https://file.aiquickdraw.com/mathbot/file/1726348967637whfpv9hk.jpg

The function given in the problem is \( f(x) = x^2 - 2 \).

### Analysis:
- This function is **nonlinear** because it contains a quadratic term, \( x^2 \), and is therefore not a straight line (linear functions have the form \( f(x) = mx + b \)).
- A nonlinear function is not constant either because its value changes with different inputs of \( x \).

### Graph:
The correct graph should show a **parabola** that opens upwards since the coefficient of \( x^2 \) is positive (1 in this case), and it has been shifted downward by 2 units (due to the \( -2 \) in the equation).

In the options shown:
- **Option A** is the correct graph. It represents a parabola that opens upward and is shifted down by 2 units, matching the equation \( f(x) = x^2 - 2 \).

### Conclusion:
- The function \( f(x) = x^2 - 2 \) is **nonlinear**.
- The correct graph is **Option A**, showing a parabola.

Would you like a detailed explanation on quadratic functions, or do you have further questions?

Here are 5 questions for additional exploration:
1. What makes a function linear or nonlinear?
2. How do you find the vertex of a quadratic function like \( f(x) = x^2 - 2 \)?
3. What transformations occur when adding constants to quadratic functions?
4. How can we graph other quadratic functions like \( f(x) = -x^2 + 3 \)?
5. How do you find the axis of symmetry for quadratic functions?

**Tip**: A quadratic function always graphs as a parabola, and its general form is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) define the parabola's shape and position.

Determine if f(x) = x^2 - 2 is a linear or nonlinear function, and choose the correct graph representing it.

Explore how to determine if f(x) = x^2 - 2 is a nonlinear function and how to graph it. This problem is ideal for students in Grades 8-10, covering quadratic equations and function graphs.

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