☆ Summary ☆
I. Basic concepts
1. equation, its solution (root), its solution, its solution (group)
2. Classification:
Second, the basis of solving equation-the nature of equation
1.a=b←→a+c=b+c
2.a=b←→ac=bc (c≠0)
Third, the solution
1. Solution of linear equation with one variable: remove denominator → remove brackets → move terms → merge similar terms →
The coefficient becomes 1→ solution.
2. Solution of linear equations: ① Basic idea: "elimination method" ② Method: ① Replacement method.
② addition and subtraction
Fourth, a quadratic equation
1. Definition and general form:
2. Solution: (1) direct leveling method (pay attention to characteristics)
(2) Matching method (pay attention to the step-inferring the root formula)
(3) Formula method:
(4) factorization method (feature: left =0)
3. The discriminant of the root:
4. The relationship between root and coefficient top:
Inverse theorem: If, then the quadratic equation with one root is:.
5. Common equation:
5. Equations that can be transformed into quadratic equations
1. Fractional equation
(1) definition
(2) Basic ideas:
⑶ Basic solution: ① Denominator removal ② Substitution method (such as).
(4) Root test and method
2. Unreasonable equation
(1) definition
(2) Basic ideas:
(3) Basic solution: ① Multiplication method (pay attention to skills! ! (2) substitution method (example), (4) root test and method.
3. Simple binary quadratic equation
A binary quadratic equation consisting of a binary linear equation and a binary quadratic equation can be solved by method of substitution.
Six, column equation (group) to solve application problems
summary
Solving practical problems by using equations (groups) is an important aspect of integrating mathematics with practice in middle schools. The specific steps are as follows:
(1) review the questions. Understand the meaning of the question. Find out what is a known quantity, what is an unknown quantity, and what is the equivalent relationship between problems and problems.
⑵ Set an element (unknown). ① Direct unknowns ② Indirect unknowns (often both). Generally speaking, the more unknowns, the easier it is to list the equations, but the more difficult it is to solve them.
⑶ Use algebraic expressions containing unknowns to express related quantities.
(4) Find the equation (some are given by the topic, some are related to this topic) and make the equation. Generally speaking, the number of unknowns is the same as the number of equations.
5] Solving equations and testing.
[6] answer.
To sum up, the essence of solving application problems by column equations (groups) is to first transform practical problems into mathematical problems (setting elements and column equations), and then the solutions of practical problems (column equations and writing answers) are caused by the solutions of mathematical problems. In this process, the column equation plays a role of connecting the past with the future. Therefore, the column equation is the key to solve the application problem.
Two commonly used equality relations
1. Travel problem (uniform motion)
Basic relationship: s=vt
(1) Meeting problem (at the same time):
+ = ;
(2) Follow-up questions (start at the same time):
If A starts in t hours, B starts, and then catches up with A at B, then
(3) sailing in the water:
2. batching problem: solute = solution × concentration
Solution = solute+solvent
3. Growth rate:
4. Engineering problems: Basic relationship: workload = working efficiency × working time (workload is often considered as "1").
5. Geometric problems: Pythagorean theorem, area and volume formulas of geometric bodies, similar shapes and related proportional properties.
Third, pay attention to the relationship between language and analytical formula.
Such as more, less, increase, increase to (to), at the same time, expand to (to), ...
Another example is a three-digit number, where A has 100 digits, B has 10 digits and C has one digit. Then this three-digit number is: 100a+ 10b+c, not abc.
Fourth, pay attention to writing equal relations from the language narrative.
For example, if X is greater than Y by 3, then x-y=3 or x=y+3 or X-3 = Y, and if the difference between X and Y is 3, then x-y=3. Pay attention to unit conversion
Such as the conversion of "hours" and "minutes"; Consistency of s, v and t units, etc.
Seven, application examples (omitted)
Chapter VI One-dimensional Linear Inequalities (Groups)
★ Emphasis ★ Properties and Solution of One-dimensional Linear Inequality
☆ Summary ☆
1. Definition: A > B, A < B, a≥b, a≤b, A ≠ B.
2. Unary linear inequality: ax > b, ax < b, ax≥b, ax≤b, ax≠b(a≠0).
3. One-dimensional linear inequalities;
4. the essence of inequality: (1) a > b ←→ a+c > b+c
⑵a & gt； b←→AC & gt； BC (c>0)
⑶a & gt； b←→AC & lt； BC (c<0)
(4) (transitivity) a>b, b & gtc→a & gt；; c
⑸a & gt； b，c & gtd→a+c & gt； b+d。
5. Solution of one-dimensional linear inequality, solution of one-dimensional linear inequality
6. Solution of one-dimensional linear inequality group, solution of one-dimensional linear inequality group (representing the solution set on the number axis)
7. Application examples (omitted)
Chapter VII Similarity
★ Focus★ similar triangles's judgment and nature
☆ Summary ☆
First of all, there are two sets of theorems in this chapter.
The first group (proportion of related properties):
Concepts involved: ① The fourth proportional item ② The front and back items of the third proportional item, the inner item and the outer item ④ The golden section, etc.
The second episode:
Note: ① the meaning of the word "correspondence" in the theorem;
② Parallel → Similar (proportional line segment) → Parallel.
Second, the nature of similar triangles
1. The corresponding line segment ...; 2. Corresponding circumference ...; 3. Corresponding areas ...
Third, correlation mapping.
(1) as the fourth proportion; (2) As a proportional term.
Four, card (solution) problem method, auxiliary line
1. Change "equal product" to "proportion" and find "similarity" in "proportion".
2. If you can't find similarity, find the middle proportion. Methods: Express the ratio of the left and right sides of the equation. ⑴
⑵
⑶
3. Adding auxiliary parallel lines is an important way to obtain proportional line segments and similar triangles.
4. The common method to deal with the ratio problem is to look at K; For the equal ratio problem, the common solution is to set the "common ratio" to K.
5. For complex geometric figures, the method of "extracting" some needed figures (or basic figures) is adopted.
Application examples of verbs (abbreviation of verb) (omitted)
Chapter 8 Functions and Their Images
★ Emphasis★ Positive and negative proportional functions, images and properties of linear and quadratic functions.
☆ Summary ☆
First, the plane rectangular coordinate system
1. Coordinate characteristics of points in each quadrant
2. Coordinate characteristics of each point on the coordinate axis
3. About the characteristics of coordinate axis and symmetry point.
4. The corresponding relationship between points on the coordinate plane and ordered real number pairs.
Second, function
1. Representation method: (1) analysis method; (2) List method; (3) Image method.
2. The principle of determining the range of independent variables: (1) makes algebraic expressions meaningful; (2) The practical problems in manufacturing are as follows
Meaning.
3. Draw a function image: (1) list; (2) tracking points; (3) connection.
Third, several special functions.
(Definition → Image → Attribute)
1. proportional function
⑴ definition: y=kx(k≠0) or y/x = k.
⑵ image: straight line (through the origin)
⑶ nature: ① k > 0，…②k & lt； 0,…
2. Linear function
⑴ definition: y=kx+b(k≠0)
⑵ Image: The straight line passes through the intersection of point (0, b)- and Y axis and the intersection of point (-b/k, 0, b)- and X axis.
⑶ nature: ① k > 0，…②k & lt； 0,…
(4) Four situations of images:
3. Quadratic function
(1) Definition:
In particular, they are all quadratic functions.
⑵ Image: parabola (tracing points: first determine the vertex, symmetry axis and opening direction, and then trace points symmetrically). If the configuration method is changed to, the vertex is (h, k); The symmetry axis is a straight line x = h；; A>0, the opening is upward; A<0, opening down.
⑶ Nature: a>0, on the left and right side of the symmetry axis; A<0, on the left … and right … of the symmetry axis.
4. Inverse proportional function
⑴ Definition: or xy=k(k≠0).
⑵ Image: hyperbola (two branches)-drawn by tracing points.
⑶ nature: ① k > 0, the image is at …, y follows x …; ②k & lt； 0, the image is at …, y follows x …; ③ Two curves are infinitely close to the coordinate axis but can never reach the coordinate axis.
Fourth, important problem-solving methods
1. Use the undetermined coefficient method to find the analytical formula (solving the sequence equation [group]). To find the analytic formula of quadratic function, we should reasonably choose the general formula or vertex type, make full use of the characteristics of parabola about the axis of symmetry, and find the coordinates of new points. As shown in the figure below:
2. K and B represent the linear (proportional) function, inverse proportional function and quadratic function in the image; The symbols of a, b and C.