Moving a figure changes only where it is or which way it faces, while its shape and size stay the same.
A translation slides a figure by adding the same amount (a, b) to every point, yet in the figure's equation you put x − a and y − b in place of x and y, which is easy to get backwards.
A reflection mirrors the figure in the x-axis, the y-axis, the origin or the line y = x, flipping the sign of a coordinate or swapping x and y.
In the first picture the triangle moves with the amount of translation; in the second you can compare side by side how the vertex coordinates change under the four reflections.
Translation: same shape, new place
📦 Like sliding a box across a desk
①Slide a box across a desk and its shape, size and orientation stay the same
②Only its position changes; that is a translation
③In maths, the same amount (a, b) is added to every point of the figure
④So all three vertices move the same distance in the same direction
seegongsik.com
The original triangle (blue) and the triangle translated by (a, b) (gold), with the coordinates of each vertex
2
1
Translating a point
(x, y) → (x + a, y + b)
Add the same amount (a, b) to every point
Translating an equation: why subtract?
Translating the equation of a figure
f(x, y) = 0 → f(x − a, y − b) = 0
Points get a and b added, but in the equation you put x − a and y − b
🤔 Why the signs are reversed
①Let (X, Y) be a point on the moved figure
②Before the move, this point was at (X − a, Y − b)
③That earlier point lies on the original figure, so f(X − a, Y − b) = 0
④So the moved figure has equation f(x − a, y − b) = 0
⑤In other words, subtracting the shift from the new coordinates takes you back to where the point came from
Example: translating a circle
x² + y² = r² → (x − a)² + (y − b)² = r²
The centre moves from the origin to (a, b)
Example: translating a parabola
y = x² → y − b = (x − a)²
The vertex moves from (0, 0) to (a, b)
Reflection: a mirror image
seegongsik.com
Buttons choose reflection in the x-axis, the y-axis, the origin or the line y = x, comparing the vertex coordinates of the original and reflected triangles
Reflection rules
Four reflections
Reflection in the x-axis
Only the sign of y changes
(x, y) → (x, −y)
Reflection in the y-axis
Only the sign of x changes
(x, y) → (−x, y)
Reflection in the origin
Both signs change
(x, y) → (−x, −y)
Reflection in the line y = x
x and y swap places
(x, y) → (y, x)
💡 How reflections work
①Reflecting in the x-axis turns the figure upside down, so the sign of y changes
②Reflecting in the y-axis flips left and right, so the sign of x changes
③Reflecting in the origin is the same as a half turn (180°) about the origin, so both signs change
④Reflecting in the diagonal line y = x swaps x and y; the graphs of a function and its inverse are related in exactly this way
Doing one move after another
🔗 Rules for combining moves
①Two translations in a row add their shifts, and the order does not matter
②Reflecting in the x-axis then the y-axis is the same as reflecting in the origin
③When translations and reflections are mixed, the order can change the result
④For a question such as 'reflect (2, 3) in the x-axis, then translate it by 1 in the x-direction and −2 in the y-direction', do the steps one at a time in the order given
Combining reflections
Two reflections in a row
Order
Change in coordinates
Same as
x-axis → y-axis
(x,y)→(−x,−y)
reflection in the origin
y-axis → x-axis
(x,y)→(−x,−y)
reflection in the origin
x-axis → y = x
(x,y)→(−y,x)
rotation of 90° anticlockwise about the origin
y = x → x-axis
(x,y)→(y,−x)
rotation of 90° clockwise about the origin
Worked examples and unit test
Example 1
Translate the point (3, −2) by 4 in the x-direction and by −5 in the y-direction. Find the coordinates of the new point.
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Translating a point simply adds the shifts to its coordinates.
(3 + 4, −2 + (−5))
2
Work it out.
(7, −7)
▸ (7, −7)
A shift of −5 in the y-direction means moving down 5 units.
Example 2
Find the equation of the line obtained by translating y = 2x + 1 by 3 in the x-direction.
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Moving a figure 3 units in the x-direction means putting x − 3 in place of x in its equation.
y = 2(x − 3) + 1
2
Expand and simplify.
y = 2x − 6 + 1 = 2x − 5
▸ y = 2x − 5
The point (0, 1) on the original line moves to (3, 1), and indeed putting x = 3 into y = 2x − 5 gives y = 1.
Summary
Translating a point
(x + a, y + b)
Points: add the shift
Translating an equation
f(x − a, y − b) = 0
Equations: subtract the shift
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)
1
Reflection across y = x swaps the x- and y-coordinates.
(a, b) → (b, a)
2
Swap the coordinates.
(4, −3) → (−3, 4)
🎯 Points that often come up
①A point moves to (x + a, y + b), while the equation of a figure becomes f(x − a, y − b) = 0: the signs are opposite
②Reflecting in the x-axis replaces y with −y; reflecting in the y-axis replaces x with −x
③Reflecting in y = x swaps x and y, which links to the graph of an inverse function
④Translating a circle adds the shift to its centre and leaves the radius unchanged
⑤For several moves, do them one at a time in the given order; changing the order can change the result