Worksheet on transforming quadratic functions with graphs and equations.

To solve this problem, we need to match each quadratic function (numbered 1–8) with its correct transformation description (lettered A–T). Each letter corresponds to a specific verbal description of how the parent function $ f(x) = x^2 $ was transformed.

We’ll go one by one and analyze the transformations in each equation. Remember:

- Vertical stretch/compression: coefficient in front of $x^2$ → if |a| > 1, stretch; if 0 < |a| < 1, compression.
- Reflection over x-axis: negative sign in front of $x^2$.
- Horizontal shift: inside the parentheses with $x$, like $(x - h)^2$ → shifts right by $h$; $(x + h)^2$ → shifts left by $h$.
- Vertical shift: constant added/subtracted at the end → up or down.

Also note: Some functions have both horizontal and vertical shifts — we must identify both.

Let’s start:

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1. $ f(x) = x^2 - 4 $

→ This is just the parent function shifted down 4 units.

Look for: “vertical translation down 4” → That’s O

✔ Match: 1 → O

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2. $ f(x) = 4x^2 + 3 $

→ Coefficient 4 → vertical stretch by factor 4
→ +3 → vertical shift up 3

Look for: “vertical stretch by a factor of 4 and vertical translation up 3” → That’s U

✔ Match: 2 → U

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3. $ f(x) = \frac{1}{5}x^2 $

→ Coefficient $\frac{1}{5}$ → vertical compression by factor $\frac{1}{5}$

Look for: “vertical compression by a factor of $\frac{1}{5}$” → That’s P

✔ Match: 3 → P

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4. $ g(x) = -(x + 2)^2 $

→ Negative sign → reflection over x-axis
→ $(x + 2)$ → horizontal shift left 2

Look for: “reflection about the x-axis and horizontal translation left 2” → That’s N

Wait — let’s check the options again.

Actually, looking at the list:

> N: reflection about the x-axis and horizontal translation left 2 → YES!

But wait — option N says: “reflection about the x-axis and horizontal translation left 2”

Yes! So:

✔ Match: 4 → N

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5. $ f(x) = 2x^2 $

→ Vertical stretch by factor 2

Look for: “vertical stretch by a factor of 2” → That’s L

✔ Match: 5 → L

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6. $ f(x) = -\frac{1}{2}(x - 3)^2 - 8 $

Break it down:

- Negative sign → reflection over x-axis
- $\frac{1}{2}$ → vertical compression by factor $\frac{1}{2}$
- $(x - 3)$ → horizontal shift right 3
- $-8$ → vertical shift down 8

So: reflection, vertical compression by 1/2, horizontal shift right 3, vertical shift down 8.

Look for that combo.

Check option Q: “reflection about the x-axis, vertical compression by a factor of 1/2, horizontal translation right 3, and vertical translation down 8” → PERFECT MATCH.

✔ Match: 6 → Q

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7. $ g(x) = (x + 5)^2 + 9 $

→ $(x + 5)$ → horizontal shift left 5
→ +9 → vertical shift up 9

Look for: “horizontal translation left 5 and vertical translation up 9” → That’s R

✔ Match: 7 → R

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8. $ f(x) = -\frac{1}{4}(x + 2)^2 + 1 $

Breakdown:

- Negative → reflection over x-axis
- $\frac{1}{4}$ → vertical compression by factor $\frac{1}{4}$
- $(x + 2)$ → horizontal shift left 2
- +1 → vertical shift up 1

Look for: “reflection about the x-axis, vertical compression by a factor of 1/4, horizontal translation left 2, and vertical translation up 1”

That’s M

✔ Match: 8 → M

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Now let’s double-check all matches:

1. $x^2 - 4$ → down 4 → O ✔
2. $4x^2 + 3$ → stretch 4, up 3 → U ✔
3. $\frac{1}{5}x^2$ → compress 1/5 → P ✔
4. $-(x+2)^2$ → reflect, left 2 → N ✔
5. $2x^2$ → stretch 2 → L ✔
6. $-\frac{1}{2}(x-3)^2 -8$ → reflect, compress 1/2, right 3, down 8 → Q ✔
7. $(x+5)^2 +9$ → left 5, up 9 → R ✔
8. $-\frac{1}{4}(x+2)^2 +1$ → reflect, compress 1/4, left 2, up 1 → M ✔

All match correctly.

Final Answer:
1-O, 2-U, 3-P, 4-N, 5-L, 6-Q, 7-R, 8-M