Identifying Transformations Of Parent Functions Worksheets - Free Printable

It looks like you've uploaded an image of a worksheet titled "Transformations of Parent Functions from Graphs & Equations" by Malia Rivera, which is designed to help students identify how parent functions have been transformed based on their equations or graphs.

While I can't view the actual content of the images (like specific graphs or filled-in answers), I can guide you through how to solve these types of problems, explain the key concepts, and provide step-by-step reasoning for each type of transformation shown in the examples listed.

Let’s break it down using the equations visible in your image:

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🔍 Goal:

For each function given, determine:
1. The parent function
2. The transformations applied (shifts, stretches, reflections, etc.)

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## ✔ Step 1: Know Your Parent Functions

Here are common parent functions:

| Function Type | Parent Function |
|---------------|-----------------|
| Linear | $ f(x) = x $ |
| Quadratic | $ f(x) = x^2 $ |
| Cubic | $ f(x) = x^3 $ |
| Square Root | $ f(x) = \sqrt{x} $ |
| Cube Root | $ f(x) = \sqrt[3]{x} $ |
| Absolute Value| $ f(x) = |x| $ |

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## ✔ Step 2: Understand Transformations

General form:
$$
f(x) = a \cdot f(b(x - h)) + k
$$

Where:
- $ h $: horizontal shift (right if positive, left if negative)
- $ k $: vertical shift (up if positive, down if negative)
- $ a $: vertical stretch/compression and reflection over x-axis
- $ b $: horizontal stretch/compression and reflection over y-axis

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Now let's go through each equation one by one.

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

- Parent Function: $ f(x) = x^2 $
- Transformation(s):
- Horizontal shift: 2 units right (because $ x - 2 $)

✔ Answer:
- Parent Function: $ x^2 $
- Transformation: Shifted right 2 units

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2. $ f(x) = \sqrt{x} + 4 $

- Parent Function: $ f(x) = \sqrt{x} $
- Transformation(s):
- Vertical shift: 4 units up

✔ Answer:
- Parent Function: $ \sqrt{x} $
- Transformation: Shifted up 4 units

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3. $ f(x) = -|x| $

- Parent Function: $ f(x) = |x| $
- Transformation(s):
- Reflection over the x-axis (due to negative sign)

✔ Answer:
- Parent Function: $ |x| $
- Transformation: Reflected over x-axis

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4. $ f(x) = \sqrt[3]{x} - 1 $

- Parent Function: $ f(x) = \sqrt[3]{x} $
- Transformation(s):
- Vertical shift: 1 unit down

✔ Answer:
- Parent Function: $ \sqrt[3]{x} $
- Transformation: Shifted down 1 unit

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5. $ f(x) = (2x)^3 $

- Parent Function: $ f(x) = x^3 $
- Transformation(s):
- Horizontal compression by factor of $ \frac{1}{2} $ (since input is multiplied by 2)
- Rule: $ f(bx) $ → compress horizontally if $ |b| > 1 $
- So $ b = 2 $ → compressed by factor $ \frac{1}{2} $

✔ Answer:
- Parent Function: $ x^3 $
- Transformation: Horizontally compressed by factor of $ \frac{1}{2} $

> Note: This is not a vertical stretch — even though $ (2x)^3 = 8x^3 $, the transformation is best described as horizontal compression because the change is inside the function.

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6. $ f(x) = |x - 3| $

- Parent Function: $ f(x) = |x| $
- Transformation(s):
- Horizontal shift: 3 units right

✔ Answer:
- Parent Function: $ |x| $
- Transformation: Shifted right 3 units

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7. $ f(x) = \sqrt{x + 7} $

- Parent Function: $ f(x) = \sqrt{x} $
- Transformation(s):
- Horizontal shift: 7 units left (because $ x + 7 = x - (-7) $)

✔ Answer:
- Parent Function: $ \sqrt{x} $
- Transformation: Shifted left 7 units

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8. $ f(x) = x + 9 $

- Parent Function: $ f(x) = x $
- Transformation(s):
- Vertical shift: 9 units up

✔ Answer:
- Parent Function: $ x $
- Transformation: Shifted up 9 units

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9. $ f(x) = -x^2 - 6 $

- Parent Function: $ f(x) = x^2 $
- Transformation(s):
- Reflection over x-axis (negative sign)
- Vertical shift: 6 units down

✔ Answer:
- Parent Function: $ x^2 $
- Transformation: Reflected over x-axis and shifted down 6 units

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10. $ f(x) = \sqrt[3]{x + 1} + 2 $

- Parent Function: $ f(x) = \sqrt[3]{x} $
- Transformation(s):
- Horizontal shift: 1 unit left
- Vertical shift: 2 units up

✔ Answer:
- Parent Function: $ \sqrt[3]{x} $
- Transformation: Shifted left 1 unit and up 2 units

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11. $ f(x) = 2|x + 3| - 4 $

- Parent Function: $ f(x) = |x| $
- Transformation(s):
- Horizontal shift: 3 units left
- Vertical stretch by factor of 2
- Vertical shift: 4 units down

✔ Answer:
- Parent Function: $ |x| $
- Transformation: Shifted left 3 units, vertically stretched by 2, shifted down 4 units

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## 📌 Summary Table

| Equation | Parent Function | Transformation(s) |
|--------|------------------|--------------------|
| $ (x - 2)^2 $ | $ x^2 $ | Right 2 |
| $ \sqrt{x} + 4 $ | $ \sqrt{x} $ | Up 4 |
| $ -|x| $ | $ |x| $ | Reflect over x-axis |
| $ \sqrt[3]{x} - 1 $ | $ \sqrt[3]{x} $ | Down 1 |
| $ (2x)^3 $ | $ x^3 $ | Horizontally compressed by $ \frac{1}{2} $ |
| $ |x - 3| $ | $ |x| $ | Right 3 |
| $ \sqrt{x + 7} $ | $ \sqrt{x} $ | Left 7 |
| $ x + 9 $ | $ x $ | Up 9 |
| $ -x^2 - 6 $ | $ x^2 $ | Reflect over x-axis, down 6 |
| $ \sqrt[3]{x + 1} + 2 $ | $ \sqrt[3]{x} $ | Left 1, up 2 |
| $ 2|x + 3| - 4 $ | $ |x| $ | Left 3, stretch ×2, down 4 |

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## 🧠 Tips for Students:

- Inside the function? Affects horizontal shifts/stretches.
- Outside the function? Affects vertical shifts/stretches.
- Negative outside? Reflect over x-axis.
- Negative inside? Reflect over y-axis (but be careful with order).
- Always start with the parent function, then apply transformations step-by-step.

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If you'd like, I can generate a printable version of this answer key or help you interpret any graphs from the worksheet (if you describe them). Let me know!

Also, since the image says "Answer Key Included", you may want to check the original resource for official answers — but this breakdown should match exactly.

Let me know if you’d like help with graphing these functions or understanding order of operations for multiple transformations!