Inverse proportional function of induction of mathematics knowledge points in grade three
A function in the form of 1.y=k/x(k≠0) or y = kx- 1 is called an inverse proportional function, and k is called an inverse proportional coefficient. It looks like a hyperbola. -1 means negative once.
2. In the function y=k/x(k≠0), when k >; 0, the symbols of X and Y in the expression are the same, and the points (X, Y) are in the first and third quadrants, so the image of function y=k/x(k≠0) is in the first and third quadrants; When k < 0, the signs of x and y in the expression are opposite, and the points (x, y) are in the second and fourth quadrants, so the image of the function y=k/x(k≠0) is in the second and fourth quadrants.
3. in y=k/x(k≠0), when k >; 0, in the first quadrant, y decreases with the increase of x; If the value of Y increases with the value of X, the range of K is K.
4. Let P(a, b) be any point on the inverse proportional function y=k/x(k≠0), then the value of ab is equal to k. After passing any point p on the inverse proportional function, the rectangular area is k; When point P is perpendicular to X axis or Y axis and connected with OP, the area of triangle is k/2.
quadratic function
1. the shape is y = ax 2+bx+c (a ≠ 0, a, b and c are constants). The function of is called quadratic function, and its image is like a parabola.
2. The vertex coordinates of the quadratic function y = ax 2+bx+c (a ≠ 0) are (-b/2a, 4ac-b 2/4a), and the symmetry axis is a straight line x=-b/2a.
3. for the quadratic function y = ax 2+bx+c (a ≠ 0), when a >; 0, the quadratic function image opens upward; When a<0, the parabola opens downward. The coordinate of the intersection of the image and the Y axis is (0, c).
4. The solution of the unary linear equation AX 2+BX+C = 0 (A ≠ 0) can be regarded as the abscissa of the intersection of the image of the function Y = AX 2+BX+C (A ≠ 0) and the X axis.
When b 2-4ac > 0, the function image has two intersections with the x axis.
When b 2-4ac = 0, the function image intersects with the x axis.
When b 2-4ac
5. when a>0 and x=-b/2a, the function y = ax 2+bx+c (a ≠ 0) takes the minimum value, which is equal to 4ac-b 2/4a; When a<0 and x=-b/2a, the value of function y = ax 2+bx+c (a ≠ 0) is equal to 4ac-b 2/4a.
6. The symmetry axis of parabola y = ax 2+c (a ≠ 0) is the y axis.
7. For the quadratic function y = ax 2+bx+c (a ≠ 0), if A and B have the same sign, the symmetry axis is on the right side of the Y axis, and B has a different sign, and the symmetry axis is on the left side of the Y axis.
8. Parabolic Y = AX 2+BX+C (A ≠ 0), if a>0, when x≤-b/2a, Y decreases with the increase of X; When x≥-b/2a, y increases with the increase of x, if a<0, when x≤-b/2a, y increases with the increase of x; When x≥-b/2a, y decreases with the increase of x.
9. For parabola y = a (x-m) 2+k, when it is translated left and right, it is only related to m, adding to the left and subtracting to the right; When translating up and down, it is only related to k, with the upper being added and the lower being subtracted.
similar triangles
1. If the ratio of two numbers is equal to the ratio of the other two numbers, it is said that these four numbers are proportional.
2. If a/b=c/d, then ad = bc If ad=bc, bd≠0, then A/B = C/D; If a/b=c/d, then (a+b)/b = (c+d)/d. No one can be 0. 0 is meaningless.
3. Generally speaking, if the three numbers A, B and C satisfy the proportional formula a:b=b:c, then B is called the proportional mean of A and C ... (If it is a line segment, it can only be positive; If it is a number, it can be positive or negative).
4. golden section
Divide a line segment into two parts so that the ratio of one part to the total length is equal to the ratio of the other part to this part. The ratio is (√5- 1)/2, and the approximate value of the first three digits is 0.6 18.
5. Proof method of triangle similarity;
(1) A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides), and the triangle formed is similar to the original triangle;
(2) If two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar;
(3) Two triangles are similar if the ratio of the corresponding sides of the two groups is equal and the corresponding included angles are equal;
(4) If the ratios of three groups of corresponding edges are equal, then two triangles are similar;
(5) Two triangles with equal corresponding angles and proportional corresponding sides are called similarity.
monadic quadratic equation
1. General form of quadratic equation with one variable: When a≠0, ax2+bx+c=0 is called the general form of quadratic equation with one variable. When learning the related problems of quadratic equation in one variable, most of the exercises should be converted into general formulas first, in order to determine A, B and C in the general formulas. Where a, b and c can be specific numbers or algebraic expressions with undetermined letters or specific formulas.
2. Solution of quadratic equation with one variable: The four solutions of quadratic equation with one variable require flexible application, among which the direct leveling method is simple, but its application scope is small; Although the formula method has a wide range of applications, it is complicated in calculation and prone to calculation errors. Factorization is the first choice for its wide application and simple calculation. Matching method is less used.
3. Discriminant of the root of a quadratic equation with one variable: When ax2+bx+c=0 (a≠0), δ = B2-4ac is called the discriminant of the root of a quadratic equation with one variable. Please note the following equivalent propositions:
δ& gt； 0<=> has two unequal real roots; δ= 0 & lt； => has two equal real roots; δ& lt； 0<=> has no real root;
4. The average growth rate problem-one of the application problems (let the growth rate be x):
(1) The first year is A, the second year is a( 1+x), and the third year is a( 1+x)2.
(2) The equation is often expressed by the following equation: the third year = the third year or the first year+the second year+the third year = the sum.
Classroom learning method of mathematics in grade three. Prepare textbooks, notebooks and other stationery needed for class before class, and take time to simply recall and review what you learned last class. We should go to class with a strong thirst for knowledge, hoping to learn new knowledge from teachers and solve new problems in class. You should listen attentively in class. As soon as the bell rings, you should immediately enter an active learning state and consciously eliminate all kinds of distractions. When listening to the class, you should look up, keep your eyes on the teacher's every move, and listen attentively to every word of the teacher. We should firmly grasp the teacher's thinking, pay attention to the logic of the teacher's narration of the problem, how the problem is raised, and analyze the methods and steps to solve the problem. Classroom is the key link to understand and master basic knowledge, skills and methods. "Learning is not enough", students who have taught themselves before class can concentrate more, and they know where to be detailed and where to omit; Where to carve carefully, where to pass by and where to record, instead of copying all the records, pay attention to one thing and lose another.
Listening in class is very important, and 45 minutes should be effective: don't think I'm joking, who can't listen in class? Actually, it's not. By listening, I mean listening completely, seriously and carefully. For every formula and theorem that the teacher talks about in class, we should go deep into its source, so that even if we forget the formula in the exam, we can solve the problem well and avoid inner panic and tension. In addition, we should make full use of the short 45 minutes in class and try our best to absorb the knowledge learned in class, so as to further carry out the next step of study after returning home and save time.
Consolidate the foundation in an all-round way: grasp the problem-solving law of choosing fill-in-the-blank questions and ensure that the basic part gets full marks in previous exams, that is, get the full marks you deserve. In a round of review, all students should concentrate on getting through the basic barrier of choosing fill-in-the-blank questions, otherwise the college entrance examination will be very sad 100. In reality, many students have devoted themselves to aimless, varied and strange questions from the beginning. In order to achieve this goal in a round of review, students with poor foundation can take the initiative to give up large-scale complex exercises and put the saved time and energy back into choosing fill-in-the-blank questions to further consolidate their foundation; Students with a good foundation should not focus too much and mainly on solving problems in a large area, but should study one or two big problems in detail in a small amount every day, but by topic and stage, and gradually accumulate experience and rules in solving problems, and must not spread out a lot. We should know that the accumulation of problem-solving experience and law is a long process from quantitative change to qualitative change, and persistence is more important than influence.