Transformation Worksheets - Free Printable

Here are the step-by-step solutions for each transformation problem on the worksheet.

Problem 1: Translation

Rule: Translate +5 units vertically (up) and +4 units horizontally (right).
* Formula: Add 4 to the x-coordinate and add 5 to the y-coordinate $(x + 4, y + 5)$.

* Point A $(0, -3)$:
* $x$: $0 + 4 = 4$
* $y$: $-3 + 5 = 2$
* New Point: $(4, 2)$

* Point B $(0, -5)$:
* $x$: $0 + 4 = 4$
* $y$: $-5 + 5 = 0$
* New Point: $(4, 0)$

* Point C $(2, -5)$:
* $x$: $2 + 4 = 6$
* $y$: $-5 + 5 = 0$
* New Point: $(6, 0)$

* Point D $(3, -3)$:
* $x$: $3 + 4 = 7$
* $y$: $-3 + 5 = 2$
* New Point: $(7, 2)$

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Problem 2: Rotation ($90^\circ$ Counterclockwise)

Rule: Rotate $90^\circ$ counterclockwise about the origin.
* Formula: Switch the coordinates and change the sign of the new x-coordinate $(-y, x)$.

* Point A $(-4, 6)$:
* New $x$: $-(6) = -6$
* New $y$: $-4$
* New Point: $(-6, -4)$

* Point B $(-8, 2)$:
* New $x$: $-(2) = -2$
* New $y$: $-8$
* New Point: $(-2, -8)$

* Point C $(-5, 5)$:
* New $x$: $-(5) = -5$
* New $y$: $-5$
* New Point: $(-5, -5)$

* Point D $(-3, 7)$:
* New $x$: $-(7) = -7$
* New $y$: $-3$
* New Point: $(-7, -3)$

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Problem 3: Rotation ($90^\circ$ Clockwise)

Rule: Rotate $90^\circ$ clockwise about the origin.
* Formula: Switch the coordinates and change the sign of the new y-coordinate $(y, -x)$.

* Point A $(6, 2)$:
* New $x$: $2$
* New $y$: $-(6) = -6$
* New Point: $(2, -6)$

* Point B $(4, 5)$:
* New $x$: $5$
* New $y$: $-(4) = -4$
* New Point: $(5, -4)$

* Point C $(0, 2)$:
* New $x$: $2$
* New $y$: $-(0) = 0$
* New Point: $(2, 0)$

* Point D $(1, 7)$:
* New $x$: $7$
* New $y$: $-(1) = -1$
* New Point: $(7, -1)$

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Problem 4: Reflection across the Y-axis

Rule: Reflect across the vertical Y-axis.
* Formula: Keep the y-coordinate the same, but flip the sign of the x-coordinate $(-x, y)$.

* Point A $(4, -9)$:
* New $x$: $-4$
* New $y$: $-9$
* New Point: $(-4, -9)$

* Point B $(8, 6)$:
* New $x$: $-8$
* New $y$: $6$
* New Point: $(-8, 6)$

* Point C $(2, 5)$:
* New $x$: $-2$
* New $y$: $5$
* New Point: $(-2, 5)$

* Point D $(7, 3)$:
* New $x$: $-7$
* New $y$: $3$
* New Point: $(-7, 3)$

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Problem 5: Reflection across the line $Y = 3$

Rule: Reflect across a horizontal line. The x-coordinate stays the same. To find the new y, calculate the distance from the point to the line ($3$), and move that same distance to the other side.
* Shortcut Formula: New $y = (\text{Line Value} \times 2) - \text{Old } y$. Here: $6 - y$.

* Point A $(-5, 7)$:
* New $x$: $-5$
* New $y$: $6 - 7 = -1$
* New Point: $(-5, -1)$

* Point B $(6, 5)$:
* New $x$: $6$
* New $y$: $6 - 5 = 1$
* New Point: $(6, 1)$

* Point C $(9, 9)$:
* New $x$: $9$
* New $y$: $6 - 9 = -3$
* New Point: $(9, -3)$

* Point D $(-1, 8)$:
* New $x$: $-1$
* New $y$: $6 - 8 = -2$
* New Point: $(-1, -2)$

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Problem 6: Translation

Rule: Translate -8 units on the X-axis (left) and -6 units on the Y-axis (down).
* Formula: Subtract 8 from x and subtract 6 from y $(x - 8, y - 6)$.

* Point A $(4, 4)$:
* New $x$: $4 - 8 = -4$
* New $y$: $4 - 6 = -2$
* New Point: $(-4, -2)$

* Point B $(2, 7)$:
* New $x$: $2 - 8 = -6$
* New $y$: $7 - 6 = 1$
* New Point: $(-6, 1)$

* Point C $(7, 2)$:
* New $x$: $7 - 8 = -1$
* New $y$: $2 - 6 = -4$
* New Point: $(-1, -4)$

* Point D $(-2, 8)$:
* New $x$: $-2 - 8 = -10$
* New $y$: $8 - 6 = 2$
* New Point: $(-10, 2)$

Final Answer:
Problem 1 (Translation):
A' = (4, 2)
B' = (4, 0)
C' = (6, 0)
D' = (7, 2)

Problem 2 (Rotate 90° CCW):
A' = (-6, -4)
B' = (-2, -8)
C' = (-5, -5)
D' = (-7, -3)

Problem 3 (Rotate 90° CW):
A' = (2, -6)
B' = (5, -4)
C' = (2, 0)
D' = (7, -1)

Problem 4 (Reflect Y-axis):
A' = (-4, -9)
B' = (-8, 6)
C' = (-2, 5)
D' = (-7, 3)

Problem 5 (Reflect Line Y=3):
A' = (-5, -1)
B' = (6, 1)
C' = (9, -3)
D' = (-1, -2)

Problem 6 (Translation):
A' = (-4, -2)
B' = (-6, 1)
C' = (-1, -4)
D' = (-10, 2)