Definition:
The quadratic function is f (x) = ax 2+bx+c, where a \ neq 0.
Picture:
The image of a quadratic function is a parabola with an upward or downward opening.
The opening direction of parabola is determined by the sign of coefficient a, if a >；; 0, the opening is upward; If a
Vertex:
The vertex of a parabola is its highest or lowest point, and its coordinates can be calculated by the formula (-\frac{b}{2a}, f(-\frac{b}{2a})).
Symmetry axis:
The symmetry axis is a straight line perpendicular to the X axis, and its equation is x = -\frac{b}{2a}.
Geometric meaning of coefficient:
Answer: Control the opening size and direction of parabola.
B: It's related to the position of the parabola axis of symmetry.
C: indicates the intercept of parabola on the y axis.
Root:
The root of the quadratic function is the value of x when the function value is 0, that is, the equation AX 2+BX+C = 0 is solved.
Discriminant:
The discriminant \ delta = b 2-4ac is used to judge the roots of quadratic equations.
If \Delta > 0, there are two unequal real roots.
If \Delta = 0, there are two equal real roots (multiple roots).
If \ delta
Factorization:
Sometimes the quadratic function can be expressed as ax 2+bx+c = a (x-X _ 1) (x-X _ 2), where x _1and x _ 2 are roots.
Application:
Quadratic function is widely used in physics, engineering, economy and other fields, such as projectile motion and optimization problems.
Translation of images:
By changing the constant term in the quadratic function, the position of parabola on the Y axis can be changed.
Image scaling:
By changing the coefficient a in the quadratic function, the opening size of parabola can be changed.
Increase or decrease of functions:
When a > 0, the parabola decreases on the left side of the symmetry axis and increases on the right side of the symmetry axis.
When a < 0, the parabola increases on the left side of the symmetry axis and decreases on the right side of the symmetry axis.
The above are some basic concepts and properties of quadratic function, and mastering these knowledge points is very important for solving related mathematical problems.