I. Definitions and definitions:

Independent variable x and dependent variable y have the following relationship:

y=kx+b

It is said that y is a linear function of x at this time.

In particular, when b=0, y is a proportional function of x.

Namely: y=kx(k is a constant, k0)

Second, the properties of linear function:

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(k is any non-zero real number b, take any real number)

2. When x=0, b is the intercept of the function on the y axis.

Iii. Images and properties of linear functions:

1. Practice and graphics: Through the following three steps.

(1) list;

(2) tracking points;

(3) Connecting lines can make straight lines into images of linear functions. So the image of a function only needs to know two points and connect them into a straight line. (Usually find the intersection of the function image with the X and Y axes)

2. Nature:

Any point P(x, y) on the (1) linear function satisfies the equation: y = kx+b.

(2) The coordinate of the intersection of the linear function and the Y axis is always (0, b), and the image of the proportional function always intersects the origin of the X axis at (-b/k, 0).

3. Quadrant where K, B and function images are located:

K0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;

When k0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.

When b0, the straight line must pass through the first and second quadrants;

When b=0, the straight line passes through the origin.

When b0, the straight line must pass through three or four quadrants.

Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.

At this time, when k0, the straight line only passes through the first and third quadrants; K0, the straight line only passes through two or four quadrants.

Fourth, determine the expression of a linear function:

Known point A(x 1, y1); B(x2, y2), please determine the expressions of linear functions passing through points A and B. ..

(1) Let the expression (also called analytic expression) of a linear function be y = kx+b.

(2) Since any point P(x, y) on the linear function satisfies the equation y = kx+b, two equations can be listed: y 1 = kx 1+B 1 and y2 = kx2+B2.

(3) Solve this binary linear equation and get the values of K and B. ..

(4) Finally, the expression of the linear function is obtained.

Five, the application of linear function in life:

1. When the time t is constant, the distance s is a linear function of the velocity v .. s=vt.

2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t .. Set the original water quantity in the pool. G = S- feet.

6. Commonly used formula: (incomplete, I hope someone will add it)

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: |x 1-x2|/2.

3. Find the midpoint of the line segment parallel to the Y axis: |y 1-y2|/2.

4. Find the length of any line segment: (x 1-x2) 2+(y 1-y2) 2 (note: the sum of squares of (x 1-x2) and (y 1-y2) under the root sign).

Expand:

( 1)

The positional relationship between a straight line and a plane:

There are only three positional relationships between a straight line and a plane: within the plane, intersecting the plane and parallel to the plane.

A straight line on the (1) plane has countless common points.

② There is only one common point when a straight line intersects a plane.

Angle between a straight line and a plane: the acute angle formed by the diagonal of a plane and its projection on the plane.

Esp。 Space vector method (finding the normal vector of a plane)

Provisions: a, when the straight line is perpendicular to the plane, the angle formed is a right angle; B, when the lines are parallel or in a plane, the angle formed is 0.

The value range of the angle formed by the straight line and the plane is [0,90]

Minimum angle theorem: the angle formed by the diagonal line and the plane is the smallest angle between the diagonal line and any straight line in the plane.

Three Verticality Theorems and Inverse Theorems: If a straight line in a plane is perpendicular to the projection of a diagonal line in this plane, it is also perpendicular to this diagonal line.

Esp。 This line is perpendicular to the plane.

Definition of vertical line and plane: If straight line A is perpendicular to any straight line in the plane, we say that straight line A and plane are perpendicular to each other. The straight line A is called the perpendicular of the plane, and the plane is called the vertical plane of the straight line A. ..

Theorem for judging whether a straight line is perpendicular to a plane: If a straight line is perpendicular to two intersecting straight lines in a plane, then the straight line is perpendicular to the plane.

Theorem of the property that straight lines are perpendicular to a plane: If two straight lines are perpendicular to a plane, then the two straight lines are parallel.

(3) when the straight line is parallel to the plane, there is no common point.

Definition of parallelism between straight line and plane: If straight line and plane have nothing in common, then we say that straight line and plane are parallel.

Theorem for determining the parallelism between a straight line and a plane: If a straight line out of the plane is parallel to a straight line in this plane, then this straight line is parallel to this plane.

Theorem of parallelism between straight lines and planes: If a straight line is parallel to a plane and the plane passing through it intersects with this plane, then the straight line is parallel to the intersection line.

(2)

(1) inclination angle of straight line

Definition: The angle between the positive direction of the X axis and the upward direction of the straight line is called the' inclination angle' of the straight line. In particular, when a straight line is parallel or coincident with the X axis, we specify that its inclination angle is 0 degrees. Therefore, the range of inclination angle is 0 ≤α 180.

(2) the slope of the straight line

① Definition: A straight line whose inclination is not 90, and the tangent of its inclination is called the slope of this straight line. The slope of a straight line is usually represented by k, that is. Slope reflects the inclination of straight line and axis. At that time, at that time,; It didn't exist then.

② Slope formula of straight line passing through two points:

Pay attention to the following four points: (1) At that time, the right side of the formula was meaningless, the slope of the straight line did not exist, and the inclination angle was 90;

(2)k has nothing to do with the order of P 1 and P2;

(3) The slope can be obtained directly from the coordinates of two points on a straight line without inclination angle;

(4) To find the inclination angle of a straight line, we can find the slope from the coordinates of two points on the straight line.

(3) Linear equation

(1) point inclination:

The slope k of the straight line passes through this point.

Note: When the slope of the straight line is 0, k=0, and the equation of the straight line is y=y 1. When the slope of the straight line is 90, the slope of the straight line does not exist, and its equation can not be expressed by point inclination. But because the abscissa of each point on L is equal to x 1, its equation is x=x 1.

② Oblique section: the slope of the straight line is k, and the intercept of the straight line on the Y axis is b..

③ Two-point formula: () Two points on a straight line,

(4) Cutting torque type:

Where the straight line intersects the axis at the point and intersects the axis at the point, that is, the intercepts with the axis and the axis are respectively.

⑤ General formula: (A, B are not all 0)

⑤ General formula: (A, B are not all 0)

Note: ○ 1 scope of application.

○2 Special equations such as straight lines parallel to the X axis:

(b is constant); A straight line parallel to the y axis:

(a is a constant);

(4) Linear system equation: that is, a straight line with some * * * property.

(1) parallel linear system

A linear system parallel to a known straight line (a constant that is not all zero): (c is a constant)

(2) A linear system passing through a fixed point

(i) Linear system with slope k: a straight line passes through a fixed point;

(2) The equation of the straight line system where two straight lines intersect is (as a parameter), where the straight line is not in the straight line system.

(5) Two straight lines are parallel and vertical.

At that time, attention should be paid to the existence of slope when judging the parallelism and verticality of straight lines with slope.

(6) The intersection of two straight lines

stride

The coordinates of the intersection point are a set of solutions of the equation. These equations have no solution; The equation has many solutions and coincidences.

(7) Distance formula between two points: Let it be two points in the plane rectangular coordinate system, then

(8) Distance formula from point to straight line: distance from point to straight line.

(9) Distance formula of two parallel straight lines: take any point on any straight line, and then convert it into the distance from that point to the straight line to solve it.