Q: What is the significance of the discriminant in a quadratic function?

The discriminant of a quadratic function, given by the expression b^2 - 4ac, provides important information about the nature of the roots of the quadratic equation. In Cambridge IGCSE Mathematics, if the discriminant is positive, the equation has two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If the discriminant is negative, the equation has no real roots, indicating that the parabola does not intersect the x-axis.

Question 1

What is a quadratic function in the context of Cambridge IGCSE Mathematics?

Accepted Answer

In Cambridge IGCSE Mathematics, a quadratic function is a type of polynomial function that can be expressed in the standard form y = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient a.

Question 2

How do you find the vertex of a quadratic function?

Accepted Answer

To find the vertex of a quadratic function given in the form y = ax^2 + bx + c, you can use the formula for the x-coordinate of the vertex, x = -b/(2a). Once you have the x-coordinate, substitute it back into the original equation to find the y-coordinate. The vertex is the point (x, y) and represents the maximum or minimum point of the parabola.

Question 3

What are the methods to solve a quadratic equation?

Accepted Answer

In Cambridge IGCSE Mathematics, quadratic equations can be solved using several methods: factoring, completing the square, and using the quadratic formula. Factoring involves expressing the quadratic equation as a product of two binomials. Completing the square involves rewriting the equation in the form (x - p)^2 = q. The quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), can be used when other methods are not feasible.

Question 4

How can you determine the direction in which a parabola opens?

Accepted Answer

The direction in which a parabola opens is determined by the sign of the coefficient a in the quadratic function y = ax^2 + bx + c. If a > 0, the parabola opens upwards, resembling a U-shape. If a < 0, the parabola opens downwards, resembling an upside-down U-shape. This is an important concept in Cambridge IGCSE Mathematics when analyzing the graph of quadratic functions.

Question 5

What is the significance of the discriminant in a quadratic function?

Accepted Answer

The discriminant of a quadratic function, given by the expression b^2 - 4ac, provides important information about the nature of the roots of the quadratic equation. In Cambridge IGCSE Mathematics, if the discriminant is positive, the equation has two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If the discriminant is negative, the equation has no real roots, indicating that the parabola does not intersect the x-axis.

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