The discriminant, given by b² - 4ac, plays a crucial role in determining the nature of the roots of a quadratic equation. In real-world applications, the discriminant can provide insights into the feasibility or number of solutions to a problem. For instance, in physics, when calculating the time an object takes to hit the ground, a negative discriminant might indicate that there's no real-time solution, perhaps due to incorrect data. Similarly, in finance or economics, the discriminant can indicate possible break-even points or the number of solutions to an optimisation problem.

No, a quadratic function can have at most two x-intercepts. The x-intercepts, also known as roots or zeros, are the points where the graph of the function intersects the x-axis. Since a quadratic is a second-degree polynomial, it can have up to two real roots. However, depending on the discriminant, it might have two distinct real roots, one repeated real root, or no real roots at all (in which case it would have two complex conjugate roots). The discriminant (b² - 4ac) determines the nature of these roots.

The value of 'a' in the quadratic function y = ax² + bx + c directly affects the width and direction of the parabola. If the absolute value of 'a' is large, the parabola will be narrower, and if the absolute value of 'a' is small (but not zero), the parabola will be wider. Essentially, 'a' determines the "stretch" or "compression" of the parabola. A positive 'a' value means the parabola opens upwards, while a negative 'a' value means it opens downwards. By changing the value of 'a', one can control how spread out or steep the curve of the parabola is.

The nature of the extremum (maximum or minimum) of a quadratic function is determined by the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, and the vertex represents the minimum value of the function. Conversely, if 'a' is negative, the parabola opens downwards, and the vertex represents the maximum value. This is crucial in optimisation problems where one might need to find the highest or lowest value of a function under given constraints. For instance, in business, it can help determine the optimal price to charge to maximise revenue.

The graph of a quadratic function is always a parabola due to the degree of the polynomial. A quadratic function is a second-degree polynomial, and the highest power of the variable is 2. This results in a curve that is symmetric about its axis of symmetry. The coefficient of the x² term determines the direction of the parabola. If the coefficient is positive, the parabola opens upwards, and if it's negative, it opens downwards. The nature of the squared term ensures that as x moves away from the vertex in both directions, y values increase or decrease quadratically, giving the characteristic U-shape.