https://file.aiquickdraw.com/mathbot/file/1727818772809olgar3t5.jpg

We are given two functions: 
- \( f(x) = x^2 \), which is a basic parabola.
- \( g(x) = (x + 5)^2 + 1 \), which is a transformation of \( f(x) \).

### Describing the Transformation:
- The term \( (x + 5) \) indicates a **horizontal translation**. The graph is shifted 5 units to the **left**.
- The \( +1 \) indicates a **vertical translation**. The graph is shifted 1 unit **up**.

So, the graph of \( g(x) \) is a translation 5 units **left** and 1 unit **up** from the graph of \( f(x) \).

### Identifying the Graph:
Looking at the options in the image:
- We need to find the graph where \( g(x) \) (the red parabola) is 5 units to the left and 1 unit higher than \( f(x) \) (the blue parabola).
  
By analyzing the provided graphs:
- The **third graph (bottom left)** shows the red graph \( g(x) \) shifted 5 units to the left and 1 unit up from the blue graph \( f(x) \), which matches the transformation.

Thus, the correct graph is the **third one (bottom left)**.

Would you like more details on transformations or help with another problem?

### Related Questions:
1. How do you find the vertex of a transformed quadratic function?
2. What is the effect of adding a constant outside the parentheses on the graph of a function?
3. What other types of transformations can be applied to \( f(x) = x^2 \)?
4. How do horizontal and vertical translations affect the domain and range of the function?
5. Can you describe the reflection of \( f(x) = x^2 \) across the x-axis or y-axis?

### Tip:
Always remember that a transformation inside the parentheses (e.g., \( (x + 5) \)) affects the graph horizontally, while a transformation outside (e.g., \( +1 \)) affects the graph vertically.

Describe the transformation of f(x) = x^2 represented by g(x) = (x + 5)^2 + 1. Identify the graph of each function.

Learn how to describe the transformation of the quadratic function f(x) = x^2 to g(x) = (x + 5)^2 + 1, involving horizontal and vertical shifts. Includes graph identification and explanation. Suitable for Grades 8-10.

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