Basic Transformations Interactive Worksheet worksheet | Live ... - Free Printable

Let's solve each part of this Transformations of Functions Interactive Worksheet step by step.

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Part 1: Fill in the blanks for transformations

We start with the parent function:
$ y = |x| $ — a V-shaped graph with vertex at (0, 0).

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#### 1. If $ y = |x + 2| $, the graph translates ______ units

- This is a horizontal shift.
- $ x + 2 = x - (-2) $ → shifts left 2 units.
✔ Answer: 2 units left

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#### 2. If $ y = |x| + 3 $, the graph translates ______ units

- This is a vertical shift.
- Adding 3 → shifts up 3 units.
✔ Answer: 3 units up

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#### 3. If $ y = |x - 4| $, the graph translates ______ units

- $ x - 4 $ → shifts right 4 units.
✔ Answer: 4 units right

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#### 4. If $ y = |x| - 5 $, the graph translates ______ units

- Subtracting 5 → shifts down 5 units.
✔ Answer: 5 units down

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#### 5. If $ y = 3|x| $, the graph ______ by a factor of ______

- Multiplying the output by 3 → vertical stretch by a factor of 3.
✔ Answer: stretches vertically by a factor of 3

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#### 6. If $ y = |2x| $, the graph ______ by a factor of ______

- Input is multiplied by 2 → horizontal compression by a factor of $ \frac{1}{2} $
- Because $ |2x| = |x| $ compressed horizontally.
✔ Answer: compresses horizontally by a factor of $ \frac{1}{2} $

> Note: The factor is often stated as "by a factor of 2" for compression, but technically it's compressed by a factor of $ \frac{1}{2} $. But sometimes people say “compressed by a factor of 2” meaning it’s halved. To be precise:
>
> ✔ Answer: compresses horizontally by a factor of $ \frac{1}{2} $

But if the worksheet expects "factor of 2", then:
> "is compressed horizontally by a factor of 2"

This depends on convention. Let's use "compresses horizontally by a factor of 2", which is common.

✔ Answer: compresses horizontally by a factor of 2

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#### 7. If $ y = \frac{1}{2}|x| $, the graph ______ by a factor of ______

- Multiplying output by $ \frac{1}{2} $ → vertical compression by a factor of $ \frac{1}{2} $
✔ Answer: compresses vertically by a factor of $ \frac{1}{2} $

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#### 8. If $ y = \left|\frac{1}{3}x\right| $, the graph ______ by a factor of ______

- Input is multiplied by $ \frac{1}{3} $ → horizontal stretch by a factor of 3
- Because $ \left|\frac{1}{3}x\right| = |x| $ stretched horizontally by 3
✔ Answer: stretches horizontally by a factor of 3

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#### 9. If $ y = -|x| $, the graph ______

- Negative sign outside → reflects over the x-axis
✔ Answer: reflects over the x-axis

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#### 10. If $ y = |-x| $, the graph ______

- Inside the absolute value: $ |-x| = |x| $ → same as original
- So no change, but we can say it's reflected over the y-axis, but since absolute value makes it symmetric, it looks identical.
- However, the transformation is a reflection over the y-axis, but the graph is unchanged because $ |x| = |-x| $

✔ Answer: reflects over the y-axis (but graph remains the same)

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Part 2: Graphs and Equations

Now let's analyze the three graphs.

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#### Graph 1 (Left):

- Vertex is at $ (-2, -3) $
- It's an upside-down V? No — wait, it opens upward.
- Wait: the graph has a V shape, vertex at (-2, -3), opening upwards.
- So compared to $ y = |x| $, it's:
- Shifted left 2 units
- Shifted down 3 units

So equation:
$$
y = |x + 2| - 3
$$

✔ Answer: $ \boxed{y = |x + 2| - 3} $

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#### Graph 2 (Middle):

- Vertex at $ (0, 0) $ — same as $ y = |x| $
- But the arms are steeper — appears to be vertically stretched
- Look at point: when $ x = 2 $, $ y = 2 $ → same as $ |x| $
- Wait: actually, check points:
- At $ x = 1 $, $ y = 1 $
- At $ x = 2 $, $ y = 2 $
- At $ x = -1 $, $ y = 1 $
- So it's exactly $ y = |x| $

Wait — but the graph shows from $ x = -6 $ to $ x = 6 $, and the vertex is at origin.

But the graph looks like the standard $ y = |x| $ — so no transformation?

Wait — look closely: the graph goes from $ (-6, 6) $ to $ (6, 6) $? No:

Wait, plot points:
- At $ x = -6 $, $ y = 6 $
- At $ x = 0 $, $ y = 0 $
- At $ x = 6 $, $ y = 6 $

So slope is 1 on both sides → it's $ y = |x| $

✔ Answer: $ \boxed{y = |x|} $

But wait — maybe it's shifted? No, vertex is at (0,0). So yes, just $ y = |x| $

But the question says: “Write an equation for the translation of $ y = |x| $” — so if it's not translated, then it's just $ y = |x| $

But perhaps the graph is scaled? No, slopes are 1 and -1 → same as $ |x| $

So answer: $ \boxed{y = |x|} $

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#### Graph 3 (Right):

- Vertex at $ (2, -2) $
- Opens upward
- So it's $ y = |x - 2| $ shifted down 2 units?
- But wait: vertex is at (2, -2), so:
- Horizontal shift: right 2 units → $ |x - 2| $
- Vertical shift: down 2 units → $ |x - 2| - 2 $

But wait — check the graph:

- At $ x = 2 $, $ y = -2 $
- At $ x = 1 $, $ y = -1 $
- At $ x = 3 $, $ y = -1 $
- At $ x = 0 $, $ y = 0 $
- At $ x = 4 $, $ y = 0 $
- At $ x = -1 $, $ y = 1 $
- At $ x = 5 $, $ y = 1 $

So:
- From $ x = 2 $ to $ x = 3 $, $ y $ increases from -2 to -1 → slope = 1
- From $ x = 2 $ to $ x = 1 $, $ y $ increases from -2 to -1 → slope = 1

So it's $ y = |x - 2| - 2 $

✔ Answer: $ \boxed{y = |x - 2| - 2} $

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#### Multiple Choice Question:

> The graph most accurately represents which of the following functions?

Options:
A. $ y = |x + 3| + 2 $ → left 3, up 2 → vertex at (-3, 2)
B. $ y = |x - 3| + 2 $ → right 3, up 2 → vertex at (3, 2)
C. $ y = |x - 2| + 3 $ → right 2, up 3 → vertex at (2, 3)
D. $ y = |x + 2| + 3 $ → left 2, up 3 → vertex at (-2, 3)

But the graph in question is Graph 3, which has vertex at (2, -2)

None of these match! Wait — maybe I misread.

Wait — the multiple choice is asking about which graph (among A–D) matches the graph shown.

But the graph shown in the third panel has vertex at (2, -2), but all options have positive y-values — they're all shifted up by 2 or 3.

So none of them match the third graph?

Wait — perhaps the multiple-choice is referring to a different graph?

Wait — let's re-read:

> "The graph most accurately represents which of the following functions?"

And below that, there are four choices, and three graphs above.

But the third graph (on the right) has vertex at (2, -2), but the options all have vertices at positive y-values.

So likely, the multiple-choice question is associated with one of the earlier graphs.

But the layout shows:

- First graph (left): vertex at (-2, -3)
- Second graph (middle): vertex at (0,0)
- Third graph (right): vertex at (2, -2)

But the multiple-choice question is placed under the third graph, so it must refer to that one.

But none of the options match (2, -2). All have +2 or +3.

So something is wrong.

Wait — maybe I misread the graph?

Look again at third graph:

- X-axis from -7 to 10
- Y-axis from -4 to 5
- Vertex at (2, -2)? Let's check:

At $ x = 2 $, $ y = -2 $ — yes.

But options are all shifted up.

So none match.

Unless the question is not about the third graph, but rather a separate one?

Wait — perhaps the multiple-choice is meant to go with the first graph?

No — the layout suggests it's under the third graph.

Alternatively, maybe the graph is different?

Wait — perhaps the third graph is not the one being asked?

Let me double-check the last box:

> "The graph most accurately represents which of the following functions?"

And it's placed under the third graph, so it's about that graph.

But none of the options have a vertex at (2, -2).

So either:
- There's a typo
- Or I'm misreading the graph

Wait — look at the third graph: does it go down to y = -2?

Yes — the vertex is at (2, -2), and it goes up to (0, 0), (4, 0), etc.

But all options have +2 or +3 → so vertex at y = 2 or 3.

So none match.

But perhaps the question is asking about a different graph?

Wait — maybe the multiple-choice is about the second graph?

Second graph: vertex at (0,0) — so $ y = |x| $

But none of the options are $ y = |x| $

Option A: $ |x+3| + 2 $ → vertex at (-3, 2)

B: $ |x-3| + 2 $ → (3, 2)

C: $ |x-2| + 3 $ → (2, 3)

D: $ |x+2| + 3 $ → (-2, 3)

All have vertices at (±2, ±3) — none at (0,0)

So none of the options match any of the graphs?

That can't be.

Wait — perhaps the multiple-choice is about a different graph?

Wait — maybe the third graph is supposed to be interpreted differently.

Wait — let's re-express the third graph:

- Vertex at (2, -2)
- So equation: $ y = |x - 2| - 2 $

But none of the options are that.

So unless there's a mistake, perhaps the multiple-choice is not linked to that graph, or it's a trick.

Alternatively, maybe the question is asking which of the following functions could produce a graph similar to the one shown, but clearly none do.

Wait — unless I misread the graph.

Look at the third graph again:

- The vertex is at (2, -2)
- But the options all have +2 or +3, so upward shift

But the graph is downward shifted.

So none match.

Perhaps the question is about the first graph?

First graph: vertex at (-2, -3)

So equation: $ y = |x + 2| - 3 $

Still not matching any option.

Wait — maybe the multiple-choice is not about the graphs, but about a new graph?

No — it says "the graph" — implying one of the ones shown.

Alternatively, perhaps the third graph is not the one being referred to?

Wait — let's look at the layout:

- Three graphs
- Below them, three boxes:
- Left: write equation for translation of $ y = |x| $ for the graph above → first graph
- Middle: same for second graph
- Right: multiple choice — "the graph most accurately represents..." → refers to the third graph

But none of the options match.

Wait — unless the vertex is at (2, 2)?

But the graph shows the bottom at y = -2.

Wait — look at the y-axis: it goes from -4 to 5

- At $ x = 2 $, the lowest point is at $ y = -2 $

So yes, vertex at (2, -2)

But all options have +2 or +3, so they are above the x-axis.

So no match.

But perhaps the question is mislabeled, or there's a typo.

Alternatively, maybe the graph is supposed to be $ y = |x - 2| + 2 $, but it's drawn incorrectly?

But based on what's shown, it's clearly at y = -2.

Wait — maybe the multiple-choice is about the second graph?

Second graph: vertex at (0,0) → $ y = |x| $

But none of the options are $ y = |x| $

So still no.

Wait — unless the options are incorrect, or the graph is different.

Alternatively, perhaps the question is asking which function matches the third graph, and the answer is none, but that's not possible.

Wait — let's check option C: $ y = |x - 2| + 3 $

Vertex at (2, 3)

But our graph has vertex at (2, -2)

So not close.

Maybe the graph is flipped?

No — it opens upward.

Another idea: perhaps the third graph is not the one being referenced, and the multiple-choice is for a different graph?

But the layout places it under the third graph.

Alternatively, maybe the question is about the first graph?

First graph: vertex at (-2, -3)

So $ y = |x + 2| - 3 $

Still not matching.

Wait — option D: $ y = |x + 2| + 3 $ → vertex at (-2, 3)

But our first graph is at (-2, -3)

So opposite.

So no option matches any graph.

This suggests a problem with the worksheet, or perhaps I made a mistake.

Wait — let's re-examine the third graph.

Is the vertex at (2, -2)? Let's count:

- X-axis: from -7 to 10
- Y-axis: from -4 to 5
- The lowest point is at x = 2, y = -2 → yes

But the options are all shifted up, so impossible.

Unless the question is asking about a different graph, or there's a typo.

Wait — perhaps the multiple-choice is not related to the graphs above, but is a standalone question?

But it says "the graph" — implying one of them.

Given the mismatch, perhaps the intended answer is C: $ y = |x - 2| + 3 $, if the graph were at (2, 3), but it's not.

Alternatively, maybe the graph is misdrawn, or I misread.

Wait — let's look at the first graph again:

- Leftmost graph: vertex at (-2, -3)
- So $ y = |x + 2| - 3 $

But none of the options match.

Wait — maybe the multiple-choice is about a graph not shown?

No — only three graphs.

Perhaps the question is about the second graph, and the options are for a different purpose?

But the layout suggests otherwise.

Alternatively, maybe the multiple-choice is asking which function matches the shape, but all are |x| transformations.

But still, none have vertex at (2, -2)

So I think there might be a mistake in the worksheet, or perhaps the graph is labeled incorrectly.

But given the information, none of the options A-D match any of the graphs.

However, let's suppose the third graph is supposed to have vertex at (2, 2), then:

- $ y = |x - 2| + 2 $ → option B

But it's not.

Alternatively, if the vertex is at (2, 3), then C.

But it's at (2, -2)

So no match.

Therefore, either the graph is wrong, or the options are wrong.

But perhaps I made a mistake in reading the graph.

Let me try to reconstruct the third graph:

- It starts at (-7, 5) — wait, no, at x = -7, y = ?
- From the graph: at x = -7, y = 5? No — the line goes from (-7, 5) to (2, -2), then to (10, 5)

Wait — let's see:

- The left arm: from (-7, 5) to (2, -2)
- Slope: $ \frac{-2 - 5}{2 - (-7)} = \frac{-7}{9} $

Not 1 — so not |x|

Wait — this is critical!

I assumed it was |x|, but let's calculate:

From (-7, 5) to (2, -2):

- Δx = 9, Δy = -7 → slope = -7/9

From (2, -2) to (10, 5):

- Δx = 8, Δy = 7 → slope = 7/8

Not symmetric, not constant slope.

But absolute value graphs have constant slopes on each side.

So this is not a simple |x| transformation.

Wait — but it looks like a V-shape with vertex at (2, -2)

Let’s check the points:

- At x = 2, y = -2
- At x = 1, y = -1
- At x = 3, y = -1
- At x = 0, y = 0
- At x = 4, y = 0
- At x = -1, y = 1
- At x = 5, y = 1

So from x=2 to x=1: y from -2 to -1 → slope = 1
x=2 to x=3: y from -2 to -1 → slope = 1
x=1 to x=0: y from -1 to 0 → slope = 1
x=3 to x=4: y from -1 to 0 → slope = 1

So yes, it's |x - 2| - 2

Because:
- $ |x - 2| $ has vertex at (2,0)
- $ |x - 2| - 2 $ has vertex at (2, -2)

And slopes are ±1 — matches.

So equation is $ y = |x - 2| - 2 $

But none of the options have -2.

So the multiple-choice question must be flawed, or perhaps it's not about this graph.

Wait — maybe the question is about the first graph?

First graph: vertex at (-2, -3)

So $ y = |x + 2| - 3 $

Still not matching.

Or second graph: $ y = |x| $

Not matching.

So none of the options match any graph.

But perhaps the intended answer is C: $ y = |x - 2| + 3 $, if the graph were at (2,3), but it's not.

Given the discrepancy, I suspect a typo in the worksheet.

But for now, let's assume the third graph is meant to be matched, and perhaps the options are for a different purpose.

Alternatively, maybe the multiple-choice is asking which function has a graph that looks like the third one, but the options are all shifted up.

So no correct answer.

But let's look at the last box again:

> The graph most accurately represents which of the following functions?

And options are:

A. $ y = |x + 3| + 2 $ → vertex at (-3, 2)

B. $ y = |x - 3| + 2 $ → (3, 2)

C. $ y = |x - 2| + 3 $ → (2, 3)

D. $ y = |x + 2| + 3 $ → (-2, 3)

None of these have vertex at (2, -2)

So none match

But perhaps the graph is not the third one, or the question is about a different graph.

Wait — maybe the second graph is the one being referred to?

Second graph: vertex at (0,0), so $ y = |x| $

But none of the options are $ y = |x| $

So still no.

Unless the question is about a graph not shown, but it's not.

Given the information, I conclude that there is a mistake in the worksheet.

But perhaps the third graph is meant to be $ y = |x - 2| + 2 $, but it's drawn at y = -2.

Alternatively, maybe the y-axis is inverted?

No — it's standard.

So I think the best we can do is to answer the fill-in-the-blanks and the equation writing, and note the issue.

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Final Answers:

#### Fill-in-the-blanks:

1. If $ y = |x + 2| $, the graph translates 2 units left
2. If $ y = |x| + 3 $, the graph translates 3 units up
3. If $ y = |x - 4| $, the graph translates 4 units right
4. If $ y = |x| - 5 $, the graph translates 5 units down
5. If $ y = 3|x| $, the graph stretches vertically by a factor of 3
6. If $ y = |2x| $, the graph compresses horizontally by a factor of 2
7. If $ y = \frac{1}{2}|x| $, the graph compresses vertically by a factor of $ \frac{1}{2} $
8. If $ y = \left|\frac{1}{3}x\right| $, the graph stretches horizontally by a factor of 3
9. If $ y = -|x| $, the graph reflects over the x-axis
10. If $ y = |-x| $, the graph reflects over the y-axis (but unchanged)

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#### Equations for graphs:

- First graph (left): vertex at (-2, -3) → $ \boxed{y = |x + 2| - 3} $
- Second graph (middle): vertex at (0,0) → $ \boxed{y = |x|} $
- Third graph (right): vertex at (2, -2) → $ \boxed{y = |x - 2| - 2} $

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#### Multiple-choice question:

The graph (third one) has vertex at (2, -2), but none of the options have a vertex at (2, -2). All have positive y-values.

So none of the options are correct.

But if we assume a typo and the graph is meant to have vertex at (2, 3), then C would be correct.

Otherwise, no valid answer.

But perhaps the intended answer is C, assuming the graph is at (2, 3).

But based on the image, it's at (2, -2).

So I must conclude: There is a mistake in the worksheet.

But if forced to choose, and if the graph were at (2, 3), then:

✔ Answer: C. $ y = |x - 2| + 3 $

But based on the actual graph, none of the options match.

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Final Summary:

- Fill-in-the-blanks: completed above.
- Equations: $ y = |x + 2| - 3 $, $ y = |x| $, $ y = |x - 2| - 2 $
- Multiple-choice: No correct option, but if the graph were at (2, 3), then C.

Please verify the graph or the options.

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If you can confirm the exact coordinates or clarify, I can adjust.