2. The essence of quadratic root: it is a non-negative arithmetic square root. Therefore √a≥0.
3. Two formulas: (√ a) 2 = a (a ≥ 0); √a2=∣a∣.
4. Quadratic radical multiplication and division: √a×√b=√ab(a≥0, b ≥ 0); √a÷√b=√a/b(a≥0，b & gt0).
5. The simplest quadratic root formula: (1) The number of square roots does not contain denominator; ⑵ The number of square roots does not contain factors that can be opened.
6. Addition and subtraction of quadratic roots: first convert quadratic roots into the simplest quadratic roots, and then merge quadratic roots with the same number of roots.
7. Use the formula: (a+b) (a-b) = A2-B2; (a b)2=a2 2ab+b2。
Chapter 22 One-variable quadratic equation
1, definition: an equation in the form of ax2+bx+c=0(a≠0) is called a quadratic equation.
① is an integral equation, ② is quadratic with the highest number of unknowns, ③ contains only one unknown, and ④ the coefficient of quadratic term is not zero.
2. The general form of unary quadratic equation: descending order, the coefficient of quadratic term is usually positive, and the right end is zero.
3. The root of a quadratic equation: substitution makes the equation hold.
4, the solution of a quadratic equation:
① Matching method: shift term → convert quadratic term coefficient into linear term coefficient → add half linear term coefficient on both sides at the same time → formula → square root → write the solution of the equation.
② formula: x = (-b √ B2-4ac)/2a.
③ Factorization method: the right end is zero, and the left end is decomposed into the product of two factors.
5. Discriminating formula ① for roots of quadratic equation with one variable when △ >; 0, the equation has two unequal real roots.
② When △=0, the equation has two equal real roots, ③ When △
Note: the prerequisite for application is: a≠0.
6. The relationship between the root and the coefficient of a quadratic equation: x 1+x2=-b/a, x1* x2 = c/a. 。
Note: the prerequisite for application is: a≠0, delta ≥ 0.
7. Solving application problems with column equations: examining questions and setting elements → column algebra and column equations → arranging them into a general form → solving equations → testing and answering questions.
Chapter 23 Rotation
1. Three elements of rotation: rotation center, rotation direction and rotation angle.
2. The essence of rotation: ① The distance from the corresponding point to the center of rotation is equal; ② The included angle between the corresponding point and the connecting line of the rotation center is equal to the rotation angle; ③ The numbers before and after rotation are the same.
Key: Find the corresponding line segment and the corresponding angle.
3. Center symmetry: rotate the figure around a certain point 180. If it can coincide with another graph, then the two graphs are symmetric or central about this point.
4. The essence of central symmetry: ① Two figures with central symmetry, the connecting line segments of the corresponding points all pass through the symmetrical center and are equally divided by the symmetrical center. (2) congruence of two graphs with central symmetry.
5. Centrally symmetric figure: rotate the figure around a certain point 180. If the rotated figure can coincide with the original figure, then this figure is called a central symmetric figure.
6. The coordinate law of the symmetrical point: ① Axis symmetry about X: the abscissa is unchanged, but the ordinate is opposite; ② Axis symmetry about Y: the abscissa is opposite and the ordinate is unchanged; ③ Symmetry of origin: the abscissa and ordinate are opposite.
Chapter 24 Circle
1. Conditions for determining a circle: center → position, radius → size.
2. Concepts related to circle: chord diameter, arc semicircle, upper arc, lower arc, central angle, circumferential angle and chord center distance.
3. Symmetry of the circle: The circle is both an axisymmetric figure and a centrally symmetric figure.
4. Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.
Inference: The diameter (not the diameter) of bisecting the chord is perpendicular to the chord and bisects the two arcs opposite the chord.
5. Relationship among central angle, arc, chord and chord center distance: In the same circle or within the same circle, the central angle of the circle has equal arc, chord and chord center distance.
Extension: among these four groups, as long as one group is equal, the other groups are equal.
6. Theorem of circumferential angle: ① The circumferential angle is equal to half of the central angle of the same arc,
(2) 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; Equal circumferential angles face equal arcs,
(3) The circumference angle subtended by the semicircle (or diameter) is a right angle, and the chord subtended by the circumference angle of 90 is the diameter.
7. Inner heart and outer heart: ① The inner heart is the intersection point of the bisector of the inner angle of the triangle, and its distance to the three sides of the triangle is equal.
(2) The epicenter is the intersection of the perpendicular lines of the three sides of the triangle, and its distance to the three vertices of the triangle is equal.
8. positional relationship between straight line and circle: intersection →d
9. Determination of tangent: "A small point is connected with the center of the circle" → Prove verticality. "Nothing is vertical" → Prove D = R.
The nature of the tangent: the tangent of the circle is perpendicular to the radius passing through the tangent point.
10, tangent length theorem: two tangents of a circle are drawn from a point outside the circle, and their tangent lengths are equal. The connecting line between this point and the center of the circle bisects the included angle of the two tangents.
1 1, the properties of inscribed quadrangles: the diagonals of inscribed quadrangles are complementary, and each outer angle is equal to its inner diagonal.
12, the nature of the circumscribed quadrilateral: the sum of the opposite sides of the circumscribed quadrilateral is equal.
13, the positional relationship between circles: outward → d > R+R. Circumtangent → D = R+R. Intersection → R-R-R.
14, regular polygon and circle: radius → circumscribed circle radius, central angle → central angle of each side, apothem → distance from center to one side.
15, arc length and sector area: L=n∏R/ 180. S sector =n∏R2/360.
16, lateral area and total area of cone: cone. Bus length = sector radius, cone bottom circumference = sector arc length, cone lateral area = sector area, and cone total area = sector area+bottom circle area.
Chapter 25 Preliminary Probability
1, three kinds of events: random events, impossible events and inevitable events.
2. Probability: P(A)=p.0≤P(A)≤ 1.
3. Solution of classical probability: ① enumeration method (indicating all possible results), ② list method, ③ tree diagram.
4. Estimate probability by frequency: According to the constant that the frequency of a random event is gradually stable, the probability of this event can be estimated.
Chapter 26 Quadratic Function
1. definition: a function in the form of y=ax2+bx+c(a≠0, a, b and c are constants) is called a quadratic function.
2. Classification of quadratic function: ①y=ax2: vertex coordinate: origin; Symmetry axis: y axis;
②y=ax2+c: vertex coordinates: (0, c); Symmetry axis: y axis;
③y=a(x-h)2: vertex coordinates: (h, 0); Symmetry axis: straight line x = h；;
④y=a(x-h)2+k: vertex coordinates: (h, k); Symmetry axis: straight line x = h；;
⑤y=ax2+bx+c: vertex coordinates: (-b/ 2a, 4ac-b2/4a); ; Symmetry axis: straight line x=-b/ 2a
3. Determination of symbols A, B and C: A: the opening direction is upward → A > 0; Opening direction downward → a
B: The left and right are different from A, the symmetry axis is on the left side of Y axis, and A and B have the same sign; The symmetry axis is on the right side of the Y axis, and the signs of A and B are different.
C: positive semi-axis intersecting with Y axis, C >；; 0; Negative semi-axis c intersecting with y axis
B2 -4ac: the number of times intersecting with the X axis, △ > 0→ two intersections, △
3. Translation methods: "positive left and negative right" and "positive upper and negative lower".
Premise: the formula is in the form of y = a (x-h) 2+k.
4. Determine the function relationship with undetermined coefficient method: ① Select Y = AX2 as the vertex at the origin;
(2) Vertex Y = AX2+C on the Y axis;
③ Select Y = AX2+BX； by coordinate origin;
④ Knowing that the vertex is selected on the X axis as y = a (x-h) 2;
⑤ Choose y = a (x-h) 2+k when the coordinates of vertices are known;
⑥ If you know the coordinates of three points, choose Y = AX2+BX+C.
5. Other applications: finding the intersection point with the X axis → solving a quadratic equation with one variable; The intersection with the y axis is (0, c).
6, symmetry law:
① Two parabolas are symmetrical about X axis: A, B and C all become their opposites.
② Two parabolas are symmetrical about Y axis: A and C remain unchanged, and B becomes its opposite number.
7. Practical problem: profit = sales volume-total purchase price-other expenses, profit = (selling price-purchase price) * sales volume-other expenses.
Junior high school ninth grade mathematics book 2 knowledge points 1, acute angle trigonometric function
1. sine: in rt△abc, the ratio of the opposite side A to the hypotenuse of acute angle ∠a is called the sine of ∠a, which means that the opposite side/hypotenuse of sina = ∠ A = A/C;
2. Cosine: In rt△abc, the ratio of the adjacent side B of acute angle ∠a to the hypotenuse is called the cosine of ∠a, and it is recorded as cosa, that is, the adjacent side/hypotenuse of cosa=∠a = B/C;
3. Tangent line: In rt△abc, the ratio of the opposite side to the adjacent side of acute angle ∠a is called the tangent line of ∠a, and it is written as tana, that is, the opposite side of tana =∠A/∠A's adjacent side = A/B.
(1) ①tana is a complete symbol, indicating the tangent of ∠a, and the symbol "∞" is used to omit the angle in the symbol;
(2) (2) Tana has no unit, which represents a ratio, that is, the ratio of the opposite side to the adjacent side of ∠a in a right triangle;
③tana doesn't mean "tan" multiplied by "a";
④ The greater the ④④tana value, the steeper the step and the greater the ∠a; ∠a is larger, the step is steeper, and the value of tana is larger.
4. Cotangent: Definition: In rt△abc, the ratio of the adjacent side to the opposite side of acute angle ∠a is called cotangent of ∠a, that is, cota = the adjacent side of ∠ A/the opposite side of ∠ A = B/A;
5. Sine, cosine, tangent and cotangent of acute angle are equal to cosine, sine, cotangent and tangent of other angles respectively. We usually call it sine and cosine complementary function. Similarly, tangent and cotangent are complementary functions, which can be summarized as follows: the trigonometric function of an acute angle is equal to the complementary function of the other angles) expressed by equations:
If ∠a is an acute angle, ① sina = cos (90 ∠ A) and so on.
6. Remember that the trigonometric functions of special angles are 0, 30, 45, 60, 90.
7. When the angle changes from 0 to 90, the sine value and tangent value increase (or decrease) with the increase (or decrease) of the angle; Cosine value and cotangent value decrease (or increase) with the increase (or decrease) of angle. 0≤sinα≤ 1，0≤cosα≤ 1 .
The relationship between trigonometric functions with the same angle;
tanα cotα= 1，
tanα=sinα/cosα，
cotα=cosα/sinα，sin2α+cos2α= 1
Second, solve the right triangle.
1. Solving right triangle: the process of finding unknown elements from known elements in right triangle.
2. The relationship used in solving right-angled triangles: (In △abc, ∠c is a right angle, while the opposite sides of ∠a, ∠b and ∠c are A, B and C respectively).
(1) Tripartite relationship: A2+B2 = C2；; (Pythagorean theorem)
(2) The relationship between two acute angles: ∠ A+∠ B = 90;
(3) The relationship between edge and angle:
Sina = a/c;
cosa = b/c；
tana=a/b .
Sina =cosb
cosa=sinb
Sina =cos(90 -a)
sin2α+cos2α= 1
Junior high school ninth grade mathematics book 3 knowledge points 1, projection
1. projection: generally, it is the shadow obtained on a certain plane (ground, wall, etc.) when an object is irradiated by light. ) is called the projection of an object, the irradiated light is called the projection line, and the plane where the projection is located is called the projection plane.
2. Parallel projection: the projection formed by parallel rays is parallel projection. (The light source is particularly far away)
3. Central projection: The projection formed by the same point (light emitted by a point light source) is called central projection.
4. Orthographic projection: The projection generated by the projection line perpendicular to the projection plane is called orthographic projection. The shape and size of an object's orthographic projection are related to its position relative to the projection plane.
5. When a plane of an object is parallel to the projection plane, the orthographic projection of this plane is exactly the same as the shape and size of this plane. When the top of a certain surface of an object is inclined to the projection surface, the orthogonal projection of this surface becomes smaller. When the plane of an object is perpendicular to the projection plane, the orthogonal projection of this plane becomes a straight line.
Liangsanguan
1. Three views: a graph drawn by an observer observing the same spatial geometry from three different positions (front, horizontal and side). Three views are the front view, top view and left view. In addition, as an aid, there are cross-sectional drawings and semi-cross-sectional drawings, which can basically express the structure of the object completely.
2. Front view: the view from front to back of the object obtained in front.
3. Top view: A view of an object from top to bottom on a horizontal plane.
4. Left view: a view of the object viewed from left to right on the side.
5. The positional relationship of the three views:
① The front view is at the top, the top view is at the bottom, and the left view is on the right;
② The front view and top view indicate the length of the object, the front view and left view indicate the height of the object, and the left view and top view indicate the width of the object.
(3) The front view and the top view are aligned, the front view and the left view are flush, and the width of the left view and the top view is equal.
6. Drawing method: the outline of the visible part is drawn as a solid line, and the outline of the invisible part is drawn as a dotted line because it is blocked by other parts.