https://file.aiquickdraw.com/mathbot/file/1726683149406cvin0q2n.jpg

Let's analyze the given problem step by step.

The function is:
\[
f(x) = (x - 2)^2 - 5
\]

### a) Choose the correct parent function:
The structure of \((x - 2)^2 - 5\) indicates that the parent function is \(y = x^2\) because it's a quadratic function. The transformations applied to it involve shifting and translations.

So, the correct parent function is **\(y = x^2\)**.

### b) Choose the correct transformation (Reflections):
Since there are no negative signs in front of the expression, no reflections have been applied. Therefore, the correct answer for reflections is **None**.

### c) Choose the correct transformation (Stretches/Compressions):
There is no coefficient affecting the \(x^2\) term, which means there are no stretches or compressions. The answer is **None**.

### d) Choose the correct transformation (Vertical Shifts):
The term \(-5\) indicates a **vertical shift down by 5 units**.

### e) Choose the correct transformation (Horizontal Shifts):
The term \((x - 2)\) inside the squared part indicates a **horizontal shift to the right by 2 units**.

### Summary:
- a) \(y = x^2\)
- b) Reflections: None
- c) Stretches/Compressions: None
- d) Vertical Shift: Down by 5 units
- e) Horizontal Shift: Right by 2 units

Would you like further details or clarifications on any of these points?

Here are five related questions to expand your understanding:
1. How does the vertical shift affect the range of the function?
2. What would happen if there was a negative sign in front of \((x - 2)^2\)?
3. Can you describe the graph of the parent function \(y = x^2\) and how it transforms based on the shifts?
4. How would the function change if the coefficient of the squared term were different from 1?
5. What is the vertex of the function \(f(x)\)?

**Tip:** When analyzing function transformations, always observe how changes inside parentheses affect horizontal shifts, while changes outside impact vertical behavior.

Determine the parent function from which the graph of the function f(x) = (x - 2)^2 - 5 can be obtained. Identify each transformation that can be applied to the parent function in order to obtain the graph of the function shown below.

Learn how to determine the parent function for f(x) = (x - 2)^2 - 5 and apply the correct transformations. This problem covers function transformations, including horizontal and vertical shifts, with detailed steps for students in Grades 9-11.

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