x=-b/2a .

The intersection of symmetry axis and parabola is the vertex p of parabola.

Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).

2. The parabola has a vertex p, and the coordinates are

P(-b/2a，(4ac-b'2)/4a)

-b/2a=0, p is on the y axis; When δ = b' 2-4ac = 0, P is on the X axis.

3. Quadratic coefficient A determines the opening direction and size of parabola.

When a>0, the parabola opens upwards; When a<0, the parabola opens downward.

The larger the |a|, the smaller the opening of the parabola.

4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.

When A and B have the same number (ab>0), the symmetry axis is on the left side of Y axis;

When a and b have different numbers (i.e. AB

5. The constant term c determines the intersection of parabola and Y axis.

The parabola intersects the Y axis at (0, c)

6. Number of intersections between parabola and X axis

δ= b ' 2-4ac & gt； 0, parabola and x axis have two intersections.

When δ = b' 2-4ac = 0, the parabola has 1 intersection points with the X axis.

δ= b ' 2-4ac & lt； 0, the parabola has no intersection with the x axis. The value of x is an imaginary number (the reciprocal of the value of x =-b √ b' 2-4ac, multiplied by the imaginary number I, and the whole formula is divided by 2a).

Mathematics learning methods in senior one.

1. Learning mentality.

Most middle school students are expected to improve their math scores. On the one hand, with a certain foundation and hard work, there is nothing wrong with students' attitude, but they lack direction and appropriate methods. On the other hand, there is still enough time to prepare for the exam and adjust and optimize. Therefore, give yourself more positive psychological hints on weekdays and stick to practicing the learning methods that suit you.

2. the direction of preparing for the exam.

What is the preparation direction? The so-called preparation direction is the examination direction. When doing the problem at ordinary times, we should find out what kind of knowledge framework and question type the problem in front of us is, what is the method of this question type, and what is this question type? Wait a minute.

Questions and knowledge points are limited. As long as we look for ways to solve problems and carry out reasonable training according to the questions we often take, it is easy to improve our math scores.

3. Training methods.

Everyone's actual situation is different, and the training methods are different. The good results obtained in the exam are the result of reasonable training before the exam. Many students complain that there is not enough time, and they are exhausted after finishing their homework every day. In the face of a bunch of problems, especially math problems, we can pay attention to the following angles:

(1) Know your own needs. For example, if you get the homework assigned by the teacher, whether it is a test paper or a textbook exercise, if you do it with emotion, the effect will definitely be bad. First of all, we must understand our own needs, such as which of these topics are of good quality? What don't you get? Which ones have appeared frequently before? Are you sure what you want to do? Wait, which problem do you want to solve most?

(2) Set goals. If dealing with teachers to do problems will undoubtedly lead to poor quality, then you should set certain goals before doing them. As mentioned above, what questions do you use to train the correct rate? What topics are used to practice speed? What topics are used to improve the steps and so on. With the goal, we can achieve it better. In this process, you will certainly gain a lot.

two

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, k≠0)

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) The connection can be the image of a function-a straight line. 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. Property: any point P(x, y) on the (1) linear function satisfies the equation: y = kx+b (2) The coordinates of the intersection of the linear function and the y axis are always (0, b), and the image of the (-b/k, 0) proportional function always crosses the origin.

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

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 b>0, the straight line must pass through the first and second quadrants;

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

When b<0, 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 k>0, the straight line only passes through the first and third quadrants; When k < 0, the straight line only passes through the second and fourth 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 of linear function (also called analytic expression) 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+b … ②.

(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, and the original water quantity in the pool is S.g=S-ft.

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 (x1-x2) and (y1-y2) under the root sign).