The question you're working on asks you to graph the function $g(x) = x^2 - 4$, and to find the domain and range of the function while using techniques like shifting, compressing, stretching, and/or reflecting.

Let's break it down:

1. Understanding the Function $g(x) = x^2 - 4$:

• The function $g(x)$ is a quadratic function, which normally has the form $f(x) = x^2$.
• The basic graph of $f(x) = x^2$ is a parabola opening upwards with its vertex at $(0, 0)$.
• In this case, the function is $g(x) = x^2 - 4$, which means the graph of $f(x) = x^2$ has been shifted down by 4 units. This is a vertical shift, and the vertex moves from $(0, 0)$ to $(0, -4)$.

2. Domain:

• The domain of any quadratic function is all real numbers, since you can input any $x$-value into the function. Thus, the domain is: $\text{Domain} = (-\infty, \infty)$

3. Range:

• Since the parabola opens upwards and the vertex is at $(0, -4)$, the lowest point on the graph is $-4$. Therefore, the range is all values of $y$ greater than or equal to $-4$. Thus, the range is: $\text{Range} = [-4, \infty)$

4. Graphing the Function:

• Plot the vertex at $(0, -4)$.
• The parabola opens upwards and is symmetrical about the y-axis.
• The shape of the parabola will be the same as $y = x^2$, just shifted down by 4 units.

Would you like more details about how to graph the parabola on the tool or how to handle shifts and transformations in general?

Here are five follow-up questions related to this concept:

1. How does adding or subtracting a constant to $x^2$ affect the graph of the function?
2. What is the difference between a horizontal and vertical shift in a graph?
3. How would the graph change if the function were $g(x) = -x^2 + 4$?
4. What does it mean for a function to be reflected across the x-axis or y-axis?
5. How can you determine the vertex of a quadratic function in standard form?

Tip: When graphing transformations, always identify how the constants affect the basic function's shape, position, and direction.