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The problem asks which equation would vertically stretch the quadratic function \( f(x) = x^2 \) by a factor of 3 and shift it 4 units to the right.

### Key transformations to apply:
1. **Vertical Stretch by a factor of 3:** This means multiplying the entire function by 3, so the function becomes \( f(x) = 3x^2 \).
2. **Shift 4 units to the right:** To shift a function to the right by 4, you replace \( x \) with \( (x - 4) \). So, the function becomes \( f(x) = 3(x - 4)^2 \).

### Now, let's check the given options:
- **Option a)**: \( f(x) = (x - 4)^2 + 3 \) → This shifts the function 4 units to the right, but it doesn't apply a vertical stretch of 3. The +3 shifts the function upwards.
- **Option b)**: \( f(x) = (x + 3)^2 + 4 \) → This shifts the function 3 units to the left and 4 units upward, but no vertical stretch is applied.
- **Option c)**: \( f(x) = (x + 4)2 + 3 \) → This shifts the function to the left and also doesn't apply a vertical stretch properly.
- **Option d)**: \( f(x) = 3(x - 4)^2 \) → This applies both the vertical stretch by 3 and the horizontal shift 4 units to the right.

### Conclusion:
The correct answer is **d)** \( f(x) = 3(x - 4)^2 \).

Would you like more details, or do you have any questions? Here are five related questions:

1. How does shifting a quadratic function affect its graph?
2. What is the general form of a quadratic function after a vertical stretch and horizontal shift?
3. Can you explain how the vertex of the quadratic function moves after a horizontal shift?
4. What happens to the graph of \( f(x) = x^2 \) if it's shifted 4 units to the left instead?
5. How would the graph change if you vertically compressed the function by a factor of \(\frac{1}{2}\) instead of stretching it?

**Tip:** Always remember that shifting horizontally affects the \( x \)-values inside the parentheses, while stretching affects the entire function's output.

Which equation would vertically stretch the quadratic function f(x) = x^2 by a factor of 3 and shift it 4 to the right?

Explore how to transform the quadratic function f(x) = x^2 by applying a vertical stretch by a factor of 3 and shifting it 4 units to the right. Learn about quadratic function transformations with this step-by-step explanation.

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