Let's analyze the problem step by step.

Problem Overview

The given problem involves the quadratic function $f(x) = x^2$ and how it is transformed into $g(x) = -2x^2 + 4x + 30$. The task includes:

(a) Describing the transformations applied to $f(x) = x^2$ in the correct order.
(b) Finding the point on $g(x)$ that corresponds to the given point $A(2, 4)$ on $y = f(x)$.

Part (a): Sequence of Transformations

To transform $f(x) = x^2$ into $g(x) = -2x^2 + 4x + 30$, we analyze the function step by step:

1. Start with $f(x) = x^2$:

   • Begin with the base quadratic.

2. Introduce the coefficient of $-2$:

   • Scaling by $-2$ reflects $f(x)$ across the $x$-axis and stretches it vertically by a factor of 2. The function becomes $-2x^2$.

3. Add $4x$:

   • Adding $4x$ represents a horizontal shift and tilts the parabola. The function now becomes $-2x^2 + 4x$.

4. Add $30$:

   • Finally, adding $30$ shifts the function vertically upwards by 30 units, giving the final form $g(x) = -2x^2 + 4x + 30$.

Final Transformations (in order):

1. Reflect over the $x$-axis and stretch by a factor of 2.
2. Add $4x$, introducing a horizontal shift.
3. Shift the function up by 30.

Part (b): Corresponding Point

The given point $A(2, 4)$ lies on $y = f(x) = x^2$. To find the corresponding point on $y = g(x)$, follow these steps:

1. Start with $A(2, 4)$.

   • Since the transformations affect the function, we substitute $x = 2$ into $g(x)$ to calculate the corresponding $y$-value.

2. Substitute $x = 2$ into $g(x) = -2x^2 + 4x + 30$:
   $g(2) = -2(2^2) + 4(2) + 30$ $g(2) = -2(4) + 8 + 30 = -8 + 8 + 30 = 30$

3. The corresponding point is $B(2, 30)$.

Final Answers:

(a) The transformations are:

1. Reflect $f(x)$ over the $x$-axis and stretch by a factor of 2.
2. Add $4x$ to shift horizontally.
3. Shift up by 30.

(b) The corresponding point is $B(2, 30)$.

Would you like a deeper explanation of any step or further assistance?

Here are five related questions to explore:

1. How do horizontal shifts impact quadratic functions?
2. What is the geometric meaning of adding $4x$ to a quadratic equation?
3. How would the transformation change if $g(x)$ included $-4x$ instead?
4. How can we derive the vertex of $g(x) = -2x^2 + 4x + 30$?
5. What is the impact of a reflection over the $x$-axis on a function's graph?

Tip: When working with transformations, always apply them sequentially to understand their cumulative effects!