Linear slope formula: k=(y2-y 1)/(x2-x 1)

The product of slopes of two vertically intersecting lines is-1:k1* k2 =-1.

The slope of curve y=f(x) at point (x 1, f(x 1)) is the derivative of function f(x) at point x 1.

When the slope of the straight line L exists, the formula of oblique section y=kx+b is k=0 and y = b.

When the slope of the straight line L exists, the point inclination angle Y2-Y 1 = K (X2-X 1),

When the straight line L has a non-zero intercept on two coordinate axes, there is an intercept formula X/a+y/b= 1.

For any point on any function, its slope is equal to the included angle between its tangent and the positive direction of X axis, that is, tanα.

(1) As the name implies, "slope" means "degree of inclination". In the past, when we were learning to solve the right triangle, the textbook said: the ratio I between the vertical height H and the horizontal width L of the slope is called the slope; If the included angle α between the slope and the horizontal plane is called slope, then; The greater the slope.

The slope k we are studying now is equal to the tangent of the inclination angle (only one) α of the corresponding straight line (there are countless parallel lines), which can reflect the inclination degree of this kind of straight line to the X axis. In fact, the concept of "slope" is consistent with the "slope" in engineering problems.

(2) In analytic geometry, it is necessary to study straight lines through the coordinates of points and the equations of straight lines, which are obtained through coordinate calculation, so that the equations are simpler in form. If only the concept of inclination angle is used, it is actually equivalent to the arctank function, which is difficult to get directly through coordinate calculation, making the equation form complicated.

(3) In the coordinate plane, every straight line has a unique inclination, but not every straight line has a slope, and the straight line with an inclination of 90 (that is, the perpendicular to the X axis) has no slope. In the future study, it is often necessary to discuss whether a straight line has a slope point.

The slope of a point on the curve reflects the changing speed of the variable of the curve at that point.

The changing trend of the slope curve can still be described by the slope of the tangent of a point on the curve, that is, the derivative. The geometric meaning of derivative is the tangent slope of function curve at this point.

F'(x)>0, the function increases monotonically in this interval, and the curve shows an upward trend; f '(x)& lt； 0, the function monotonically decreases in this interval, and the curve shows a downward trend.

In (a, b) f'' (x)

Extended data

We can see the slope, which is a very important concept for middle school students to learn. Why is it important? We can look at it from the following aspects:

First, from the perspective of curriculum standards, we can know that in the compulsory education stage, we have learned a function whose geometric meaning is expressed by a straight line, and the coefficient of a linear term is the slope of the straight line, but it cannot be expressed when the straight line is perpendicular to the X axis. Although the term slope is not clearly given, in fact, the idea has penetrated into it.

In senior high school, compulsory one and compulsory two discussed the problem of straight line, and elective one and elective two also mentioned some problems related to straight line. The contents listed above actually involve the concept of slope, so it can be said that the concept of slope is one of the important mathematical concepts that students gradually accumulate.

Secondly, from a mathematical point of view, we can understand how to describe the inclination of a straight line relative to the X axis in a rectangular coordinate system from the following four angles. First of all, from a practical point of view, slope is what we call slope, which is the average change rate of height. Slope is used to describe the slope of the road.

In other words, the ratio of the tangential height to the horizontal length of the slope is equivalent to moving one kilometer in the horizontal direction and rising or falling tangentially. This ratio actually indicates the size of the slope. There are actually many such examples, such as the slope of stairs and roofs.

Secondly, the tangent value from the inclination angle; There is also the angle between the vector in the upward direction of the straight line and the unit vector in the X axis direction.

Finally, the concept of slope is re-recognized from the perspective of derivative. Slope is actually the instantaneous rate of change of a straight line. Understanding the concept of slope not only plays an important role in future study, but also helps to learn some important mathematical problem solving methods in the future.

Third, from the perspective of textbooks.

(1) From the outline, when dealing with the knowledge of the slope of a straight line, the textbook first talks about the inclination of the straight line, then the slope of the straight line, and then introduces the derivation of the slope formula of two points on the straight line; Judging from the new curriculum standard, we can see that the first thing in the textbook A of People's Education Edition is the inclination angle of a straight line.

Then we will talk about the slope of a straight line, but in the form of questions. First of all, there can be countless straight lines passing through point P, so they all pass through point P, thus forming a straight line cluster. What is the difference between these straight lines? It is easy to see that their dip angles are different, so how to describe the dip angles of these straight lines?

When the straight line L intersects with the X axis, the angle α formed by the direction of the X axis and the upward direction of the straight line L is defined as the inclination angle of the straight line L, and then the range of the inclination angle is discussed, and then the quantity related to the inclination degree in daily life is put forward, so that students can illustrate it by themselves, such as AG ratio; For example, if you compare two liters and three liters with two liters, the former will be steeper.

If we use the concept of inclination angle, then we will see that the slope is actually the tangent of inclination angle α, which describes the inclination degree of a straight line. It is particularly emphasized here that all straight lines with an inclination angle other than 90 degrees have slopes.

Because the slope of a straight line is different with different inclination angles, the inclination angle can be used to express the inclination degree of a straight line, and then students can be guided to explore how to derive the slope formula of a straight line by using two points on the straight line. The slope expressions of different values from 0 to 90 degrees, 90 degrees, 90 degrees to 180 degrees are also involved here.

Let's look at the mathematics of the People's Education Edition again. Here, the concept of straight line slope is mentioned, but Group B of the general review questions only involves the formulation of slope. At this time, the introduction of slope formula is mentioned in the form of vector.

Fourthly, it is needed to solve and calculate the average speed, instantaneous speed and acceleration in physics learning.

Fifth, the slope can help us better understand, deduce and understand the formula.

References:

Baidu slope encyclopedia