Several graphs of the quadratic function ƒ(x) = ax² + bx + c are shown below. For the given restrictions on a, b, and c, select the corresponding graph from choices A–F. (Hint: Use the discriminant.) A > 0; b² - 4ac > 0

Verified step by step guidance

1. Recall the quadratic function is given by \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
2. Understand the given conditions: \(a > 0\) means the parabola opens upward, so the graph should be a 'U' shape opening upwards.
3. Use the discriminant formula \(\Delta = b^2 - 4ac\) to determine the nature of the roots. Since \(b^2 - 4ac > 0\), the quadratic has two distinct real roots.
4. Interpret these conditions graphically: the parabola opens upward and crosses the x-axis at two distinct points (because of two real roots).
5. From the given graphs A–F, select the one that shows a parabola opening upward and intersecting the x-axis at two distinct points.

Key Concepts

Quadratic Function and Its Graph

A quadratic function is a polynomial of degree two, expressed as ƒ(x) = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. The coefficients a, b, and c determine the shape and position of the parabola on the coordinate plane.

Discriminant of a Quadratic Equation

The discriminant, given by b² - 4ac, indicates the nature of the roots of a quadratic equation. If the discriminant is greater than zero, the equation has two distinct real roots, meaning the parabola intersects the x-axis at two points. This helps in identifying the correct graph based on root behavior.

Effect of Coefficient 'a' on Parabola Orientation

The coefficient 'a' in the quadratic function determines the direction the parabola opens. When a > 0, the parabola opens upward, resembling a 'U' shape. This is crucial for selecting the correct graph since the problem specifies a > 0, so the parabola must open upwards.