving a Package

①Shape, size, orientation stay the same

②Only the position changes — that's translation

③Mathematically: add the same vector (a, b) to every point

④The whole figure 'slides' to a new spot

x-shift a2

y-shift b1

Translation of a Point

(x, y) → (x + a, y + b)

Add the same vector (a, b) to every point

Translating Equations — Why Subtract?

Translation of an Equation

f(x, y) = 0 → f(x − a, y − b) = 0

Points add (+a, +b), but the equation uses −a, −b

🤔 Key Question: Why minus?

①Let (X, Y) be a point on the translated figure

②It came from (X−a, Y−b) on the original

③Original satisfies f(x, y) = 0

④So f(X−a, Y−b) = 0 is the equation of the translated figure

⑤i.e., subtract the shift to recover the original position

Example: Translating a Circle

x² + y² = r² → (x−a)² + (y−b)² = r²

Centered at origin → centered at (a, b)

Example: Translating a Parabola

y = x² → y − b = (x − a)²

Vertex moves from (0, 0) to (a, b)

Reflection — Mirror Image

Reflection Formulas

ListFour Reflections

About x-axis

Flip y only

(x, y) → (x, −y)

About y-axis

Flip x only

(x, y) → (−x, y)

About origin

Flip both x and y

(x, y) → (−x, −y)

About y=x

Swap x and y

(x, y) → (y, x)

💡 Why Reflections Work

①About x-axis: x-axis is the mirror → top↔bottom → y flips

②About y-axis: y-axis mirror → left↔right → x flips

③Through origin: 180° rotation → both flip

④About y=x: 45° diagonal mirror → swap x and y (inverse function!)

Composing Transformations — Does Order Matter?

🔗 Composition Rules

①Translation + Translation: just add vectors (order doesn't matter)

②Reflection + Reflection: x-axis then y-axis = origin

③Translation + Reflection: order changes the result!

④Exam: 'reflect (2,3) over x-axis, then translate by (1,−2)' → do each step in order!

Composed Reflections

ChartResult of Composing Reflections

Composition	Result	Equivalent
x-axis → y-axis	(x,y)→(−x,−y)	Origin reflection
y-axis → x-axis	(x,y)→(−x,−y)	Origin reflection
x-axis → y=x	(x,y)→(−y,x)	90° CCW rotation
y=x → x-axis	(x,y)→(y,−x)	90° CW rotation

Work It Out

Example 1

Translate the point (3, −2) by 4 along the x-axis and −5 along the y-axis. Find the image.

Translation simply adds the shift to each coordinate.

(3 + 4, −2 + (−5))

Compute.

(7, −7)

▸ (7, −7)

Translating a point just adds the shift to its coordinates.

Example 2

Translate the line y = 2x + 1 by 3 along the x-axis. Find its equation.

Shifting a figure by 3 along the x-axis replaces x with x − 3 in the equation.

y = 2(x − 3) + 1

Expand and simplify.

y = 2x − 6 + 1 = 2x − 5

▸ y = 2x − 5

For a figure, substitute x − a (the opposite of what a point seems to do).

Wrap-up

Point Translation

(x+a, y+b)

Points: add

Equation Translation

f(x−a, y−b)=0

Equations: subtract

Grade-10 school exam type

Reflect the point (4, −3) across the line y = x. What is the image?

①(−4, 3)

②(3, −4)

③(−3, 4)

④(4, 3)

⑤(−4, −3)

▸ ③ (−3, 4)

Reflection across y = x swaps the x- and y-coordinates.

(a, b) → (b, a)

Swap the coordinates.

(4, −3) → (−3, 4)

🎯 Exam Points

①Point: (x+a, y+b); equation: f(x−a, y−b)=0 — opposite signs!

②x-axis reflection → substitute −y for y; y-axis → −x for x

③y=x reflection → swap x and y (inverse function!)

④Translating a circle: add the vector to the center

⑤Composition: apply step by step; order matters!

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Equations of a Circle

Sets & Propositions

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