Attribute: It is non-negative;

2 multiplication and division of quadratic root:

3 addition and subtraction of quadratic roots: when adding and subtracting quadratic roots, first merge the simplest quadratic roots of Huawei, and then merge the quadratic roots with the same number of roots.

Helen-Qin Jiushao formula: S is the area of triangle, and P is.

1 unary quadratic equation: an equation with algebraic expressions on both sides of the equal sign, only one unknown, and the highest degree is 2.

2 the solution of quadratic equation in one variable

Matching method: match one side of the equation into a completely flat way, and then square both sides;

Factorization: the product of two factors on the left and zero on the right.

3 The application of quadratic equation in one variable in practical problems

Vieta Theorem: Let it be the two roots of the equation, then there is.

1: graphic transformation in which a graphic rotates by an angle around a certain point.

Property: the distance from the corresponding point to the rotation center is equal;

The included angle between the corresponding point and the line segment connecting the rotation center is equal to the rotation angle.

Graphic consistency before and after rotation.

2. Center symmetry: if one graph rotates around a point by 180 degrees and coincides with another graph, then the two graphs are symmetrical about the center of this point;

Centrally symmetric figure: a figure rotates 180 degrees around a certain point and can coincide with the original figure, so it is called centrosymmetric;

Three coordinates of a point with symmetrical origin.

1 Definition of circle, center, radius, diameter, arc, chord and semicircle

2 Diameter perpendicular to the chord

A circle is an axisymmetric figure, and any straight line with a diameter is its axis of symmetry;

The diameter perpendicular to the chord divides the chord in two and squares the two arcs opposite to the chord;

The diameter of the chord is perpendicular to the chord, and the two arcs opposite the chord are equally divided.

3 Arc, chord and central angle

In the same circle or in the same circle, arcs and chords with equal central angles are equal.

4 circle angle

In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half the central angle of the arc;

The semicircle (or diameter) faces the right angle, and the 90-degree angle faces the diameter.

The positional relationship between five points and a circle

D>r, the point outside the circle

Points on the circle d=r

Points in circle d

Theorem: Three points that are not on a straight line determine a circle.

6 the positional relationship between a straight line and a circle

Intersection d < r

Tangent d=r

D & gtr

The property theorem of tangent: the tangent of a circle is perpendicular to the radius of the tangent point;

Judgment theorem of tangent: the straight line passing through the outer end of the circle and perpendicular to this radius is the tangent of the circle;

Tangent Length Theorem: Two tangents leading to a circle from a point outside the circle are equal in length, and the connecting line between the point and the center of the circle bisects the included angle of the two tangents.

The inscribed circle of a triangle: the circle tangent to each side of the triangle is its inscribed circle, and the center of the circle is the intersection of bisectors of three angles of the triangle, which is the heart of the triangle.

7 the positional relationship between circles

External d & gtR+r

Circumscribed =R+r

Intersecting r-r

Internal incision d=R-r

Include d < R-r

Regular polygons and circles

Center of regular polygon: the center of the circumscribed circle.

Radius of regular polygon: radius of circumscribed circle

The central angle of a regular polygon: the central angle of an edge.

Vertex of regular polygon: distance from center to edge

9 arc length and sector area

Arc length:

Sector area:

Side area and total area of 10 cone

Transverse area:

Total area:

1 1 intersection chord theorem and section line theorem

Probabilistic significance of 1: In a large number of repeated experiments, the frequency of event A is stable near a certain constant p, so this constant p is called an event.

The probability of a.

2 use enumeration method to find probability

Generally speaking, in an experiment, there are n possible results, and their probability of occurrence is equal. Event A contains m possible results, so the probability of occurrence of event A is p(A)= 1

3 Use frequency to estimate probability

1 quadratic function =

A>0, the opening is upward; A<0, opening downward;

Axis of symmetry:;

Vertex coordinates:;

The translation of an image can refer to the translation of vertices.

2 from the perspective of function to see a quadratic equation

3 Quadratic function and practical problems

1 Similarity of graphs

Similar polygons have equal proportions of corresponding sides and equal corresponding angles;

If the corresponding angles of two polygons are equal and the ratio of corresponding edges is equal, the two polygons are similar;

Similarity ratio: the ratio of corresponding edges of similar polygons.

Two similar triangles

Judge:

The straight line parallel to one side of the triangle intersects with the other two sides, forming a triangle similar to the original triangle;

If the ratio of two sets of corresponding sides of two triangles is equal and the corresponding included angles are equal, then two triangles are similar;

If two angles of one triangle are equal to two angles of another triangle, two triangles are similar.

The circumference and area of similar triangles.

The ratio of the perimeters of similar triangles (polygon) is equal to the similarity ratio;

The area ratio of similar triangles (polygon) is equal to the square of similarity ratio.

4-bit similarity

Potential diagram: two polygons are similar, the connecting lines of corresponding vertices intersect at one point, and the corresponding edges are parallel to each other. These two graphs are called potential graphs, and the intersection points are called potential centers.

1 acute trigonometric function: sine, cosine and tangent;

2 Solving Right Triangle

1 projection: parallel projection, central projection and orthographic projection.

Three views: top view, front view and left view.

3 drawing of three views

1 The main contents of this unit.

The concept of quadratic equation in one variable; Solution method of quadratic equation in one variable; Application problems of quadratic equation in one variable.

The position and function of this unit in teaching materials.

On the basis of studying univariate quadratic equation, bivariate quadratic equation and fractional equation, the univariate quadratic equation is studied. It is also a method of mathematical modeling. Learning quadratic equation of one variable well is essential to learn quadratic function well, and it is the basic project to learn high school mathematics well. It should be said that the quadratic equation of one variable is the key content of this book.

Understand the unary quadratic equation and related concepts; Master the order reduction of collocation method, formula method and factorization method-solve the quadratic equation of one variable; Master the method of establishing mathematical model of quadratic equation with one variable according to practical problems; Apply the above knowledge to solve problems.

Through abundant examples, let students discuss in cooperation, teachers comment and analyze, and establish a mathematical model. According to the mathematical model, the concept of quadratic equation with one variable is given properly. Combined with the related concepts in the eighth volume "Algebraic Formula", the concept of derivative of quadratic equation with one variable such as quadratic term is introduced. By mastering the direct opening method, the solution of the quadratic equation with one variable has no elementary term, the collocation method is introduced to solve the quadratic equation with one variable, and the collocation method is consolidated to solve the quadratic equation with one variable through a lot of practice. 0，b2-4ac=0，B2-4ac & lt； 0. By reviewing the fifth section of "Algebraic Expressions" in the first volume of Grade 8, the knowledge is transferred, and the quadratic equation of one variable is solved by factorization, which is consolidated by practice. Put forward problems, analyze problems, establish a mathematical model of quadratic equation with one variable, and use this model to solve practical problems.

3 Emotions, attitudes and values

Through the process of abstracting concepts such as quadratic equation of one variable from factual problems, students realize that quadratic equation of one variable is also an effective mathematical model to describe the quantitative relationship in the real world. famous book

By using collocation method, formula method and factorization method to solve a linear equation, students can experience mathematical ideas such as transformation; Experience and set up rich problem scenarios, so that students can build mathematical models to solve them.

The process of practical problems, so as to better understand the meaning and function of the equation and stimulate students' interest in learning.

1 unary quadratic equation and other related concepts.

Reduce the order by collocation method, formula method and factorization method-solve the quadratic equation of one variable.

3. Establish a mathematical model of quadratic equation with practical problems to solve this problem.

1 solution of quadratic equation in one variable.

Discussion on solving quadratic equation with one variable by formula method.

3. Establish the mathematical model of practical problems of quadratic equation with one variable; The difference between the solution of equation and the solution of practical problem.

1 Analyze practical problems and how to establish a mathematical model of a quadratic equation with one variable.

The steps of solving a quadratic equation with one variable by collocation method.

Derivation of formula method for solving quadratic equation of one variable.

The teaching time of this unit is about 16 class hours, and the specific distribution is as follows:

22 1 quadratic equation with one variable 2 class hours

222 Degeneration & Solving quadratic equation in one variable for 7 class hours

223 practical problems and quadratic equation of one variable for 5 class hours

Discover the relationship between the roots and coefficients of a quadratic equation in one variable for 2 class hours.

1 quadratic radical

The formula is called quadratic radical, which must satisfy the following requirements: it contains quadratic radical sign ""; The root sign a must be non-negative.

2 The simplest quadratic root

If the square root satisfies: the factor of the square root is an integer and the factor is an algebraic expression; The square root number contains no factor or can be completely opened. Such a quadratic root is called the simplest quadratic root.

Methods and steps of transforming quadratic form into simplest quadratic form;

If the square root of the quotient is a fraction (including a decimal) or a fraction, it is written in the form of a fraction by using the properties of the arithmetic square root of the quotient and then simplified by the denominator. If the number of roots is an integer or algebraic expression, first decompose it into factors or factors, and then extract the factors or factors that can be fully explored.

Three similar quadratic roots

After several quadratic roots are transformed into the simplest quadratic roots, if the number of roots is the same, these quadratic roots are called similar quadratic roots.

4 the properties of quadratic roots

Fifth-order square root mixed operation

The mixed operation of quadratic root is in the same order as that in real number. Multiply first, then divide, and finally add and subtract. If there are brackets, count them first (or remove them first).

1 unary quadratic equation

An integral equation with an unknown number and the highest degree of the unknown number is 2 is called a quadratic equation.

2 the general form of quadratic equation in one variable

It is characterized in that there are eleven quadratic polynomials on the left side of the equation about the unknown x, and the right side of the equation is zero, which is called quadratic term, and a is called quadratic term coefficient; Bx is called a linear term, and b is called a linear term coefficient; C is called a constant term.

Solution of quadratic equation in one variable

1 direct Kaiping method

Using the definition of square root to find the solution of quadratic equation in one variable is called direct Kaiping method. The direct Kaiping method is suitable for solving quadratic equations with one variable. According to the definition of square root, it is the square root of b, when B < 0, the equation has no real root.

2 matching method

Matching method is an important mathematical method, which is not only suitable for solving quadratic equations with one variable, but also widely used in other fields of mathematics. The theoretical basis of the matching method is the complete square formula. If a in the formula is regarded as an unknown x and replaced by x, there is.

3 formula method

Formula method is a method to solve the quadratic equation of one variable by finding the root formula, and it is a general method to solve the quadratic equation of one variable.

The formula for finding the root of quadratic equation with one variable;

4 factorization method

Factorization is to find the solution of the equation by factorization. This method is simple and easy to use, and it is the most commonly used method to solve the quadratic equation of one variable.

Discriminating formula of roots of quadratic equation with one variable

discriminant

In the unary quadratic equation, the discriminant called the root of the unary quadratic equation is usually represented by "",that is,

The relationship between roots and coefficients of quadratic equation with one variable.

If the two real roots of the equation are, then. That is to say, for the real root of any unary quadratic equation, the sum of the two roots is equal to the reciprocal of the quotient obtained by dividing the coefficient of the first term of the equation by the coefficient of the second term; The product of two roots is equal to the quotient obtained by dividing the constant term by the coefficient of quadratic term.

1 definition

The graphic transformation that rotates a graph by an angle around a certain point o is called rotation, where o is called rotation center and the rotation angle is called rotation angle.

2 attributes

The distance between the corresponding point and the center of rotation is equal. The included angle between the corresponding point and the straight line connecting the rotation center is equal to the rotation angle.

1 definition

Rotate the graph around a point 180. If the rotated figure can coincide with the original figure, then this figure is called a central symmetric figure, and this point is its symmetric center.

2 attributes

On the congruence of two graphs with central symmetry. For two graphs with central symmetry, the straight lines connecting the symmetrical points pass through and are equally divided by the symmetrical center. For two figures with symmetrical centers, the corresponding line segments are parallel (or on the same straight line) and equal.

3 judgment

If a straight line connecting the corresponding points of two graphs passes through a point and is equally divided by the point, then the two graphs are symmetrical about the point.

Four-center symmetrical figure

Rotate the figure around a point 180. If the rotated graph can coincide with the original graph, then this graph is called a central symmetric graph, and this shop is its symmetric center.

Test five, the characteristics of symmetrical points in the coordinate system (3 points)

On the characteristics of 1 symmetry point of origin

When two points are symmetrical about the origin, the signs of their coordinates are opposite, that is, the symmetrical point of point P(x, y) about the origin is P'(-x, -y).

2 on the characteristics of the axis symmetry point of X.

When two points are axisymmetrical about X, in their coordinates, X is equal and the sign of Y is opposite, that is, the point where point P(x, y) is axisymmetrical about X is P'(x, -y).

3 on the characteristics of Y-axis symmetry points

When two points are symmetrical about Y, Y is equal, and the sign of X is opposite in its coordinates, that is, the point where P(x, y) is symmetrical about Y is P'(-x, y).

Definition of 1 circle

In each plane, the line segment OA rotates around its fixed end point O, the figure formed by the rotation of the other end point A is called a circle, the fixed end point O is called a center, and the line segment OA is called a radius.

Geometric representation of two circles

The circle centered on point O is marked as "⊙O" and pronounced as "circle O"

Definition of chord, arc, etc. Related to the circle

(1) chord

A line segment connecting any two points on a circle is called a chord. (AB in the figure)

(2) Diameter

The chord passing through the center of the circle is called the diameter. (such as the upcoming CD)

The diameter is equal to twice the radius.

(3) semicircle

The two endpoints of a circle with any diameter are divided into two arcs, and each arc is called a semicircle.

(4) Arc, Upper Arc and Lower Arc

The part between any two points on a circle is called an arc.

The arc is represented by the symbol "⌒", and the arc with A and B as endpoints is marked as "",which is pronounced as "arc AB" or "arc AB".

An arc larger than a semicircle is called an optimal arc (usually represented by three letters); An arc smaller than a semicircle is called a bad arc (usually represented by two letters).

Vertical Diameter Theorem and Its Inference

Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the arc opposite to the chord.

Inference 1: bisect the diameter of the chord (not the diameter) perpendicular to the chord and bisect the two arcs opposite the chord. The perpendicular bisector of the chord bisects the two arcs opposite the chord through the center of the circle. The diameter of the arc bisecting the chord bisects the chord vertically and bisects another arc opposite the chord.

Inference 2: The arcs sandwiched by two parallel chords of a circle are equal.

The vertical diameter theorem and its inference can be summarized as follows:

Over the center of the circle

Perpendicular to the chord

The diameter bisects the chord, and the upper arc and the lower arc bisects.

Axisymmetry of 1 circle

A circle is an axisymmetric figure, and every straight line passing through the center of the circle is its axis of symmetry.

2 The center of a circle is symmetrical

A circle is a central symmetrical figure with the center of the circle as the symmetrical center.

Theorem of the relationship between arc, chord, chord center distance and central angle

1 central angle

The angle of the vertex at the center of the circle is called the central angle.

2 chord center distance

The distance from the center of the circle to the chord is called the chord center distance.

Theorem of the relationship between arc, chord, chord center distance and central angle.

In the same circle or in the same circle, the arcs with equal central angles are equal, the chords are equal, and the chord distance is equal.

Inference: In the same circle or equal circle, if one set of quantities in two circles, two arcs, two chords' central angles or two chords' central distances are equal, the corresponding other set of quantities are equal respectively.

Theorem of Circle Angle and Its Inference

1 circle angle

The angle whose vertex is on the circle and whose two sides intersect the circle is called the circumferential angle.

2 circle angle theorem

An arc subtends a circumferential angle equal to half the central angle it subtends.

Inference 1: the circumferential angles of the same arc or equal arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

Inference 2: the circumferential angle of a semicircle (or diameter) is a right angle; A chord with a circumferential angle of 90 is a diameter.

Inference 3: If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.

Position relationship between point and circle

Let the radius of ⊙O be r and the distance from point P to the center of O be d, then there are:

D<r point p is within ⊙O;

D = ⊙ o on point p;

D>r point P is outside ⊙ O.

1 circle passing through three points

Three points that are not on the same straight line determine a circle.

Circumscribed circle of triangle

A circle passing through the three vertices of a triangle is called the circumscribed circle of the triangle.

The outer center of a triangle

The center of the circumscribed circle of a triangle is the intersection of the perpendicular lines of the three sides of the triangle, which is called the center of the triangle.

Properties of quadrilateral inscribed by four-point circle (judging conditions of four-point * * * circle)

Diagonal complementarity of quadrilateral inscribed in a circle.

First, assume that the conclusion in the proposition is not valid, and then through reasoning, lead to contradictions and judge that the hypothesis is incorrect, so as to get the original proposition to be valid. This method of proof is called reduction to absurdity.

The positional relationship between straight line and circle

There are three positional relationships between a straight line and a circle, as shown below:

Intersection: When a straight line and a circle have two common points, it is called the intersection of the straight line and the circle. At this time, the straight line is called the secant of the circle, and the common point is called the intersection point; Tangency: When a line and a circle have a unique common point, they are said to be tangent. At this time, the straight line is called the tangent of the circle, and it is separated. When a line and a circle have nothing in common, it is said that the line and the circle are separated.

If the radius ⊙O is r and the distance from the center o to the straight line l is d, then:

Intersection of straight lines l and o

The straight line l is tangent to ⊙O, and d = r；;

The straight line l is separated from ⊙O d >; r；

The Theorem of Tangent of 1

The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.

2 property theorem of tangent line

The tangent of a circle is perpendicular to the radius passing through the tangent point.

1 tangent length

On the tangent of a circle passing through a point outside the circle, the length of the line segment between the point and the tangent point is called the tangent length from the point to the circle.

2 tangent length theorem

Two tangents drawn from a point outside the circle are equal in length, and the connecting line between the center of the circle and the point bisects the included angle of the two tangents.

inscribed circle of a triangle

1 inscribed circle of triangle

A circle tangent to all sides of a triangle is called the inscribed circle of the triangle.

The center of a triangle

The center of the inscribed circle of a triangle is the intersection of three bisectors of the triangle, which is called the heart of the triangle.

1 Relationship between Circle and its Position

If there is nothing in common between two circles, then they are said to be separated, and separation can be divided into external and internal.

If two circles have only one common point, they are said to be tangent, and tangency can be divided into circumscribed and inscribed.

If two circles have two common points, they are said to intersect.

2 center distance

The distance between two centers is called the distance between two centers.

3 the nature and judgment of the position relationship between circles

Let the radii of two circles be r and r, respectively, and the distance between the centers be d, then

D & gtR+r

Circumscribed circle d=R+r

Two circles intersect r-r

The inscribed circle d = r-r (r >); r)

Two circles contain d

Four important properties of tangency and intersection of two circles

If two circles are tangent, then the tangent point must be on the connecting line, they are axisymmetric figures, and the symmetry axis is the connecting line of two circles; The intersection of two circles bisects the common chord of the two circles vertically.

Definition of 1 regular polygon

Polygons with equal sides and angles are called regular polygons.

2 The relationship between regular polygon and circle

As long as a circle is divided into equal arcs, the inscribed regular polygon of this circle can be made, and this circle is the circumscribed circle of this regular polygon.

1 center of regular polygon

The center of the circumscribed circle of a regular polygon is called the center of this regular polygon.

2 radius of regular polygon

The radius of the circumscribed circle of a regular polygon is called the radius of this regular polygon.

Vertex of regular polygon

The distance from the center of a regular polygon to one side of the regular polygon is called the vertex of the regular polygon.

4 central angle

The central angle of the circumscribed circle opposite to each side of a regular polygon is called the central angle of this regular polygon.

Axisymmetry of 1 regular polygon

Regular polygons are all axisymmetric figures. A regular n-polygon has n symmetry axes, and each symmetry axis passes through the center of the regular n-polygon.

2 the central symmetry of regular polygons

A regular polygon with an even number of sides is a central symmetric figure, and its symmetric center is the center of the regular polygon.

3 drawing method of regular polygon

First divide the circle into equal parts with a protractor or ruler, and then make a regular polygon.

1 arc length formula

The formula for calculating the arc length l corresponding to the central angle n is

2 sector area formula

Where n is the degree of central angle of the sector, r is the radius of the sector, and l is the arc length of the sector.

Three-cone lateral area

Where l is the generatrix length of the cone and r is the grounding radius of the cone.

Supplement: (This knowledge is beyond the requirements of the syllabus, but it is of great help to develop students' intelligence and improve students' mathematical thinking mode)

1 intersection chord theorem

⊙O, the chord AB intersects the chord CD, and at point E, AEBE=CEDE.

2 chord tangent angle theorem

Chord tangent angle: the angle formed by the tangent of a circle and the chord passing through the tangent point is called chord tangent angle.

Chord tangent angle theorem: Chord tangent angle is equal to the circumferential angle between chord and tangent arc.

Namely: ∠BAC=∠ADC

3 secant theorem PL:PA⊙O tangent, PBC⊙O secant, then