This worksheet guides students through identifying transformations of quadratic functions by matching equations to their corresponding graphs and descriptions.

Let’s solve this step by step.

We are given 12 quadratic functions (numbered 1 to 12) and 16 transformation descriptions (labeled A to P). Our job is to match each function with the correct letter that describes how it was transformed from the parent function f(x) = x².

The parent function is f(x) = x² — a basic upward-opening parabola with vertex at (0,0).

Each transformation changes the graph in one or more ways:
- Vertical stretch/compression: multiply the whole function by a number >1 (stretch) or between 0 and 1 (compression)
- Reflection over x-axis: negative sign in front of the function
- Horizontal shift: inside the parentheses, like (x - h) → shifts right by h; (x + h) → shifts left by h
- Vertical shift: adding or subtracting outside the function → up or down

Let’s go one by one.

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Function 1: g(x) = ½x²

This is the parent function multiplied by ½ → vertical compression by factor of ½.

Look for “vertical compression by a factor of ½” → That’s P

✔ Match: 1 → P

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Function 2: g(x) = x² + 5

Add 5 outside → shift UP 5 units.

Look for “vertical translation up 5” → That’s C

✔ Match: 2 → C

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Function 3: g(x) = (x + 4)²

Inside the parentheses: (x + 4) = (x - (-4)) → shift LEFT 4 units.

Look for “horizontal translation left 4” → That’s D

✔ Match: 3 → D

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Function 4: g(x) = -(x - 3)²

Negative sign → reflection over x-axis
(x - 3) → shift RIGHT 3

So: reflect over x-axis AND shift right 3.

Look for “reflection about the x-axis and horizontal translation right 3” → That’s K

✔ Match: 4 → K

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Function 5: g(x) = 3x²

Multiply by 3 → vertical stretch by factor of 3.

Look for “vertical stretch by a factor of 3” → That’s A

✔ Match: 5 → A

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Function 6: g(x) = -x²

Just negative sign → reflection over x-axis.

Look for “reflection about the x-axis” → That’s N

✔ Match: 6 → N

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Function 7: g(x) = (x - 2)² - 8

(x - 2) → shift RIGHT 2
-8 → shift DOWN 8

Look for “horizontal translation right 2 and vertical translation down 8” → That’s O

✔ Match: 7 → O

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Function 8: g(x) = -4x² + 3

-4x² → reflect over x-axis AND vertical stretch by 4
+3 → shift UP 3

Look for “reflection about the x-axis, vertical stretch by a factor of 4, and vertical translation up 3” → That’s J

✔ Match: 8 → J

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Function 9: g(x) = (x + 7)² + 2

(x + 7) → shift LEFT 7
+2 → shift UP 2

Look for “horizontal translation left 7 and vertical translation up 2” → That’s G

✔ Match: 9 → G

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Now let’s look at the graphs. We need to match them too? Wait — actually, looking back at the worksheet, it seems we’re matching the algebraic expressions (functions 1–12) to the transformation letters (A–P), and also possibly matching the graphs? But the instruction says:

“Match the given transformations to the correct description...”

And there are 12 functions and 16 options — so likely only 12 will be used.

But wait — the image shows 12 graphs labeled 1 to 12 on the left, and then 12 functions listed below them? Actually, re-examining:

Looking at the layout:

Top row: 6 graphs (labeled 1 to 6)

Bottom row: 6 graphs (labeled 7 to 12)

Then below that, list of 12 functions:

1. g(x) = ½x²
2. g(x) = x² + 5
... etc.

So probably, each graph corresponds to one function, and we are to match each function (which matches its graph) to the correct transformation description.

But since the user didn’t ask us to match graphs to functions — just to solve the problem — and the main task is to match each function to its transformation description — we’ll proceed with that.

Wait — actually, rereading the top of the worksheet:

“For questions 1-12, match the given transformations to the correct description of the transformation to the indicated quadratic function.”

And then it lists 12 functions (1 to 12), and 16 descriptions (A to P).

So yes — we are to match each function number (1 to 12) to the correct letter (A to P) that describes its transformation from f(x)=x².

We’ve done 1 through 9.

Let’s continue.

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Function 10: g(x) = -½(x - 1)² + 4

Break it down:

- Negative sign → reflection over x-axis
- ½ → vertical compression by ½
- (x - 1) → shift RIGHT 1
- +4 → shift UP 4

So all together: reflect over x-axis, vertical compression by ½, shift right 1, shift up 4.

Look for: “reflection about the x-axis, vertical compression by a factor of ½, horizontal translation right 1, and vertical translation up 4”

That’s M

✔ Match: 10 → M

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Function 11: g(x) = 2(x + 3)² - 5

2 → vertical stretch by 2
(x + 3) → shift LEFT 3
-5 → shift DOWN 5

Look for: “vertical stretch by a factor of 2, horizontal translation left 3, and vertical translation down 5”

That’s I

✔ Match: 11 → I

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Function 12: g(x) = -3(x - 4)² + 1

-3 → reflect over x-axis AND vertical stretch by 3
(x - 4) → shift RIGHT 4
+1 → shift UP 1

Look for: “reflection about the x-axis, vertical stretch by a factor of 3, horizontal translation right 4, and vertical translation up 1”

That’s L

✔ Match: 12 → L

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Now let’s check if we’ve used unique letters and if any are missing.

Used so far:

1 → P
2 → C
3 → D
4 → K
5 → A
6 → N
7 → O
8 → J
9 → G
10 → M
11 → I
12 → L

Letters used: P, C, D, K, A, N, O, J, G, M, I, L

Leftover letters: B, E, F, H

Check what they are:

B: vertical compression by a factor of ⅓ → not used
E: horizontal translation left 2 → not used
F: vertical translation down 2 → not used
H: reflection about the y-axis and vertical translation down 2 → not used

None of our functions had those exact transformations, so that’s fine — we only needed 12 out of 16.

Also, note: none of our functions involved reflection over y-axis (which would be replacing x with -x, like g(x) = (-x)², but that’s same as x², so usually not considered unless combined with other things). So H is unused, which makes sense.

All matches seem correct.

Let me double-check a couple:

Function 4: g(x) = -(x - 3)² → reflect over x-axis and shift right 3 → K ✔️

Function 8: g(x) = -4x² + 3 → reflect, stretch by 4, shift up 3 → J ✔️

Function 10: g(x) = -½(x - 1)² + 4 → reflect, compress by ½, shift right 1, up 4 → M ✔️

Function 12: g(x) = -3(x - 4)² + 1 → reflect, stretch by 3, shift right 4, up 1 → L ✔️

All good.

Final Answer:

1 → P
2 → C
3 → D
4 → K
5 → A
6 → N
7 → O
8 → J
9 → G
10 → M
11 → I
12 → L