Quadratic function

[èr cì hán shù]

Common mathematical functions

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This entry is made by China Science and Technology Information Magazine Participate in editing and review Science Popularization China · Science Encyclopedia authentication.

The basic representation of a quadratic function is y=ax ²+bx+c (a ≠ 0). The highest degree of a quadratic function must be quadratic, and the image of a quadratic function is a Axis of symmetry Parallel to or coincident with the y-axis parabola 。

Quadratic function The expression is Y=ax ²+bx+c (and a ≠ 0), its definition is quadratic polynomial (or monomial).

If the value of y is equal to zero, a Quadratic equation 。 The solution of this equation is called root Or functional zero point 。

Chinese name: Unary quadratic function
Foreign name: Quadratic function
Abbreviation: Quadratic function
Function image: parabola
Function expression: Y=ax ²+bx+c (a ≠ 0 a, b, c are constants)

Axis of symmetry: Straight line x=h
Intersection formula: y=a(x-x1)(x-x2)　（a≠0）
Vertex form: y=a(x-h)²+k(a≠0）
Discipline: mathematics
Vertex coordinates: （h，k）
Vertex coordinate formula: （-b/2a,(4ac-b²)/4a）

catalog

five Function image
▪ Basic image
▪ axial symmetry
▪ vertex
▪ Opening
▪ Location determining factors
▪ Factors determining the intersection point
▪ Number of intersections with x-axis

Basic definitions

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Generally, form

(a, b, c are constant ）Of function It is called quadratic function, where a is called Coefficient of quadratic term , b is Coefficient of primary term , c is Constant term 。 X is independent variable , y is dependent variable 。 The highest degree of the argument to the right of the equal sign is 2.

Vertex coordinates

Intersection formula by y=a（x-x one ）（x-x two ）（ Only parabola with intersection with x-axis ），

The coordinate of the intersection point with the x-axis is A (X one ，0） and B（x two ,0）。

be careful : "Variable" is different from "unknown number". It cannot be said that "quadratic function means that the highest degree of unknown number is quadratic Polynomial function ”。 The "unknown number" is just a number (the specific value is unknown, but only one value is taken), and the "variable" can take any value within a certain range. The concept of "unknown number" is applicable to the equation (unknown function in functional equation and differential equation, but whether it is unknown or unknown function, it generally represents a number or function - special cases will also be encountered), but the letter in the function represents a variable, which has different meanings. The difference between the two can also be seen from the definition of function.^[1]

history

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About 480 BC, the ancient Babylonians and the Chinese had used Matching method The positive root of the quadratic equation was obtained, but no general solution method was proposed. Around 300 BC, Euclid proposed a more abstract geometric method to solve quadratic equations.

Brahmagupta of India in the 7th century was the first person who knew how to use algebraic equations. It allowed the roots of positive and negative numbers at the same time.

In the 11th century, the Arab Wallazmi independently developed a set of formulas to seek the positive number solution of the equation. In his book Liber embadorum, Abraham Bahia (also known for his Latin name Savasoda) first introduced the complete solution of quadratic equations in one variable to Europe.

It is said that Schrieder Haller was one of the first mathematicians who gave a universal solution to the quadratic equation. But this is controversial in his time. The solution rule is: multiply both sides of the equation by four times the coefficient of the quadratic term unknown; Add the square of the coefficient of the unknowns of the first order term on both sides of the equation at the same time; Then open the quadratic on both sides of the equation at the same time (quoted from Boshgalo II)

Function property

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one The image of a quadratic function is a parabola, but the parabola is not necessarily a quadratic function. A parabola with an opening up or down is a quadratic function. Parabola is an axisymmetric figure. The axis of symmetry is a straight line

。 The only intersection between the axis of symmetry and the parabola is the vertex P of the parabola. In particular, when b=0 The axis of symmetry is the y-axis (i.e. straight line x=0).

two The parabola has a vertex P, whose coordinate is P

。 When

When P is on the y-axis; When

P is on the x-axis.

three The quadratic coefficient a determines the opening direction and size of the parabola. When a>0, the parabolic opening is upward; When a<0, the parabola opening is downward| A | The smaller, the larger the opening of the parabola| A | The larger the, the smaller the opening of the parabola

four The linear term coefficient b and the quadratic term coefficient a jointly determine the position of the axis of symmetry. When a and b are the same sign (that is, ab>0), the symmetry axis is on the left side of the y-axis; When a and b are different (that is, ab<0) (it can be conveniently noted that the left is the same as the right), the axis of symmetry is on the right side of the y-axis.

five The constant term c determines the intersection of the parabola and the y-axis. The parabola intersects the y-axis at (0, c)

six Number of intersection points of parabola and x-axis:

The parabola has 2 intersections with the x-axis.

The parabola has 1 intersection with the x-axis. When

The parabola has no intersection with the x-axis.

seven When

When the function is

Get minimum value at

； stay

Is a minus function on the

Is an increasing function; The opening of the parabola is upward; The value range of the function is

。

When

When the function is

Get maximum at

； stay

Is an increasing function on

Is a minus function; The opening of the parabola is downward; The value range of the function is

。

When

When, the symmetric axis of the parabola is the y-axis, then, the function is an even function, and the analytic expression is transformed to y=ax ²+c (a ≠ 0).

8. Definition field ：R

9. Value Field : When a>0, the value field is

； When a<0, the value field is

。^[4]

Parity: when b=0, this function is even; When b is not equal to 0, this function is Non odd and non even function 。

Periodicity : None

Analytic expression:

① General formula:

⑴a≠0

⑵ If a>0, the parabola opening faces upward; If a<0, the parabola opening is downward.

⑶ Vertex:

；

⑷

If Δ>0, the function image intersects the x-axis at two points:

and

；

If Δ=0, the function image intersects the x-axis at the same point:

If Δ<0, there is no common point between the function image and the x-axis.^{[5 ]}

② Vertex:

Now the vertex is (h, k)

When the corresponding vertex is

, where,

；

③ Intersection:

The function image intersects the x-axis at

and

Two points.

expression

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Vertex form

Y=a (x-h) ²+k (a ≠ 0, a, h, k are constants), vertex coordinates are (h, k) ， The axis of symmetry is a straight line x=h, and the location characteristics of the vertex and the opening direction of the image are the same as those of the image with the function y=ax ². When x=h, the maximum (small) value of y=k. Sometimes the question will point out that you can use matching methods to convert the general formula into vertex formula.

Example: Given the vertex (1,2) of the quadratic function y and another arbitrary point (3,10), find the analytic expression of y.

Solution: Let y=a (x-1) ²+2, substitute (3,10) into the above equation, and get y=2 (x-1) ²+2.

be careful: And point at Rectangular coordinate system The translation in is different. In the vertex formula after the translation of the quadratic function, when h>0, the larger the h is, the farther the symmetry axis of the image is from the y-axis. In the positive direction of the x-axis, it cannot simply be considered as a left translation because the sign before h is negative.

It can be divided into the following situations:

When h >At 0, y = a (x- h ）² image can be represented by parabola y = ax ² Right parallel movement h Units;

When h >At 0, y = a (x+ h ）² image can be represented by parabola y = ax ² Left parallel movement h Units;

When h >0, k >When 0, the parabola y = ax ² Right parallel movement h Units, then move up k Unit, you can get y = a ( x - h )²+ k Image of;

When h >0, k >When 0, the parabola y = ax ² Left parallel movement h Units, then move down k Unit, you can get y = a ( x + h )²- k Image of;

When h ＜0, k >When 0, the parabola y = ax ² Left parallel movement| h |Units, then move up k Units can get y = a ( x - h )²+ k Image of;

When h ＜0, k When<0, change the parabola y = ax ² Left parallel movement| h |Units, then move down |k| Units can get y = a ( x - h )²+ k Image of.^[2]

Intersection formula

[Only the parabola at the intersection with the x-axis, i.e. y=0, i.e. b two -4ac≥0] .

Quadratic function

Given that the parabola intersects with the x-axis, that is, y=0, A (x one , 0) and B (x two , 0), we can set

And then substitute the third point into x and y to find a.

Steps to change from general formula to intersection formula: (Weida theorem)

Key concepts: a. B, c are constants, a ≠ 0, and a determines the opening direction of the function. a> When 0, the opening direction is upward; a absolute value The size of the opening can be determined. The larger the absolute value of a, the smaller the opening, and the smaller the absolute value of a, the larger the opening.

f(x)=f[x0]+f[x0,x1](x-x0)+f[x0,x1,x2](x-x0)(x-x1)+... f[x0,... xn](x-x0)... (x-xn-1)+Rn (x) can lead to the coefficient of the intersection formula

(y is the intercept) The right side of the quadratic function expression is usually Quadratic trinomial 。

Euler intersection formula:

If ax ²+bx+c=0, there are two real roots x one ,x two , then

The axis of symmetry of this parabola is a straight line

。

bikini

Method 1:

Given three points on the quadratic function, (x one , y one )、(x two , y two )、(x three , y three )。 Substitute the three points into Function analytic expression Y=a (x-h) ²+k (a ≠ 0, a, h, k are constants), including:

The values of a, b and c can be solved by obtaining a set of ternary linear equations.

Method 2:

Given three points on the quadratic function, (x one , y one )、(x two , y two )、(x three , y three )

Using the Lagrange interpolation method, the analytical formula of the quadratic function can be obtained as follows:

Condition of intersection with X axis:

When

There are two intersections between the function image and the x-axis, namely (x one , 0) and (x two , 0)。

When

There is only one tangent point between the function image and the x-axis, that is

。

When

The parabola has no common intersection with the x-axis. The value range of x is imaginary（

）

Function image

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Basic image

Make the quadratic function y=ax in the plane rectangular coordinate system two +Bx+c, we can see that the quadratic function image without a specific domain is an endless parabola. If the drawing is correct, the quadratic function image will be

Translated.

axial symmetry

Quadratic function image^[3]

The quadratic function image is axial symmetry graphical. The axis of symmetry is a straight line

Symmetry axis and quadratic function image only The intersection point of is the vertex P of the quadratic function image.

In particular, when b=0, the symmetry axis of the quadratic function image is Y-axis (i.e. straight line x=0). Is vertex Abscissa (i.e. x=?).

a. B is the same sign, and the symmetry axis is on the left side of y-axis;

a. B The sign is different, and the symmetry axis is on the right side of the y-axis.

vertex

The quadratic function image has a vertex P with coordinates P (h, k).

When h=0, P is on the y-axis; When k=0, P is X axis On. It can be expressed as the vertex formula y=a (x-h) two +k（a≠0）

。

Opening

The quadratic coefficient a determines the opening direction and size of the quadratic function image.

When a>0 Time, quadratic function image Up Opening; When a<0 Time, parabola down Opening.

|a| Larger , the opening of the quadratic function image Smaller 。

Location determining factors

The linear term coefficient b and the quadratic term coefficient a jointly determine the position of the axis of symmetry.

When a>0, and b with When (i.e. ab > 0), symmetry axis at Y-axis left ； Because the symmetry axis is on the left, the symmetry axis less than 0, that is - b/2a

When a>0, and b different When sign (i.e. ab<0), the axis of symmetry is at Y-axis right 。 Because the symmetry axis is on the right, the symmetry axis should greater than 0, that is, - b/2a>0, so b/2a should be less than 0, so a and b should be signed differently

Can be simply memorized as Same left but different right That is, when the symmetry axis is on the left of the y-axis, a and b are the same sign (that is, a>0, b>0 or a

In fact, b has its own geometric meaning: the slope k value of the function analytic expression (primary function) of the tangent line of the quadratic function image at the intersection of the quadratic function image and the y-axis. It can be obtained by derivative of quadratic function.

Factors determining the intersection point

The constant term c determines the intersection of the quadratic function image and the y-axis.

Quadratic function image and y-axis intersect at (0, C) point

Note: The vertex coordinates are (h, k) and intersect with the y-axis at (0, C).

Number of intersections with x-axis

a 0; k> 0 or a>0; k

k=0 There is only one intersection point between the quadratic function image and the x-axis.

Question point: when a0, there is no intersection between the quadratic function image and the x-axis.

When a>0, the function gets the minimum value at x=h

=k. In the range of xh, it is an increasing function (that is, y increases with the increase of x). The opening of the quadratic function image is upward, and the value range of the function is y>k

When a

=k. The range of xh is a subtractive function (that is, y decreases with the increase of x), the opening of the quadratic function image is downward, and the value range of the function is y

When h=0, the symmetric axis of the parabola is the y-axis, and then the function is an even function

Symmetry relation

about General formula ：

①y=ax two +Bx+c and y=ax two -Bx+c two images are symmetric about y axis

②y=ax two +Bx+c and y=- ax two -Bx-c two images are symmetric about x axis

③y=ax two +Bx+c and y=- ax two -bx+c-b two /2a About vertex symmetry

④y=ax two +Bx+c and y=- ax two +Bx-c is symmetric about the center of the origin. (i.e. the figure obtained after rotating 180 degrees around the origin)

For vertex type:

①y=a(x-h) two +K and y=a (x+h) two +K The two images are symmetric about the y-axis, that is, the vertices (h, k) and (- h, k) are symmetric about the y-axis, with opposite abscissa and the same ordinate.

②y=a(x-h) two +K and y=- a (x-h) two -K The two images are symmetric about the x axis, that is, the vertices (h, k) and (h, - k) are symmetric about the x axis, with the same abscissa and the opposite ordinate.

③y=a(x-h) two +K and y=- a (x-h) two +K is symmetric about the vertex, that is, the vertex (h, k) and (h, k) are the same, and the opening direction is opposite.

④y=a(x-h) two +K and y=- a (x+h) two -K is symmetric about the origin, that is, the vertices (h, k) and (- h, - k) are symmetric about the origin, and the abscissa and ordinate are opposite.

(Actually ① ③ ④ is the case of f (- x), - f (x), - f (- x) for f (x) ）

Five point method

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Five point sketch method, also called five point drawing method, is a common drawing method of quadratic function.

Note: Although it is a sketch, it is not a sketch.

The five points in the five point sketch method are all extremely important five points, which are: vertex, intersection with the x axis, intersection with the y axis and the symmetry point about the symmetry axis.

Ps. The formal examination also uses this method to preliminarily determine the image. But the requirement of formal examination is to list tables, take x, y, and then determine the overall image. The five point method can be used in formal examinations.

Point tracing method

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In junior high school mathematics, it is required to draw a quadratic function image with the method of tracing points.

The method is similar to the five point method:

As an example]

1. List

x	……	-1	-0.5	zero	one	two	two point five	three	……
	……	seven	three point five	one	-1	one	three point five	seven	……

Point tracing method

Take the vertex first and draw the axis of symmetry with a dotted line. Take two intersections with the x-axis (if any), the y-axis intersection and its symmetrical point (if any), and the other two points and their symmetrical points. Ps. In principle, the difference between adjacent x is equal, but the difference between points far from the vertex can be appropriately reduced

2. Draw function image according to table data, as shown in Figure 1

Equation relation

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In particular, quadratic function (hereinafter referred to as function)

When y=0 When, the quadratic function is the quadratic equation of one variable about x (hereinafter referred to as equation), that is

At this time, whether the function image intersects with the x-axis is whether the equation has real roots.

The abscissa of the intersection point between the function and the x-axis is the root of the equation.

1. Quadratic function y=ax two ，y=a(x-h) two ，y=a(x-h) two +k，y=ax two +The images of bx+c (a ≠ 0 in all kinds) have the same shape but different positions. Their vertex coordinates and symmetry axes are shown in the following table:

When h>0, the image of y=a (x-h) ^ 2 can be obtained by moving the parabola y=ax ² to the right parallel for h units,

When h

When h>0, k>0, move the parabola y=ax ^ 2 parallel to the right for h units, and then move up for k units to get the image of y=a (x-h) ²+k (h>0, k>0)

When h>0, k0, k

When h0, move the parabola y=ax ^ 2 | h | units to the left in parallel, and then move k units upward to get the image of y=a (x+h) ²+k (h0)

When h

Up or down. When translating the parabola to the left or right, it can be abbreviated as "up plus down minus, left plus right minus".

Therefore, study the parabola y=ax two +Bx+c (a ≠ 0), the general formula is changed to y=a (x-h) by formula two +The form of k can determine its vertex coordinates, axis of symmetry, and the general position of the parabola. This provides convenience for drawing images.

2. Parabola y=ax two +Image of bx+c (a ≠ 0): when a>0, the opening is upward, when a

3. Parabola y=ax two +Bx+c (a ≠ 0), if a>0, when x ≤ - b/2a, y decreases with the increase of x; When x ≥ - b/2a, y increases with the increase of x. If a

4. Parabola y=ax two +Intersection point of bx+c image and coordinate axis:

(1) The image and y-axis must intersect, and the intersection coordinates are (0, c);

(2) When

The image intersects the x-axis at two points A (x one , 0) and B (x two , 0), where x1 and x2 are quadratic equations of one variable ax two +The distance between two points bx+c=0 (a ≠ 0)

In addition, the distance of any pair of symmetric points on the parabola can be determined by

(A is twice the abscissa of one point)

When

There is only one tangent point between the image and the x-axis;

When

There is no common point between the image and the x-axis. When a>0, the image falls above the x-axis. When x is any real number, there is y>0; When a

5. Parabola y=ax two +Maximum value of bx+c: if a>0, then when

When,

； If a

When,

。

The abscissa of the vertex is the independent variable value when obtaining the maximum value, and the ordinate of the vertex is the value of the maximum value.

6. Use Undetermined coefficient method Find the analytic expression of quadratic function

(1) When the given condition is that a known image passes through three known points or three pairs of corresponding values of x and y, the analytic expression (expression) can be set as a general form:

(a≠0)

(2) When the given condition is the vertex coordinate or axis of symmetry or the maximum (minimum) value of the known image, the analytical formula can be set as the vertex formula: y=a (x-h) two +k(a≠0)。

(3) When the given condition is that the coordinates of the two intersection points of the image and the x-axis are known, the analytical formula can be set as two: y=a (x-x one )(x-x two )(a≠0)。

learning method

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Key points of knowledge

1. Understand the meaning of the function.

2. Remember several expressions of functions and pay attention to the distinction.

3. General formula, vertex formula, intersection formula, etc., distinguish the differences of symmetry axis, vertex, image, and y, which decrease (increase) (decrease) with the increase of x.

4. Understand the function image with practice.

5. When calculating, remember the value range when looking at the image.

6. It changes with the number of image understanding. Test points and examples of quadratic function

The knowledge of quadratic function is easy to be integrated with other knowledge to form more complex comprehensive problems. Therefore, comprehensive questions based on quadratic function knowledge are the hot topics in the middle school entrance examination, which often appear in the form of large questions.

Misunderstanding Reminder

(1) The concept of quadratic function is misunderstood, and the limitation that the coefficient of quadratic term is not 0 is omitted;

(2) There are mistaken ideas about the image and nature of quadratic function;

(3) Ignore the value range of the independent variable of the quadratic function;

(4) Reverse direction when translating parabola 。

Definitions and expressions

Generally, the relationship between the independent variable x and the dependent variable y is as follows:

(a, b, c are constants, a ≠ 0, and a determines the opening direction of the function. When a>0, the opening direction is upward, a

Y is called the quadratic function of x.

The right side of the quadratic function expression is usually a quadratic trinomial.

Three expressions

General formula: y=ax ²+bx+c (a, b, c are constants, a ≠ 0)

Vertex formula: y=a (x-h) ²+k [Vertex of parabola P (h, k)]

Intersection formula: y=a (x-x one )(x-x two ）[Only limited to the intersection point A (x one , 0) and B (x two Parabola of, 0)]

Note: In the mutual transformation of the three forms, there are the following relations:

，

Properties of parabola

1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line

。

The only intersection between the axis of symmetry and the parabola is the vertex P of the parabola.

In particular, when b=0, the symmetric axis of the parabola is the y-axis (that is, the straight line x=0)

2. Parabola has a top

Point P, coordinate is

When

When P is on the y-axis; When

P is on the x-axis.

3. The quadratic coefficient a determines the opening direction of the parabola, and | a | determines the opening size of the parabola.

When a>0, the parabolic opening is upward; When a

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

4. There is one intersection point between the primary term coefficient b and the secondary term coefficient a.

5. The constant term c determines the intersection point of the parabola and the y-axis.

The parabola intersects the y-axis at (0, c)

Parabola and x-axis

Number of intersections

When Δ=b ² - 4ac>0, there are two intersections between the parabola and the x-axis.

When Δ=b ² - 4ac=0, there is an intersection point between the parabola and the x-axis.

When Δ=b ² - 4ac<0, there is no intersection point between parabola and x-axis.

（ reference material:^[3] ）

Meaning of coefficient expression

A determines the opening direction and size of the parabola. When a>0, the parabola opens upwards; When a

B and a jointly determine the position of the axis of symmetry. When a and b are the same sign (that is, ab>0), the axis of symmetry is on the left of the y-axis; When a and b are different (i.e. ab<0)

C determines the intersection point of the parabola and the y-axis, and the parabola and the y-axis intersect at (0, c)

Quadratic function image and property pithy formula

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For the parabola of quadratic function, image symmetry is the key;

Openings, vertices and intersections, which determine the quadrant of the graph;

The opening and size are broken by a, and the c and Y axes meet each other. The symbol of b is special, and the symbol is related to a;

Find the vertex position first, and use the Y axis as the reference line. The left is the same as the right, and the middle is 0. Remember not to be confused;

The vertex coordinate is the most important. It can be seen in the general formula. The horizontal mark is the axis of symmetry. The vertical mark function has the maximum value shown in.

If the position of the axis of symmetry is calculated, the symbol is reverse, general, vertex, intersection, and different expressions can be interchanged.

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