The transformation that converts the graph of $f(x) = 10x^2 - 2$ into the graph of $g(x) = -10x^2 + 2$ involves two distinct transformations:

1. Vertical reflection:

   • The coefficient of $x^2$ in $f(x)$ is positive (10), while in $g(x)$ it is negative (-10). This means the graph of $f(x)$ is vertically reflected across the x-axis to obtain the graph of $g(x)$.
   • Specifically, this changes the direction in which the parabola opens: the graph of $f(x)$ opens upwards, while the graph of $g(x)$ opens downwards.

2. Vertical shift:

   • In $f(x)$, the constant term is -2, whereas in $g(x)$, the constant term is +2. This means the graph is shifted 4 units upwards. The transformation is a vertical translation of the graph of $f(x)$ by 4 units to get the graph of $g(x)$.

Summary of Transformations:

1. Vertical reflection: Reflect across the x-axis.
2. Vertical translation: Shift 4 units upwards.

Would you like more details on how to apply these transformations?

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Here are five related questions to explore further:

1. What are the general steps for reflecting a graph vertically?
2. How would you shift the graph of $f(x)$ if the constant term changed by more than 4 units?
3. What is the effect of changing the coefficient of $x^2$ in a quadratic function?
4. How do horizontal transformations differ from vertical ones?
5. Can you combine multiple transformations in one step?

Tip: When reflecting a graph, remember that a vertical reflection changes only the sign of the output (y-values), not the x-values.