The concept of quadratic function:

The concept of 1. Quadratic function: A function with a general shape of (constant,) is called a quadratic function. What needs to be emphasized here is that, similar to the unary quadratic equation, the quadratic term coefficient can be zero, and the definition domain of the quadratic function is all real numbers.

2. The structural characteristics of quadratic function:

(1) The left side of the equal sign is the function, and the right side is the quadratic form of the independent variable, with the highest degree of 2.

⑵ is a constant, quadratic coefficient, linear coefficient and constant term.

Basic forms of quadratic and quadratic functions

Properties of the basic form of 1. quadratic function;

The greater the absolute value of a, the smaller the opening of parabola.

the symbol of

Opening direction

Vertex coordinates

axis of symmetry

nature

up

axis

When, it increases with the increase of; When, it decreases with the increase of; There is a minimum value.

downwards

axis

When, it decreases with the increase of; When, it increases with the increase of; When there is a maximum value.

2. The nature of:

Add and subtract.

the symbol of

Opening direction

Vertex coordinates

axis of symmetry

nature

up

axis

When, it increases with the increase of; When, it decreases with the increase of; There is a minimum value.

downwards

axis

When, it decreases with the increase of; When, it increases with the increase of; When there is a maximum value.

3. The nature of:

Left plus right minus.

the symbol of

Opening direction

Vertex coordinates

axis of symmetry

nature

up

X=h

When, it increases with the increase of; When, it decreases with the increase of; There is a minimum value.

downwards

X=h

When, it decreases with the increase of; When, it increases with the increase of; When there is a maximum value.

4. The nature of:

the symbol of

Opening direction

Vertex coordinates

axis of symmetry

nature

up

X=h

When, it increases with the increase of; When, it decreases with the increase of; There is a minimum value.

downwards

X=h

When, it decreases with the increase of; When, it increases with the increase of; When there is a maximum value.

Translation of Cubic and Quadratic Function Images

1. Translation steps:

Method 1: (1) Convert the parabola analytical formula into a vertex and determine its vertex coordinates;

⑵ Keep the parabola shape unchanged and translate its vertex to it. The specific translation method is as follows:

2. Translation method

On the basis of the original function, "the value moves positively to the right and negatively to the left; The value moves up and down. " .

To sum up, it is eight words "left plus right minus, up plus down minus".

Method 2:

(1) Translate along the axis: translate the unit up (down).

(or)

⑵ Translation along the axis: translate the unit to the left (right) to become (or).

Fourth, quadratic function and

From the analytical formula, sum is two different expressions, and the latter can get the former, which is one of them.

Drawing method of quintic function image

Five-point drawing method: the quadratic function is transformed into a vertex by matching method, and its opening direction, symmetry axis and vertex coordinates are determined, and then symmetrical drawing is made on both sides of the symmetry axis. Generally, the five points we choose are: the vertex, the intersection with the axis, and the intersection with the axis of symmetry (if there is no intersection with the axis, take two groups of points about the axis of symmetry).

When sketching, we should pay attention to the following points: opening direction, symmetry axis, vertex, intersection with axis and intersection with axis.

Sixth, the properties of quadratic function

1. When the parabolic opening is upward, the symmetry axis is, and the vertex coordinates are.

When, it decreases with the increase of; When, it increases with the increase of; There is a minimum value.

2. When the parabolic opening is downward, the symmetry axis is, and the vertex coordinates are. When, it increases with the increase of; When, it decreases with the increase of; When there is a maximum value.

7. Representation of quadratic resolution function.

1. General formula: (,,is a constant,);

2.Vertex: (,,is a constant,);

3. Two formulas: (,,is the abscissa of the intersection of parabola and axis).

Note: The analytic formula of any quadratic function can be transformed into a general formula or a vertex, but not all quadratic functions can be written as intersections. Only when the parabola intersects the axis can the analytical expression of the parabola be expressed as the intersection point. These three forms of quadratic resolution functions are interchangeable.

Eight, the relationship between the image of quadratic function and coefficient.

1. Quadratic coefficient

In quadratic function, as the coefficient of quadratic term, obviously.

(1) When the parabolic opening is upward, the larger the numerical value, the smaller the opening, and the smaller the numerical value, the larger the opening;

(2) When the parabola opens downwards, the smaller the numerical value, the smaller the opening, and the larger the numerical value, the larger the opening.

To sum up, it determines the size and direction of the parabolic opening, the positive and negative of the parabola determines the direction of the opening, and the size of the parabola determines the size of the opening.

2. The first coefficient

On the premise of determining the coefficient of quadratic term, the symmetry axis of parabola is determined.

(1) On the premise of,

When, that is, the symmetry axis of parabola is on the left side of the axis;

When, that is, the axis of symmetry of parabola is axis;

When, that is, the parabola symmetry axis is on the right side of the axis.

2, the conclusion is just the opposite of the above, namely

When, that is, the symmetry axis of parabola is on the right side of the axis;

When, that is, the axis of symmetry of parabola is axis;

When, that is, the parabola symmetry axis is on the left side of the axis.

To sum up, on the premise of certainty, determine the position of the parabolic axis of symmetry.

The sign of judging "left": the axis of symmetry is on the left side of the axis and on the right side of the axis, which means "left and right are different" in a word.

Summary:

3. Constant term

(1) When the intersection of parabola and axis is above the axis, that is, the ordinate of the intersection of parabola and axis is positive;

2 When the intersection of parabola and axis is the coordinate origin, that is, the ordinate of the intersection of parabola and axis is;

(3) When the intersection of the parabola and the axis is below the axis, that is, the ordinate of the intersection of the parabola and the axis is negative.

To sum up, the intersection position of parabola and axis is determined.

In short, this parabola is unique as long as everything is certain.

Determination of quadratic resolution function;

The undetermined coefficient method is usually used to determine the secondary resolution function according to the known conditions. To find the analytic formula of quadratic function by undetermined coefficient method, we must choose the appropriate form according to the characteristics of the topic, so as to make the problem solving simple. Generally speaking, there are the following situations:

1. Given the coordinates of three points on a parabola, the general formula is generally used;

2. The vertex or symmetry axis or the maximum (minimum) value of a parabola is known, and the vertex type is generally selected;

3. Given the abscissa of the intersection of parabola and axis, two formulas are generally used;

4. It is known that two points with the same ordinate on a parabola are commonly used as vertices.

Symmetry of quadratic function image

There are generally five kinds of symmetry of quadratic function images, which can be expressed by general formula or vertex.

1. About Axisymmetry

After axial symmetry, the analytical formula is:

After axial symmetry, the analytical formula is:

2. About axis symmetry

After axial symmetry, the analytical formula is:

After axial symmetry, the analytical formula is:

3. About the symmetry of the origin

After the origin is symmetrical, the analytical formula obtained is:

After the origin is symmetrical, the analytical formula obtained is:

4. Symmetry about the vertex (that is, parabola rotates around the vertex 180).

After the vertices are symmetrical, the analytical formula is:

After the vertices are symmetrical, the analytical formula is.

5. About point symmetry

After point symmetry, the analytical formula obtained is

According to the nature of symmetry, it is obvious that the shape of parabola will never change no matter what kind of symmetry transformation is made, so it will never change. When finding the expression of parabola symmetry parabola, we can choose the appropriate form according to the meaning of the question or the principle of easy operation. It is customary to determine the vertex coordinates and opening direction of the original parabola (or parabola with known expression), then determine the vertex coordinates and opening direction of its symmetrical parabola, and then write the expression of its symmetrical parabola.

Ten, quadratic function and a quadratic equation:

1. The relationship between quadratic function and unary quadratic equation (intersection of quadratic function and axis);

Quadratic equation with one variable is a special case of quadratic function with function value of time.

Number of intersections between image and axis:

(1) When the image intersects the axis at two points, there are two unary quadratic equations. The distance between these two points.

② When there is only one intersection point between the image and the axis;

③ When, the image and the axis do not intersect.

When the image falls above the axis, no matter it is any real number, there is;

When the image falls under the axis, no matter it is any real number.

2. The parabola image must intersect with the axis, and the coordinates of the intersection point are,;

3. Summary of common problem-solving methods of quadratic function:

(1) To find the coordinates of the intersection point between the image and the axis of a quadratic function, it needs to be transformed into a quadratic equation;

⑵ To find the maximum (minimum) value of quadratic function, it is necessary to transform quadratic function from general formula to vertex by collocation method;

(3) Judging the sign of, in the quadratic function according to the position of the image, or judging the position of the image according to the sign of, in the quadratic function, the combination of numbers and shapes is needed;

(4) The image of quadratic function is symmetrical about the symmetry axis. By using this property, the coordinates of a point with known symmetry can be summed up, or the coordinates of an intersection point of an axis can be summed up, and the coordinates of another intersection point can be obtained from symmetry.

5] There is also a quadratic trinomial related to a quadratic function, which itself is the quadratic function of the letters contained; Let's take time as an example to reveal the internal relationship among quadratic function, quadratic trinomial and unary quadratic equation:

Parabola and axis have two intersections.

The value of quadratic trinomial can be positive, zero or negative.

A quadratic equation with one variable has two unequal real roots.

Parabola and axis have only one intersection.

The value of quadratic trinomial is nonnegative.

A quadratic equation with one variable has two equal real roots.

The parabola has no intersection with the axis.

The value of quadratic trinomial is always positive.

A quadratic equation with one variable has no real root.

Quadratic function image reference:

XI。 Application of function

Application of quadratic function

Key points and common problems of quadratic function examination

1. Investigate the definition and properties of quadratic function. Related questions often appear in multiple-choice questions, such as:

It is known that the image of the quadratic function of the independent variable passes through the origin, and the value of is

2. Comprehensively investigate the images of direct proportion, inverse proportion, linear function and quadratic function. The characteristic of the exercise is to examine the images of two functions in the same rectangular coordinate system. The test questions are multiple choice questions, such as:

As shown in the figure, if the image of the function is in the first, second and third quadrants, then the image of the function is roughly ().

yoyoyo

1 1

0 x o- 1 x 0 x 0 - 1 x

A B C D

3. Using the undetermined coefficient method to examine the analytical formula of quadratic function, the frequency of related exercises is very high, and the types of exercises include intermediate solution questions and selective synthesis questions, such as:

It is known that a parabola passes through two points (0,3) and (4,6), and the symmetry axis is, so find the analytical expression of this parabola.

4. How to find the vertex coordinates, symmetry axis and extreme value of parabolic quadratic function by matching method is investigated. Related problems are solutions, such as:

It is known that the abscissa of parabola (a≠0) intersecting with X axis is-1, 3, and the ordinate intersecting with Y axis is-

(1) Determine the analytical formula of parabola; (2) Determine the opening direction, symmetry axis and vertex coordinates of parabola by matching method.

5. Investigate the comprehensive ability of algebra and geometry, which is often used as a special finale.

Classic example

The sign of the coefficient is determined by the position of the parabola.

Example 1 (1) The image of the quadratic function is as shown in figure 1, and the key point is ().

A. first quadrant B. second quadrant C. third quadrant D. fourth quadrant

(2) The image of a given quadratic function y=ax2+bx+c(a≠0) is shown in Figure 2, and the following conclusions are obtained: ①a and B have the same sign; ② When x= 1 and x=3, the function values are equal; ③4a+b = 0； ④ When y=-2, the value of x can only be 0. The correct number is ().

1。

( 1) (2)

Finding out the relationship between the position of parabola and coefficients A, B and C is the key to solve the problem.

Example 2. It is known that the image of quadratic function y=ax2+bx+c intersects with the x axis at points (-2, o) and (x 1, 0), and 1

A 1 B. 2 C. 3 D.4

Answer: d

Will use the undetermined coefficient method to find the quadratic resolution function.

Example 3. It is known that one root of the unary quadratic equation ax2+bx+c=3 about X is x=-2, and the symmetry axis of the quadratic function y=ax2+bx+c is a straight line x=2, then the vertex coordinate of the parabola is ().

A(2，-3) B.(2， 1) C(2，3) D.(3，2)

Answer: c

Example 4. As shown in the figure (unit: m), the isosceles triangle ABC moves to the square along the straight line L at a speed of 2m/s until AB and CD coincide. When x seconds is assumed, the overlapping area of triangle and square is ym2. ..

(1) Write the relationship between y and x;

(2) When x = 2,3.5, what is y?

(3) When the area of the overlapping part is half of the square area,

How long did the triangle move? Find the coordinates of the parabola vertex,

Symmetry axis

Example 5, the parabola y = x2+x- is known.

(1) Find its vertex coordinates and symmetry axis by matching method.

(2) If the two intersections of the parabola and the X axis are A and B, find the length of the line segment AB.

The comment on this question (1) is an examination of the "basic method" of quadratic function, and the second question mainly examines the relationship between quadratic function and quadratic equation with one variable.

Example 6, "The image of a known function passes through point A (c, -2),

It is proved that the symmetry axis of this quadratic function image is x=3. The rectangular box in the title is illegible text polluted by ink.

(1) According to the known and existing information in the conclusion, can you find the second discriminant function in the problem? If yes, please write out the solution process and draw the image of quadratic function; If not, please explain why.

(2) Please fill in a suitable condition in the rectangular box of the original question according to the existing information to complete the original question.

Comments: For the sub-topic (1), if we want to find the quadratic resolution function in the topic according to the known and existing information in the conclusion, we should take the original conclusion that the symmetry axis of the function image is x=3 as known, and combine with the condition that the image crosses point A(c, -2), we can list two equations, but there are only two unknowns in the analytical formula. For the item (2), as long as the given conditions can make the quadratic resolution function be the analytical formula in the item (1). From different angles, adding different conditions, we can give the coordinates of any point on the image, or the coordinates of the vertex or the intersection with the coordinate axis.

[Solution] (1) According to the image passing through point A (c, -2), the symmetry axis of the image is x=3, so.

solve

So the sub-resolution function is the image as shown in the figure.

(2) Let y=0 in the analytical formula, and get, get.

So you can fill in "the coordinate of the intersection of parabola and X axis is (3+)" or "the coordinate of the intersection of parabola and X axis is"

Substituting x=3 into the analytical formula, we get

So the vertex coordinates of parabola are

So you can also fill in the vertex coordinates of parabola as and so on.

Functions mainly focus on: understanding the specific features of functions in different ways (images, analytical expressions, etc.). ); Understand the function with the help of various realistic backgrounds; Function is regarded as a mathematical model of "the relationship between variables in the process of change"; The concept of osmotic function; Pay attention to the relationship between function and related knowledge.

Solving the maximum problem with quadratic function

Example 1 A square with a known side length of 4 becomes a pentagonal ABCDE (as shown in the figure), where AF=2 and BF = 1. Try to find a point p on AB to maximize the area of rectangular PNDM.

Comment on this problem is a comprehensive problem of algebra and geometry. Combining the knowledge of similar triangles and quadratic function organically, students' comprehensive application ability can be well examined. At the same time, it also leaves a thinking space for students to explore the ideas of solving problems.

Example 2 The cost of a product is 10 yuan. The relationship between the sales price of each product X (yuan) and the daily sales volume Y (pieces) of this product in the trial sale stage is as follows:

X (yuan)

15

20

30

…

Y (piece)

25

20

10

…

If the daily sales volume Y is a linear function of the sales price X 。

(1) Find the functional relationship between daily sales y (pieces) and sales price x (yuan);

(2) How much should the sales price of each product be set to maximize the daily sales profit? What is the daily sales profit at this time?

Analysis (1) Let this linear function expression be y = kx+B, then the solution is k=- 1, and b=40, that is, the linear function expression is y =-x+40.

(2) Assume that the sales price of each product should be set at X yuan, and the sales profit obtained is W yuan.

w =(x- 10)(40-x)=-x2+50x-400 =-(x-25)2+225。

The sales price of products should be set as 25 yuan, and the maximum daily sales profit at this time is 225 yuan.

Comment on the most valuable problem. The thinking of solving the application problem is similar to that of the general application problem, but there are also differences. There are two main points: (1) Set the unknown number in the question "What is the maximum (or minimum and minimum) when so-and-so is the value", "so-and-so" should be set as the independent variable and "what" should be set as the function; (2) The solution of the problem depends on collocation method or maximum formula, not solving the equation.

Quadratic function corresponds to exercises.

First, multiple choice questions

The vertex coordinate of 1. quadratic function is ()

A.(2，- 1 1) B.(-2，7) C.(2， 1 1) D. (2，-3)

2. Translate the parabola upward by 1 unit, and the parabola obtained is ().

A.B. C. D。

3. The functions and images in the same rectangular coordinate system may be () in the figure.

4. Given the quadratic function image as shown in the figure, we can draw the following conclusions: ① A and B have the same symbol; ② When summing, the function values are equal; ③ ④ At that time, the value of can only be 0. The correct number is ().

1。

5. Given the quadratic function (-1, -3.2) and the vertex coordinates of some images (as shown in the figure), we can know from the images that the two roots of a quadratic equation are () respectively.

A.- 1.3 B- 2.3 c-0.3d-3.3

6. Known quadratic function image as shown in the figure, the key point is ().

A. The first quadrant B. The second quadrant

C. The third quadrant D. The fourth quadrant

7. The number of positive roots of this equation is ()

A.0 B. 1 c.2.3

8. It is known that the parabola passes through points A (2,0), B(- 1 0) and intersects the axis at point C, OC=2. Then the analytical formula of this parabola is

A.B.

C. or D. or

Second, fill in the blanks

9. The symmetry axis of quadratic function is, then _ _ _ _ _.

10. Known parabola y=-2(x+3)? +5, if y decreases with the increase of x, then the value range of x is _ _ _ _ _.

1 1. A function has the following properties: ① image crossing point (-1, 2), ② when < 0, the function value increases with the increase of independent variables; The analytical formula of the function satisfying the above two properties is (write only one).

12. The vertex of the parabola is C, and it is known that the straight line passes through point C, so the triangle area surrounded by this straight line and two coordinate axes is.

13. The image of quadratic function is to move the image to the left by 1 unit, and then move it down by 2 units, then b=, c=.

14. As shown in the figure, the arch of a bridge is a parabola, the maximum height of the bridge is16m, the span is 40m, and the height of the bridge is (π is 3. 14) at 5m from the center on the AB line.

Third, answer questions:

Map number 15

15. It is known that the symmetry axis of the quadratic function image is the image (1, -6), and the intersection point with the axis is (0,).

(1) Find the analytic expression of this quadratic function;

(2) When x is what value, the function value of this function is 0?

(3) When x changes in what range, the function value of this function increases with the increase of x?

16. After the firecrackers are ignited, their rising height h (m) and time t (s) conform to the relationship (0

(1) How long does 15m leave the ground after the firecrackers are lit?

(2) From 1.5 seconds to 1.8 seconds after the firecrackers are ignited, judge whether the firecrackers are rising or falling, and explain the reasons.

17. As shown in the figure, the parabola passes through two intersections A and B of the straight line and the coordinate axis, the other intersection of the parabola and the axis is C, and the vertex of the parabola is D. 。

(1) Find the analytical expression of this parabola;

(2) Point P is a moving point on a parabola. Find the coordinates of point P that makes: 5: 4.

18. Hongxing Building Materials Store sells a kind of building materials for a factory (consignment here means that the manufacturer provides the goods free of charge first, and then settles the accounts after the goods are sold, and the unsold goods are handled by the manufacturer). When the price per ton is 260 yuan, the monthly sales volume is 45 tons. In order to improve the operating profit, the building materials store is going to take the way of price reduction and promotion. According to market research, when the price per ton drops 10 yuan, the monthly sales will increase by 7. Five tons. Considering various factors, the manufacturer and other expenses shall be paid for every ton of building materials sold 100 yuan. If the price per ton of materials is X (yuan), then the monthly profit of the dealer is Y (yuan).

(1) When the price per ton is 240 yuan, calculate the monthly sales at this time;

(2) Find the functional relationship between Y and X (it is not required to write the value range of X);

(3) How much does a building materials store have to sell for a ton to get the maximum monthly profit?

(4) Xiao Jing said, "When the monthly profit is the largest, the monthly sales are also the largest." Do you think this is right? Please explain the reason.

Practice the answers to questions

First, multiple-choice questions,

1.A 2。 C 3。 A 4。 B 5。 D 6。 B 7。 C 8。 C

Second, fill in the blanks,

9. 10.

Third, answer questions.

15.( 1) Let the analytical formula of parabola be, which can be obtained from the meaning of the question.

solve problems

(2) or -5 (2)

16.( 1) The known solution, which was inappropriate at the right time, was discarded. Therefore, when the firecrackers are lit, the ground 1 sec is15m. (2) From the meaning of the question, the abscissa of the vertex is downward, so after the firecrackers are lit, the firecrackers rise from 1.5 seconds to 108 seconds.

17.( 1) Find the intersection of a straight line and coordinate axes A (3 3,0) and B (0 0,3).

Therefore, the analytical expression of this parabola is. (2) The vertex D( 1, -4) of the parabola and the other intersection point C (- 1, 0) of the axis. Make p simple.

When > 0, ∴ p (4 4,5) or p (-2,5) is obtained.

When < 0, the equation has no solution. To sum up, the coordinates of the points that meet the conditions are (4,5) or (-2,5).

18.( 1) = 60 (tons). (2) Simplify: (3).

In order to obtain the maximum monthly profit, the material price should be set at 2 10 yuan per ton.

(4) I think what Xiao Jing said is wrong. Reason: Method 1: When the profit of the current month is the largest, X is 2 10 yuan. For the sales of the current month,

When x is 160 yuan, the monthly sales volume w is the largest. When x is 2 10 yuan, the monthly sales volume w is not the largest. Xiao Jing is wrong.

Method 2: When the monthly profit is maximum, X is 2 10 yuan, and the monthly sales at this time is 17325 yuan; And when X is 200 yuan, the monthly sales amount is 18000 yuan. ∫ 17325 < 18000, ∴ when the monthly profit is the largest, the monthly sales amount w is not the largest. Xiao Jing is wrong.