(1) \( f(x)=x^{2}-1 \)

The function \( f(x) = x^2 - 1 \) is a classic example of a quadratic function, which has a parabolic shape when graphed. It opens upwards and has its vertex at the point (0, -1), making it a great representation of the basic structure of quadratic functions. The roots, or x-intercepts, can be found by factoring, leading to \( f(x) = (x-1)(x+1) \). This means the function crosses the x-axis at x = 1 and x = -1! This function also has real-world applications, such as in physics, where it could represent the trajectory of a thrown object, adjusted for height. For instance, if you consider an object's height above the ground with respect to time, the equation can help model scenarios where height follows a parabolic path, like a ball being thrown or a satellite orbiting in its arc! Keep an eye out for those x-intercepts—they often signify important points in real-life contexts.