Cycle 1. Symmetry of circle
The (1) circle is an axisymmetric figure, and its symmetry axis is the straight line where the diameter lies.
(2) A circle is a figure with a symmetrical center, and its symmetrical center is the center of the circle.
(3) A circle is a rotationally symmetric figure.
2. Vertical diameter theorem
(1) bisects the chord perpendicular to its diameter and bisects the two arcs opposite the chord.
(2) Inference:
Bisect the diameter (non-diameter) of a chord, perpendicular to the chord and bisecting the two arcs opposite the chord.
Bisect the diameter of the arc and bisect the chord of the arc vertically.
3. The degree of the central angle is equal to the degree of the arc it faces. The degree of the circle angle is equal to half the radian it subtends.
(1) The circumferential angles of the same arc are equal.
(2) The circumferential angle of the diameter is a right angle; The angle of a circle is a right angle, and the chord it subtends is a diameter.
4. In the same circle or equal circle, as long as one of the five pairs of quantities, namely two chords, two arcs, two circumferential angles, two central angles and the distance between the centers of two chords, is equal, the other four pairs are also equal.
5. The two arcs sandwiched between parallel lines are equal.
(1) The center of the circle passing through two points must be on the vertical line connecting the two points.
(2) Three points that are not on the same straight line determine a circle, the center of which is the intersection of the perpendicular lines of three sides, and the distances from this point to these three points are equal.
The outer center of a right triangle is the midpoint of the hypotenuse. )
Parabola 1 related knowledge points. Definition: The trajectory from a fixed point on a plane to a point with the same distance from an alignment is called a parabola. The fixed point is called the focus of parabola, and the fixed line is called the directrix of parabola.
2. Parabola is an axisymmetric figure. The symmetry axis is a straight line x=-b/2a. The only intersection of the symmetry axis and the parabola is the vertex p of the parabola. Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
3. The parabola has a vertex p, and its coordinates are: P(-b/2a, (4ac-b 2)/4a) When -b/2a=0, p is on the Y axis; When δ = b 2-4ac = 0, p is on the x axis.
4. Quadratic coefficient A determines the opening direction and size of parabola: when a >: 0, parabola opens upwards; When a<0, the parabola opens downward. The larger the |a|, the smaller the opening of the parabola.
5. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.
When A and B have the same number (ab>0), the symmetry axis is on the left side of Y axis;
When a and b have different numbers (i.e. AB
6. The constant term c determines the intersection of parabola and Y axis. The parabola intersects the y axis at (0, c).
The characteristic of the coordinates of a point in a special position is that the ordinate of the point on the X axis is zero. The abscissa of a point on the y axis is zero.
2. The horizontal and vertical coordinates of the points on the bisector of the first quadrant and the third quadrant are equal; The horizontal and vertical coordinates of the points on the bisector of the second and fourth quadrants are opposite to each other.
3. If the abscissas of any two points are the same, the connecting line of the two points is parallel to the longitudinal axis; If the vertical coordinates of two points are the same, the straight line connecting the two points is parallel to the horizontal axis.
4. Distance from point to axis and origin
The distance from the point to the X axis is | y | The distance from the point to the Y axis is | x | The distance from the point to the origin is the square of x plus the square root of y.
Quadratic function 1. Properties of quadratic function
Especially quadratic function (hereinafter referred to as function) y=ax? +bx+c(a≠0).
When y=0, the quadratic function is a unary quadratic equation about x (hereinafter referred to as the equation), that is, ax? +bx+c=0(a≠0)
At this point, whether the function image intersects with the X axis means whether the equation has real roots.
The abscissa of the intersection of the function and the x axis is the root of the equation.
2. The range of quadratic function
Vertex coordinates (-b/2a, (4αc-b? )/4α)
The basic form of quadratic function is y=ax? +bx+c(a≠0)
When a > 0, the parabolic opening is upward and the image is above the vertex, so the range y≥(4ac-b? ) /4a, that is [(4ac-b? )/4a，+∞).
When a < 0, the parabola opens downward, and the range of the function is (-∞, (4ac-b? )/4a]
When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is an even function, and the analytical expression is deformed into y=ax? +c(a≠0).