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Let's break down the problem step by step to find the corresponding point.

1. The given point is \( (2, 4) \) on the graph of \( y = x^2 \). This means when \( x = 2 \), \( y = 4 \) because:
   \[
   y = 2^2 = 4.
   \]

2. Now, we need to find the corresponding point on the graph of \( y = (x + 3)^2 \).

   - In the equation \( y = (x + 3)^2 \), we can think of this as a horizontal shift of the graph \( y = x^2 \) by 3 units to the left.

3. To find the corresponding \( x \)-coordinate for the new graph:
   \[
   x_{\text{new}} + 3 = x_{\text{old}} \quad \text{so} \quad x_{\text{new}} = x_{\text{old}} - 3.
   \]
   Substituting \( x_{\text{old}} = 2 \):
   \[
   x_{\text{new}} = 2 - 3 = -1.
   \]

4. Now, substitute \( x_{\text{new}} = -1 \) into the equation \( y = (x + 3)^2 \) to find the new \( y \)-coordinate:
   \[
   y = (-1 + 3)^2 = 2^2 = 4.
   \]

Thus, the corresponding point on the graph of \( y = (x + 3)^2 \) is \( (-1, 4) \).

The correct answer is \( (-1, 4) \).

Would you like more details or have any questions? Here are 5 related questions to expand on this:

1. How do horizontal shifts affect the graph of a function?
2. What happens to the graph of \( y = x^2 \) if we apply a vertical shift?
3. How would you solve for the corresponding point if the graph was \( y = (x - 2)^2 \)?
4. What is the vertex of the graph \( y = (x + 3)^2 \)?
5. Can you derive the transformation of \( y = x^2 \) to \( y = (x + 3)^2 \) algebraically?

**Tip**: When analyzing transformations, always pay attention to shifts and the sign inside the parentheses—it determines the direction of the shift!

A point (2, 4) is on the graph of y = x^2. What is the corresponding point on the graph of y = (x + 3)^2?

Find the corresponding point on the graph of y = (x + 3)^2, given that the point (2, 4) is on the graph of y = x^2. Learn about quadratic transformations and horizontal shifts.

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