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.