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The question asks whether the function \( f(x) = \left(\frac{x + 3}{2}\right)^2 - 7 \) is a transformation of the parent function \( f(x) = x^2 \) that is shifted down 7 units and shifted right 3 units.

Let's break this down:

1. **Shifting down 7 units:** The term \(-7\) at the end of the function \( f(x) = \left(\frac{x + 3}{2}\right)^2 - 7 \) indicates a vertical shift downward by 7 units. This part of the statement is correct.

2. **Shifting right 3 units:** To determine the horizontal shift, let's focus on the expression inside the square, \( \left(\frac{x + 3}{2}\right) \).

   - First, consider the term inside the square, \( x + 3 \). A term like \( x + 3 \) generally indicates a horizontal shift **left** by 3 units (since it's \( x + 3 \), the opposite sign).

   - However, this expression is divided by 2, so it represents a horizontal **compression** by a factor of 2. The function \( \left(\frac{x + 3}{2}\right)^2 \) would compress the function horizontally but would not shift it by 3 units to the right. 

   To correctly shift the function right by 3 units, you would expect to see \( (x - 3) \) in the expression, which we do not.

Therefore, the given function \( f(x) = \left(\frac{x + 3}{2}\right)^2 - 7 \) is **not** shifted right 3 units but is instead shifted left by 3/2 units and compressed horizontally.

**Conclusion:** The statement is **False**.

Would you like further details on this? Here are some related questions for further exploration:

1. How do vertical and horizontal shifts affect the graph of a function?
2. What does a horizontal compression or stretch mean for a function?
3. Can a function be both shifted and compressed? How is that reflected in its equation?
4. How would you write a function that is shifted right 3 units and down 7 units?
5. What is the effect of dividing \( x \) by a number inside a function?

**Tip:** Always check both horizontal and vertical transformations separately to fully understand how a function has been modified from its parent function.

True or false: The function f(x) = ((x + 3)/2)^2 - 7 is a transformation of the parent function f(x) = x^2 shifted down 7 units and shifted right 3 units.

Determine whether the function f(x) = ((x + 3)/2)^2 - 7 is a transformation of the parent function f(x) = x^2 shifted down 7 units and right 3 units. Learn about vertical shifts, horizontal compressions, and function transformations. Suitable for Grades 9-11.

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