This appears to be a set of math problems involving transformations of quadratic functions. Let's break down the task into parts based on the provided information.

a) Describe the transformation from f(x) to g(x):
For each pair of equations $f(x)$ and $g(x)$, the transformation is typically a shift. The general form of transformations is:

• If $g(x) = f(x) + c$, it represents a vertical shift by $c$ units.
• If $g(x) = f(x + c)$, it represents a horizontal shift by $c$ units.

Now, let's look at the given equations and describe their transformations.

1. $f(x) = 3x^2 + 4$ and $g(x) = f(x) + 2$:

   • Transformation: $g(x)$ is the same as $f(x)$, but shifted up by 2 units.

2. $f(x) = \frac{1}{2}x^2 + 1$ and $g(x) = f(x) - 4$:

   • Transformation: $g(x)$ is the same as $f(x)$, but shifted down by 4 units.

3. $f(x) = -\frac{1}{4}x^2 - 6$ and $g(x) = -f(x) - 3$:

   • Transformation: First, the graph of $f(x)$ is reflected across the x-axis (due to the negative sign in front of $f(x)$), and then it is shifted down by 3 units.

4. $f(x) = 4x^2 - 5$ and $g(x) = f(x) + 7$:

   • Transformation: $g(x)$ is the same as $f(x)$, but shifted up by 7 units.

b) Write an equation that represents $g(x)$ in terms of $x$:
For each pair, the transformation can be written by directly modifying $f(x)$.

Let me know if you need further elaboration on the specific steps! Would you like me to help you with additional transformations or questions related to this topic?

Here are some related questions for further exploration:

1. How does the transformation $g(x) = -f(x)$ affect the graph of a quadratic function?
2. What is the impact of changing the coefficient of $x^2$ in the quadratic function $f(x) = ax^2 + bx + c$?
3. How would the transformation change if $g(x) = f(x + 3)$ instead of $f(x) + c$?
4. How would the graph look if $g(x) = -f(x + 2) + 5$?
5. How do horizontal shifts differ from vertical shifts in the transformation of quadratic functions?

Tip: To check your transformations, graph both $f(x)$ and $g(x)$ using a graphing calculator or software. It can be helpful in visualizing the shift directly!