Multiple Transformations Answers | PDF - Free Printable

Let’s go step by step through each problem to make sure we get the right answers.

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Problem 1: Translate ΔQRS if Q(4,1), R(1,-2), S(2,3) by the rule (x,y) → (x - 3, y - 4)

This means for every point, subtract 3 from the x-coordinate and subtract 4 from the y-coordinate.

- Q(4,1) → Q’(4-3, 1-4) = Q’(1, -3)
- R(1,-2) → R’(1-3, -2-4) = R’(-2, -6)
- S(2,3) → S’(2-3, 3-4) = S’(-1, -1)

✔ These match what’s written on the worksheet.

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Problem 2: Reflect ΔQ’R’S’ over the x-axis

Reflecting over the x-axis means you keep the x-coordinate the same, but flip the sign of the y-coordinate. So (x, y) → (x, -y)

We use the points from Problem 1:

- Q’(1, -3) → Q’’(1, -(-3)) = Q’’(1, 3)
- R’(-2, -6) → R’’(-2, -(-6)) = R’’(-2, 6)
- S’(-1, -1) → S’’(-1, -(-1)) = S’’(-1, 1)

✔ These also match what’s written.

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Problem 3: Rotate ΔCAR 180° about the origin

Rotating 180° about the origin means flipping both coordinates: (x, y) → (-x, -y)

Given: C(-1,-4), A(2,3), R(-3,-2)

- C(-1,-4) → C’(-(-1), -(-4)) = C’(1, 4)
- A(2,3) → A’(-2, -3)
- R(-3,-2) → R’(3, 2)

Wait — let’s check the worksheet answer:

It says:
C’(1, 4) ✔
A’(-2, -3) ✔
R’(3, 2) ✔

But in the image, it looks like they wrote R’ as (3, 2) — yes, that’s correct.

So all good here.

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Problem 4: Reflect ΔC’A’R’ over the line y = x

Reflecting over y = x swaps the x and y coordinates: (x, y) → (y, x)

Use the rotated points from Problem 3:

- C’(1, 4) → C’’(4, 1)
- A’(-2, -3) → A’’(-3, -2)
- R’(3, 2) → R’’(2, 3)

Check worksheet:

C’’(4, 1) ✔
A’’(-3, -2) ✔
R’’(2, 3) ✔

Perfect.

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Problem 5: What did you notice? How are the shapes related?

Look at Problems 1 & 2 together:

Original triangle QRS was translated then reflected over x-axis → ended up as Q’’R’’S’’

Now look at Problems 3 & 4:

Original triangle CAR was rotated 180° then reflected over y=x → ended up as C’’A’’R’’

BUT — compare final positions:

From Problem 2: Q’’(1,3), R’’(-2,6), S’’(-1,1)
From Problem 4: C’’(4,1), A’’(-3,-2), R’’(2,3)

Wait — actually, let’s think differently.

The question is asking: “How were the shapes related?” between problems 1&2 and 3&4.

Actually, looking at the *final* images after two transformations:

In 1&2: Start with QRS → translate → reflect over x-axis → end at Q’’R’’S’’

In 3&4: Start with CAR → rotate 180° → reflect over y=x → end at C’’A’’R’’

But notice: The final shape in 1&2 (after translation + reflection over x-axis) is NOT the same as in 3&4.

Wait — maybe the student is supposed to see that doing TWO transformations can be equivalent to one other transformation?

Let’s test this idea:

What if we take original QRS and do BOTH steps at once?

Translation: (x,y) → (x-3, y-4)
Then reflect over x-axis: (x,y) → (x, -y)

So combined: (x,y) → (x-3, -(y-4)) = (x-3, -y+4)

Try on Q(4,1): (4-3, -1+4) = (1,3) → matches Q’’

Similarly, R(1,-2): (1-3, -(-2)+4) = (-2, 2+4)=(-2,6) → matches R’’

S(2,3): (2-3, -3+4)=(-1,1) → matches S’’

So yes — combining translation and reflection gives a new rule.

Now for Problems 3&4:

Rotate 180°: (x,y) → (-x,-y)
Then reflect over y=x: (x,y) → (y,x)

So combined: (x,y) → first becomes (-x,-y), then swap → (-y, -x)

Test on C(-1,-4): → (-(-4), -(-1)) = (4,1) → matches C’’

A(2,3): → (-3, -2) → matches A’’

R(-3,-2): → (-(-2), -(-3)) = (2,3) → matches R’’

So again, combining two transformations gives a single rule: (x,y) → (-y, -x)

Now, the second part of Question 5:

> Explain how you could translate ΔQRS by the rule (x,y)→(x−3,y−4) and then reflect the image over the x-axis. Where does the final image end up?

We already did this above — it ends up at Q’’(1,3), R’’(-2,6), S’’(-1,1)

And the handwritten note says: “subtract 3 from x, subtract 4 from y, make y opposite” — which is exactly what we did.

Last part:

> How would you rotate ΔCAR 180° about the origin and then reflect it over the line y = x?

We did this too — result is (x,y) → (-y, -x)

Handwritten note says: “switch (x,y) twice for 180° then take the inside coordinate and switch the x and y coordinate”

That’s a bit confusingly worded, but essentially:

- Rotate 180°: flip signs → (-x, -y)
- Then reflect over y=x: swap → (-y, -x)

Which is correct.

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Final Answer Summary:

All calculations checked out. The key insight is that multiple transformations can sometimes be combined into one simpler rule.

For example:
- Translation followed by reflection over x-axis → (x,y) → (x-3, -y+4)
- Rotation 180° followed by reflection over y=x → (x,y) → (-y, -x)

These help us predict where any point will end up without doing each step separately.

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Final Answer:
After translating ΔQRS by (x,y) → (x−3, y−4) and reflecting over the x-axis, the final image has vertices at Q''(1,3), R''(-2,6), S''(-1,1). After rotating ΔCAR 180° about the origin and reflecting over y = x, the final image has vertices at C''(4,1), A''(-3,-2), R''(2,3). Combining transformations creates new rules: for QRS it's (x,y) → (x−3, −y+4); for CAR it's (x,y) → (−y, −x).