Table of function transformations with graphs, tables, and equations.

It looks like you've uploaded a table that includes various types of functions (Quadratic, Square Root, Cube Root, Exponential, Logarithmic, and Rational), each with the following components:

- Function: The mathematical expression.
- Graph: A blank grid where you're expected to sketch the graph.
- Table: A set of x and y values to plot.
- Transformation: A general form showing how transformations affect the function.
- Notes: Additional information or rules.

Since I can't view images directly, I’ll assume your task is to complete the table by filling in missing parts such as:
- Completing the graph (by plotting points from the table),
- Filling in missing values in the table,
- Understanding the transformations, and
- Possibly identifying key features (domain, range, asymptotes, etc.).

Let me walk through each function and explain how to complete it properly. Since this appears to be a worksheet for practicing function transformations, here’s how you would approach each one.

---

1. Quadratic Function

- Function: $ y = x^2 $
- Transformation: $ y = a(x - h)^2 + k $
- Table:

| x | y |
|---|---|
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |

✔ This is correct. Plot these points on the graph — they form a parabola opening upward with vertex at (0,0).

💡 Note: Transformations shift, stretch, or reflect the parabola. For example:
- $ h $ shifts horizontally,
- $ k $ shifts vertically,
- $ a $ affects width and direction.

---

2. Square Root Function

- Function: $ y = \sqrt{x} $
- Transformation: $ y = a\sqrt{b(x - h)} + k $
- Table:

| x | y |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |

✔ These are correct. Graph starts at (0,0) and increases slowly.

⚠️ Note: Domain is $ x \geq 0 $. Only non-negative inputs are valid.

---

3. Cube Root Function

- Function: $ y = \sqrt[3]{x} $
- Transformation: $ y = a\sqrt[3]{b(x - h)} + k $
- Table:

| x | y |
|---|---|
| -8 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 8 | 2 |

✔ Correct. Cube root is defined for all real numbers. It's symmetric about origin.

💡 Graph passes through origin and increases slowly.

---

4. Exponential Function

- Function: $ y = b^x $, $ b > 0, b \ne 1 $
- Transformation: $ y = ab^{x-h} + k $
- Table:

| x | y |
|---|---|
| -2 | 1/4 |
| -1 | 1/2 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |

Assuming $ b = 2 $, so $ y = 2^x $

✔ Values match. Graph increases rapidly for $ x > 0 $, approaches 0 as $ x \to -\infty $

💡 Horizontal asymptote: $ y = 0 $

---

5. Logarithmic Function

- Function: $ y = \log_b x $, $ b > 0, b \ne 1 $
- Transformation: $ y = a \log_b(x - h) + k $
- Table:

| x | y |
|---|---|
| 1/4 | -2 |
| 1/2 | -1 |
| 1 | 0 |
| 2 | 1 |
| 4 | 2 |

Assuming $ b = 2 $, so $ y = \log_2 x $

✔ Correct. This is inverse of $ y = 2^x $

💡 Domain: $ x > 0 $, Vertical asymptote at $ x = 0 $

---

6. Rational Function

- Function: $ y = \frac{1}{x} $
- Transformation: $ y = \frac{a}{b(x - h)} + k $
- Table:

| x | y |
|---|---|
| -2 | -1/2 |
| -1 | -1 |
| 1 | 1 |
| 2 | 1/2 |

✔ Correct values. Note: undefined at $ x = 0 $

💡 Graph has two branches:
- One in QI and QIII
- Vertical asymptote at $ x = 0 $
- Horizontal asymptote at $ y = 0 $

---

✔ Summary of Steps to Complete the Worksheet:

1. Fill in missing table values using the given function.
2. Plot the points on the provided grids.
3. Draw smooth curves based on function type.
4. Apply transformations if given (e.g., shift, stretch).
5. Label key features:
- Vertex (quadratic)
- Asymptotes (rational, log, exponential)
- Domain and Range

---

🔍 Example: Apply Transformation

Suppose you're asked to graph $ y = 2(x - 1)^2 + 3 $

- Start with base: $ y = x^2 $
- Shift right 1 → $ (x - 1)^2 $
- Stretch by 2 → $ 2(x - 1)^2 $
- Shift up 3 → $ 2(x - 1)^2 + 3 $

Vertex moves to (1, 3)

---

If you have specific questions like:
- "How do I fill in the table?"
- "What does the transformation mean?"
- "How do I draw the graph?"

Please let me know! I can help with any particular section.

Would you like me to generate a completed version of this worksheet (with graphs described)? Or focus on one function in detail?