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 point (x, y) is in the first and third quadrants, so the image of the function y=k/x(k≠0) is located 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 value range of k is k < 0.
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 and quadratic functions
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 up; 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 maximum value of the 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 the symbols of A and B are the same, the symmetry axis is on the right side of the Y axis, and the symbol of B is different, 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, increasing upwards and decreasing downwards.
Third, the nature of the circle (it's all in the knowledge book of this lesson, so I won't type O (∩_∩)o)
Fourth, 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;
According to our teacher's method, it is A-shaped and 8-shaped.
(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 their corresponding two sides is equal, the corresponding included angles are also equal.
(4) If the ratios of the three groups of corresponding sides are equal, then two triangles are similar.
(5) Two triangles with equal corresponding angles and proportional corresponding sides are called similarity.
6. There are similar figures, but our teacher said that if we don't take the exam, we won't teach.
In order to type so many words, choose me (although I copied a little from the encyclopedia, I typed almost all of them myself! )
If you are satisfied, please accept it.