Function properties:
The change value of 1.y is directly proportional to the corresponding change value of x, and the ratio is k, that is, Y = KX+B (where k and b are constants, k≠0), ∫ when x increases m, K (x+m)+B = Y+km, km/m = K.
2. When x=0, b is the point of the function on the Y axis, and the coordinate is (0, b).
When b=0 (y=kx), the image of a linear function becomes a proportional function, which is a special linear function.
4. In two linear function expressions:
When k and b in the expression of quadratic function are the same, the images of quadratic function overlap; When k is the same and b is different in the expression of quadratic function, the images of quadratic function are parallel; When k and b in two linear function expressions are different, the images of two linear functions intersect; When k is different and b is the same in quadratic function expressions, the images of quadratic functions intersect at the same point (0, b) on the Y axis. If the relationship between two variables x and y can be expressed as Y=KX+b(k, b is constant, and k is not equal to 0), then y is said to be a linear function image property of x.
1. Practice and graphics: Through the following three steps: (1) list.
(2) tracking points; [Generally, two points are taken. According to the principle of "two points determine a straight line", it can also be called "two-point method". The general image of y=kx+b(k≠0) can be drawn as a straight line passing through (0, b) and (-b/k, 0).
The image of the proportional function y=kx(k≠0) is a straight line passing through the coordinate origin, and generally takes two points (0,0) and (1, k). (3) Connecting lines can make images of linear functions-straight lines. So the image of a function only needs to know two points and connect them into a straight line. Usually, the intersections of the function image with the X axis and the Y axis are -k points B and 0, 0 and B, respectively.
2. Nature:
Any point P(x, y) on the (1) linear function satisfies the equation: y=kx+b(k≠0).
(2) The coordinates of the linear function intersecting with the Y axis are always (0, b), and the images of the proportional function intersecting with the X axis at (-b/k, 0) are all at the origin.
3. Function is not a number, it refers to the relationship between two variables in a certain change process.
4. Quadrant where K, B and function images are located:
When y=kx (that is, b is equal to 0 and y is proportional to x):
When k>0, the straight line must pass through the first and third quadrants, and Y increases with the increase of X; When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When y=kx+b:
When k>0, b>0, when the image of this function passes through the first, second and third quadrants; When k>0, b<0, when the image of this function passes through the first, third and fourth image limits; When k0, the image of this function passes through the first, second and fourth quadrants; When k < 0, b0, the straight line must pass through the first and second quadrants; When b<0, the straight line must pass through the third and fourth quadrants.
In particular, when b=0, the straight line passing through the origin o (0 0,0) represents the image of the proportional function. At this time, when k>0, the straight line only passes through the first and third quadrants, not through the second and fourth quadrants. When k < 0, the straight line only passes through the second and fourth quadrants and does not pass through the first and third quadrants.
4. Special positional relationship:
When two straight lines in the plane rectangular coordinate system are parallel, the k value of the resolution function (that is, the coefficient of the first term) is equal. When two straight lines in the plane rectangular coordinate system are perpendicular, the resolution function K is negative reciprocal (that is, the product of two K values is-1).
③ Point inclination y-y 1=k(x-x 1)(k is the slope of the straight line, (x 1, y 1) is the point where the straight line passes).
④ Two-point formula (y-Y 1)/(y2-y1) = (x-X 1)/(X2-x1) (on a known straight line, (x1,y/kloc).
1. Find the k value of the function image: (y 1-y2)/(x 1-x2) 2. Find the midpoint of the line segment parallel to the X axis: | x1-x2 |/2 3. Find the midpoint of the line segment parallel to the Y axis: | Y 1.
4. Find the length of any line segment: √ (x 1-x2) 2+(y 1-y2) 2 (note: the sum of squares of (x1-x2) and (y1-y2) under the root sign) 5. Find two ones.
Two linear functions y 1 = k1x+y1= y2 = k2x+B2 make y 1x+b 1 = k2x+b2 replace the solution value of x=x0 back to y1=
6. Find the midpoint coordinates of a line segment connected by any two points: [(x 1+x2)/2, (y 1+y2)/2].
7. Find the first resolution function of any two points: (x-x1)/(x1-x2) = (y-y1)/(y1-y2) (where denominator is 0, then numerator is 0) x y+. Negative) in the third quadrant+,-(positive and negative) in the fourth quadrant 8. If two straight lines y1= k1x+b1∑ y2 = k2x+B2, then k 1=k2, b1.Then k1× k2 =-/kloc-. Y = k (x-n)+b is a translation of n units to the right.
4. Quadrant where K, B and function images are located:
When y=kx (that is, b is equal to 0 and y is proportional to x):
When k>0, the straight line must pass through the first and third quadrants, and Y increases with the increase of X; When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When y=kx+b:
When k>0, b>0, when the image of this function passes through the first, second and third quadrants; When k>0, b<0, when the image of this function passes through the first, third and fourth image limits; When k0, the image of this function passes through the first, second and fourth quadrants; When k < 0, b0, the straight line must pass through the first and second quadrants; When b<0, the straight line must pass through the third and fourth quadrants.
Particularly, when b=0, the image of the proportional function is represented by a straight line of the origin o (0 0,0). At this time, when k>0, the straight line only passes through the first and third quadrants, not through the second and fourth quadrants. When k < 0, the straight line only passes through the second and fourth quadrants and does not pass through the first and third quadrants.
4. Special positional relationship:
When two straight lines in the plane rectangular coordinate system are parallel, the k value of the resolution function (that is, the coefficient of the first term) is equal. When two straight lines in the plane rectangular coordinate system are perpendicular, the resolution function K is negative reciprocal (that is, the product of two K values is-1).
③ Point inclination y-y 1=k(x-x 1)(k is the slope of the straight line, (x 1, y 1) is the point where the straight line passes).
④ Two-point formula (y-Y 1)/(y2-y1) = (x-X 1)/(X2-x1) (on a known straight line, (x1,y/kloc).
1. Find the k value of the function image: (y 1-y2)/(x 1-x2) 2. Find the midpoint of the line segment parallel to the X axis: | x1-x2 |/2 3. Find the midpoint of the line segment parallel to the Y axis: | Y 1.
4. Find the length of any line segment: √ (x 1-x2) 2+(y 1-y2) 2 (note: the sum of squares of (x1-x2) and (y1-y2) under the root sign) 5. Find two ones.
Two linear functions y 1 = k1x+y1= y2 = k2x+B2 make y 1x+b 1 = k2x+b2 replace the solution value of x=x0 back to y1=
6. Find the midpoint coordinates of a line segment connected by any two points: [(x 1+x2)/2, (y 1+y2)/2].
7. Find the first resolution function of any two points: (x-x1)/(x1-x2) = (y-y1)/(y1-y2) (where denominator is 0, then numerator is 0) x y+. Negative) in the third quadrant+,-(positive and negative) in the fourth quadrant 8. If two straight lines y1= k1x+b1∑ y2 = k2x+B2, then k 1=k2, b1.Then k1× k2 =-/kloc-. Y = k (x-n)+b is a translation of n units to the right.