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Transformation Effects
Using Coordinates

Master how translations, reflections, rotations, and dilations change the coordinates of 2D figures on the coordinate plane.

8th Grade Math

What Are Transformations?

A transformation is a rule that moves or changes every point of a figure to produce a new figure called the image. The original figure is called the pre-image.

Pre-image
The original figure before any transformation is applied. Points labeled \(A, B, C, \ldots\)
Image
The resulting figure after transformation. Points labeled \(A', B', C', \ldots\) (read "A prime")

The Four Types of Transformations

Translation
Slides every point the same distance and direction
Reflection
Flips every point across a line of reflection
Rotation
Turns every point around a center by a given angle
Dilation
Enlarges or shrinks every point by a scale factor from a center

Why Describe with Coordinates?

Using coordinate rules lets us predict exactly where every point lands without having to draw. We can verify our work numerically and communicate transformations precisely.

x y A(-1,0) B(0,1) C A'(1,0) B'(2,1) C' translate +2

Translation Coordinate Rule

A translation slides every point by the same amount horizontally and vertically. The coordinate rule is:

1
Add \(a\) to every \(x\)-coordinate to shift left or right.
2
Add \(b\) to every \(y\)-coordinate to shift up or down.
3
Shape and size do not change. Every side length and angle is preserved.

Reflection Coordinate Rules

A reflection flips a figure across a line of reflection. The two most common reflections on the coordinate plane:

Across the x-axis
\((x, y) \to (x, -y)\)
Negate the \(y\)-coordinate
Across the y-axis
\((x, y) \to (-x, y)\)
Negate the \(x\)-coordinate
x y P(2,3) P'(-2,3) P''(2,-3)

Rotation Coordinate Rules

A rotation turns every point around a center of rotation by a given angle. Standard rotations are counterclockwise about the origin:

\(90^\circ\) CCW
\((x, y) \to (-y, x)\)
\(180^\circ\)
\((x, y) \to (-x, -y)\)
\(270^\circ\) CCW
\((x, y) \to (y, -x)\)

Dilation Coordinate Rule

A dilation enlarges or shrinks a figure from a center point by a scale factor \(k\). When the center is the origin:

\(k > 1\)
Enlargement: the image is larger than the pre-image
\(0 < k < 1\)
Reduction: the image is smaller than the pre-image
x y O (1,0) (2,1) (3,0) (2,0) (4,2) (6,0) k = 2

Rigid vs Non-Rigid Transformations

Rigid (Isometry) Non-Rigid
Preserves Size ✓ Yes ✕ No
Preserves Shape ✓ Yes ✓ Yes
Types Translation, Reflection, Rotation Dilation
Congruent Image ✓ Congruent Similar (not congruent)

How to Apply a Coordinate Rule

1
Identify the transformation type from the problem: translation, reflection, rotation, or dilation.
2
Write the coordinate rule for that transformation (e.g., \((x, y) \to (-y, x)\) for \(90^\circ\) CCW).
3
Apply the rule to each vertex of the pre-image. Substitute the \(x\) and \(y\) values into the formula.
4
Plot and connect the image points. Verify by checking distances (rigid) or ratios (dilation).

Example 1: Translation

Translate triangle \(ABC\) with vertices \(A(1, 3)\), \(B(4, 3)\), \(C(4, 1)\) by the rule \((x, y) \to (x - 5, y + 2)\).

A
\(A(1, 3) \to (1 - 5,\; 3 + 2) =\) \(A'(-4, 5)\)
B
\(B(4, 3) \to (4 - 5,\; 3 + 2) =\) \(B'(-1, 5)\)
C
\(C(4, 1) \to (4 - 5,\; 1 + 2) =\) \(C'(-1, 3)\)

Example 2: Reflection Across the x-axis

Reflect triangle \(PQR\) with vertices \(P(2, 5)\), \(Q(6, 5)\), \(R(4, 1)\) across the x-axis.

x y P(2,5) Q(6,5) R(4,1) P'(2,-5) Q'(6,-5) R'(4,-1) x-axis

Example 3: \(90^\circ\) Rotation CCW

Rotate the point \(A(3, 1)\) by \(90^\circ\) counterclockwise about the origin.

x y O A(3,1) A'(-1,3) 90°

Notice: \(x = 3\) became the new \(y\), and \(y = 1\) was negated to become the new \(x\).

Example 4: Dilation with \(k = 3\)

Dilate triangle \(DEF\) with vertices \(D(1, 2)\), \(E(3, 2)\), \(F(2, 4)\) by scale factor \(k = 3\) centered at the origin.

D
\(D(1, 2) \to (3 \cdot 1,\; 3 \cdot 2) =\) \(D'(3, 6)\)
E
\(E(3, 2) \to (3 \cdot 3,\; 3 \cdot 2) =\) \(E'(9, 6)\)
F
\(F(2, 4) \to (3 \cdot 2,\; 3 \cdot 4) =\) \(F'(6, 12)\)

Common Mistakes to Avoid

✕ Mixing Up Axes in Reflections
Reflecting across the x-axis changes \(y\), not \(x\). Across the y-axis changes \(x\), not \(y\). Students often swap these.
✕ Wrong Rotation Direction
Standard convention is counterclockwise. Using \((x, y) \to (y, -x)\) for \(90^\circ\) CCW is incorrect: that is \(90^\circ\) clockwise.
✕ Adding Instead of Multiplying in Dilation
Dilation uses multiplication: \((kx, ky)\). Writing \((x+k, y+k)\) gives a translation, not a dilation.

Transformation Summary

Transformation Coordinate Rule Preserves
Translation by \((a, b)\) \((x+a, y+b)\) Size + Shape
Reflect over x-axis \((x, -y)\) Size + Shape
Reflect over y-axis \((-x, y)\) Size + Shape
Rotate \(90^\circ\) CCW \((-y, x)\) Size + Shape
Rotate \(180^\circ\) \((-x, -y)\) Size + Shape
Dilation by \(k\) \((kx, ky)\) Shape only

Question 1 of 5

If point \(M(4, -2)\) is translated by the rule \((x, y) \to (x - 6, y + 3)\), what are the coordinates of \(M'\)?

Question 2 of 5

Point \(N(3, -7)\) is reflected across the y-axis. What are the coordinates of \(N'\)?

Question 3 of 5

What are the coordinates of \(P(5, -2)\) after a \(90^\circ\) counterclockwise rotation about the origin?

Question 4 of 5

A triangle with vertices \(A(2, 4)\), \(B(6, 4)\), \(C(4, 8)\) is dilated by a scale factor of \(\frac{1}{2}\) from the origin. What are the coordinates of \(C'\)?

Question 5 of 5

Which transformation maps \((x, y)\) to \((-x, -y)\)?