We are given the function $f(x) = x^2$ and asked to apply three transformations:

1. Stretch vertically by a factor of 2: This transformation multiplies the function by 2, changing $f(x)$ to $2f(x)$, or $2x^2$.

2. Shift downward by 5 units: This shifts the entire graph down by 5, so we subtract 5 from the function. The new function becomes $2x^2 - 5$.

3. Shift right by 4 units: This shift requires us to adjust the input of the function by subtracting 4 from $x$. The new function becomes $2(x - 4)^2 - 5$.

Final Transformed Function:

The equation for the final transformed graph is: $g(x) = 2(x - 4)^2 - 5$

Expanded Form:

Now, let's expand this equation: $g(x) = 2(x - 4)^2 - 5$ First, expand $(x - 4)^2$: $(x - 4)^2 = x^2 - 8x + 16$ Now substitute it back into the equation: $g(x) = 2(x^2 - 8x + 16) - 5$ Distribute the 2: $g(x) = 2x^2 - 16x + 32 - 5$ Simplify: $g(x) = 2x^2 - 16x + 27$

So the final expanded form of the transformed function is: $g(x) = 2x^2 - 16x + 27$

Would you like to go over any step in detail or have further questions?

Here are 5 related questions for you:

1. How do vertical stretches affect the graph of a quadratic function?
2. What happens to the vertex of a quadratic function when it is shifted horizontally?
3. How would a horizontal stretch by a factor of 3 affect the function $f(x) = x^2$?
4. Can you describe the effect of a vertical reflection on $f(x) = x^2$?
5. How does a vertical shift compare to a horizontal shift in terms of function transformations?

Tip: When performing multiple transformations, always handle shifts inside the parentheses first, followed by stretches or compressions, and finally vertical shifts.