Graph the following function using the techniques of​ shifting, compressing,​ stretching, and/or reflecting. Start with the graph of the basic function shown to the right. Find the domain and range of the function. 
g left parenthesis x right parenthesis equals x squared minus 2

We are tasked with graphing the function \( g(x) = x^2 - 2 \), and analyzing its domain and range using transformations. Let's break this process down into steps, starting with the basic function.

### 1. **Start with the basic function:**

The basic function here is \( f(x) = x^2 \), which is a parabola that opens upwards with its vertex at the origin \( (0,0) \).

### 2. **Identify transformations:**

The function \( g(x) = x^2 - 2 \) is a transformation of the basic function \( f(x) = x^2 \). Specifically:
- **Vertical Shift**: The term \( -2 \) represents a vertical shift **down** by 2 units.

Thus, the function \( g(x) = x^2 - 2 \) will be the same parabola as \( f(x) = x^2 \), but it will be shifted downwards by 2 units. The vertex of the new parabola will be at \( (0, -2) \).

### 3. **Graphing the function:**

- Start with the parabola \( f(x) = x^2 \), which has a vertex at \( (0, 0) \) and passes through points like \( (-1, 1) \), \( (1, 1) \), \( (-2, 4) \), and \( (2, 4) \).
- Now shift the entire graph down by 2 units. This moves the vertex to \( (0, -2) \), and the graph passes through points \( (-1, -1) \), \( (1, -1) \), \( (-2, 2) \), and \( (2, 2) \).

### 4. **Domain and Range:**

- **Domain**: The domain of \( g(x) = x^2 - 2 \) is the same as the domain of \( f(x) = x^2 \), which is all real numbers. Therefore, the domain is \( (-\infty, \infty) \).
- **Range**: Since the parabola opens upwards and the lowest point is the vertex \( (0, -2) \), the range is \( [-2, \infty) \).

### Summary:
- The function \( g(x) = x^2 - 2 \) is a downward vertical shift of the basic parabola \( f(x) = x^2 \).
- The domain is \( (-\infty, \infty) \), and the range is \( [-2, \infty) \).

Do you have any questions or would you like more details on any of these steps?

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Here are 5 related questions to deepen understanding:
1. How would the graph change if we had \( g(x) = x^2 + 3 \)?
2. What would be the domain and range of \( g(x) = -x^2 - 2 \)?
3. How does the graph change when we add or subtract a constant to a function?
4. How can we reflect the graph of \( x^2 \) over the x-axis?
5. What happens to the graph of \( g(x) = x^2 \) if we compress it horizontally?

**Tip**: Always check how constants outside or inside the function (like \( +c \) or \( -c \)) affect the graph through transformations!

Graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function shown to the right. Find the domain and range of the function. g(x) = x^2 - 2

Explore how to graph the quadratic function g(x) = x^2 - 2 using transformations like vertical shifting. Understand the domain and range of this function with a detailed explanation. Ideal for students in Grades 9-11.

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