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.