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The question asks to find the equation of the function \( f(x) = x^2 \) after it has been vertically stretched by a factor of 3, shifted 5 units to the right, and 4 units up.

Let's break down the transformations:

1. **Vertical Stretch by a factor of 3**: 
   This will multiply the function by 3, turning \( f(x) = x^2 \) into \( g(x) = 3x^2 \).

2. **Shift 5 units to the right**: 
   To shift the function right, we replace \( x \) with \( (x - 5) \). So the function becomes \( g(x) = 3(x - 5)^2 \).

3. **Shift 4 units up**: 
   Finally, adding 4 units to the function shifts it upward. The equation becomes \( g(x) = 3(x - 5)^2 + 4 \).

So, the correct answer is:
\[
g(x) = 3(x - 5)^2 + 4
\]

In the provided choices, this matches the second option:
**\( g(x) = 3(x - 5)^2 + 4 \)**.

Would you like further clarification or details on any of the transformations?

Here are five related questions for further practice:
1. How would the equation change if the function were shifted 3 units left instead of right?
2. What happens to the graph if we vertically compress instead of stretch by a factor of 3?
3. How would the equation look if the function was reflected across the x-axis?
4. How does a horizontal shift affect the vertex of a parabola?
5. What is the effect of adding a constant inside the parentheses, instead of outside?

**Tip:** When transforming functions, keep track of each step by applying one transformation at a time—this makes it easier to avoid errors.

Which of the following represents the equation of the function f(x) = x^2 after it has been vertically stretched by a factor of 3 and shifted 5 units right and 4 units up?

Learn how to transform the quadratic function f(x) = x^2 by applying a vertical stretch, shifting it horizontally to the right, and moving it upwards. This problem focuses on function transformations, including a vertical stretch by a factor of 3, a horizontal shift of 5 units to the right, and a vertical shift of 4 units up. Suitable for students in Grades 9-11.

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