Question

Квадратичні функції!

Answer 1

Quadratic Functions

A quadratic function is a type of polynomial function of degree 2. It is defined by an equation of the form:

f(x) = ax^2 + bx + c

where a, b, and c are constants, and a is not equal to 0. The graph of a quadratic function is a parabola.

Quadratic functions have several important properties and characteristics:

1. Vertex: The vertex is the point on the graph where the parabola reaches its minimum or maximum value. The x-coordinate of the vertex can be found using the formula x = -b / (2a), and the y-coordinate can be found by substituting the x-coordinate into the equation.

2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The equation of the axis of symmetry is x = -b / (2a).

3. Intercepts: The x-intercepts (also known as zeros or roots) are the points where the graph intersects the x-axis. They can be found by solving the quadratic equation ax^2 + bx + c = 0. The y-intercept is the point where the graph intersects the y-axis and can be found by substituting x = 0 into the equation.

4. Discriminant: The discriminant of a quadratic function is the expression b^2 - 4ac. It determines the nature of the solutions of the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution (a repeated root). If the discriminant is negative, the equation has no real solutions (two complex conjugate roots).

5. Graph: The graph of a quadratic function is a parabola. The shape of the parabola depends on the sign of the coefficient a. If a is positive, the parabola opens upward and has a minimum value. If a is negative, the parabola opens downward and has a maximum value.

6. Applications: Quadratic functions have various applications in real-life situations, such as physics (projectile motion), engineering (optimization problems), economics (cost and revenue functions), and more.

Please let me know if there's anything specific you would like to know about quadratic functions!