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

Did you know that quadratic functions, like \( f(x) = x^2 - 1 \), can be graphed as parabolas? This specific function opens upwards and its vertex is located at the point \( (0, -1) \). The parabola will cross the x-axis at points where \( x^2 - 1 = 0 \), which means the x-intercepts are \( x = 1 \) and \( x = -1 \). In real-world scenarios, quadratic functions can model a wide variety of phenomena, including projectile motion. For example, if you were to throw a ball, its path can be described by a quadratic function similar to this one, illustrating how height changes over time. Imagine plotting its trajectory and predicting at what point it will hit the ground!