Transformations Notes and Worksheets - Lindsay Bowden - Free Printable

To determine the type of transformation for each graph, we need to look at how the original shape (pre-image) moves or changes to become the new shape (image).

1. Translation
* Observation: Look at triangle $ABC$ and triangle $A'B'C'$. The shape has simply slid from the bottom-left quadrant to the first quadrant.
* Check: The orientation is the same (it hasn't flipped or turned). The size is the same. Every point moved the same distance in the same direction.
* Conclusion: This is a Translation (or Slide).

2. Translation
* Observation: Look at the line segment $MN$ and $M'N'$. Actually, looking closely, the points $M$ and $M'$ are at the same location, as are $O$ and $O'$. Wait, let's re-examine.
* Re-evaluating Graph 2: The pre-image is the segment $MN$ (and point $O$). The image is $M'N'$ (and point $O'$).
* Point $M$ is at $(0,0)$. Point $M'$ is at $(0,0)$.
* Point $O$ is at $(3,0)$. Point $O'$ is at $(6,0)$.
* Point $N$ is at $(3,1)$. Point $N'$ is at $(6,1)$.
* Check: The shape has moved to the right. The orientation is unchanged.
* Conclusion: This is a Translation (or Slide).

3. Translation
* Observation: Look at segment $PQ$ and segment $P'Q'$. The segment has moved from the third quadrant to the first quadrant.
* Check: The slope is the same. The orientation is the same. It is just shifted up and to the right.
* Conclusion: This is a Translation (or Slide).

4. Reflection
* Observation: Look at trapezoid $ABCD$ and trapezoid $A'B'C'D'$.
* Check: The image $A'B'C'D'$ is upside down compared to $ABCD$. It looks like a mirror image across the x-axis.
* Point $A$ is at $(-4, 3)$. Point $A'$ is at $(-4, -3)$.
* Point $B$ is at $(-2, 3)$. Point $B'$ is at $(-2, -3)$.
* Conclusion: Since the y-coordinates changed signs while x-coordinates stayed the same, this is a Reflection (specifically across the x-axis).

5. Reflection
* Observation: Look at segment $DE$ and segment $D'E'$.
* Check: The shape is flipped across the y-axis.
* Point $D$ is at $(-4, 1)$. Point $D'$ is at $(4, 1)$.
* Point $E$ is at $(-1, 3)$. Point $E'$ is at $(1, 3)$.
* Conclusion: Since the x-coordinates changed signs while y-coordinates stayed the same, this is a Reflection (specifically across the y-axis).

6. Reflection
* Observation: Look at triangle $JKL$ and triangle $J'K'L'$.
* Check: The triangle is flipped across the x-axis.
* Point $J$ is at $(-2, 0)$. Point $J'$ is at $(-2, 0)$.
* Point $K$ is at $(3, 4)$. Point $K'$ is at $(3, -4)$? No, wait. Let's look closer.
* Actually, $J$ is at $(-2,0)$. $J'$ is at $(-2,0)$.
* $K$ is at $(3,4)$. $K'$ is at $(3,2)$. Wait, the size changed?
* Let's re-examine graph 6.
* Pre-image: $J(-2,0)$, $K(3,4)$, $L(3,-2)$.
* Image: $J'(-2,0)$, $K'(3,2)$, $L'(3,-1)$.
* Wait, the image is smaller. Is it a dilation?
* Let's check the distances. $JK$ length vs $J'K'$ length.
* Actually, looking at the labels, the pre-image is likely the larger triangle $JKL$ and the image is the smaller one $J'K'L'$. Or vice versa? Usually, the one without the prime is the pre-image.
* Let's look at the orientation. $K$ is "up", $L$ is "down". $K'$ is "up", $L'$ is "down". The orientation is the same.
* However, the size is different. The distance from $J$ to the line $x=3$ is 5 units. The distance from $J'$ to the line $x=3$ is 5 units.
* Let's check coordinates again.
* $J = (-2, 0)$
* $K = (3, 4)$
* $L = (3, -2)$
* $J' = (-2, 0)$
* $K' = (3, 2)$
* $L' = (3, -1)$
* This is a Dilation. The center of dilation appears to be $J(-2,0)$. The scale factor is $1/2$ because the height of the original triangle is $4 - (-2) = 6$, and the height of the new triangle is $2 - (-1) = 3$. $3/6 = 1/2$.
* Wait, let me look at the image again very carefully.
* In graph 6, the points are labeled $J, K, L$ and $J', K', L'$.
* $J$ is at $(-2,0)$. $J'$ is at $(-2,0)$.
* $K$ is at $(3,4)$. $K'$ is at $(3,2)$.
* $L$ is at $(3,-2)$. $L'$ is at $(3,-1)$.
* Yes, this is definitely a Dilation. The shape got smaller but kept the same orientation and position relative to the center point $J$.

7. Rotation
* Observation: Look at triangle $ABC$ and triangle $A'B'C'$.
* Check: The shape has turned.
* $A$ is at $(-4, -4)$. $A'$ is at $(-4, 4)$.
* $B$ is at $(-1, -3)$. $B'$ is at $(-3, 1)$.
* $C$ is at $(-2, -1)$. $C'$ is at $(-1, 2)$.
* This looks like a rotation of 90 degrees counter-clockwise around the origin? Let's check.
* Rule for 90 deg CCW: $(x, y) \rightarrow (-y, x)$.
* $A(-4, -4) \rightarrow (4, -4)$. But $A'$ is $(-4, 4)$. That doesn't match.
* Let's try 180 degrees? $(x,y) \rightarrow (-x, -y)$.
* $A(-4, -4) \rightarrow (4, 4)$. No.
* Let's look at the visual. It looks like a rotation around the origin.
* Let's check the coordinates of $A'$ again. $A'$ is at $(-4, 4)$.
* Let's check the coordinates of $B'$. $B'$ is at $(-3, 1)$.
* Let's check the coordinates of $C'$. $C'$ is at $(-1, 2)$.
* Let's check the coordinates of $A, B, C$.
* $A = (-4, -4)$
* $B = (-1, -3)$
* $C = (-2, -1)$
* Let's try a rotation of 90 degrees Clockwise. Rule: $(x, y) \rightarrow (y, -x)$.
* $A(-4, -4) \rightarrow (-4, 4)$. Matches $A'$.
* $B(-1, -3) \rightarrow (-3, 1)$. Matches $B'$.
* $C(-2, -1) \rightarrow (-1, 2)$. Matches $C'$.
* Conclusion: This is a Rotation (specifically 90 degrees clockwise about the origin).

8. Translation
* Observation: Look at trapezoid $GHIJ$ and trapezoid $G'H'I'J'$.
* Check: The shape has moved from the second quadrant to the fourth quadrant.
* $G(-3, 3)$, $H(-1, 3)$, $I(-4, 0)$, $J(0, 0)$.
* $G'(2, -1)$, $H'(4, -1)$, $I'(1, -4)$, $J'(5, -4)$.
* Let's check the shift.
* $G \rightarrow G'$: $x$ goes $-3 \rightarrow 2$ (+5). $y$ goes $3 \rightarrow -1$ (-4).
* $H \rightarrow H'$: $x$ goes $-1 \rightarrow 4$ (+5). $y$ goes $3 \rightarrow -1$ (-4).
* $I \rightarrow I'$: $x$ goes $-4 \rightarrow 1$ (+5). $y$ goes $0 \rightarrow -4$ (-4).
* $J \rightarrow J'$: $x$ goes $0 \rightarrow 5$ (+5). $y$ goes $0 \rightarrow -4$ (-4).
* Every point moved the same amount.
* Conclusion: This is a Translation (or Slide).

9. Reflection
* Observation: Look at the line $MP$ and line $M'P'$.
* Check: The lines intersect at the origin. The line $MP$ has a negative slope. The line $M'P'$ has a positive slope.
* Point $M$ is at $(-3, 3)$. Point $M'$ is at $(3, 3)$.
* Point $P$ is at $(-3, -3)$. Point $P'$ is at $(3, -3)$.
* Point $N$ is at $(-1, 0)$. Point $N'$ is at $(1, 0)$.
* The x-coordinates changed signs, y-coordinates stayed the same.
* Conclusion: This is a Reflection (specifically across the y-axis).

Final Answer:
1. Translation
2. Translation
3. Translation
4. Reflection
5. Reflection
6. Dilation
7. Rotation
8. Translation
9. Reflection