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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?

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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.

Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function: g(x) = x^2 - 4.

Learn how to graph the quadratic function g(x) = x^2 - 4 by applying vertical shifts. This problem explains how to determine the domain and range, with a focus on transformations such as shifting. Suitable for students in Grades 8-10.

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