The quadratic function Y = AX2+BX+C (A ≠ 0, A, B and C are constants) contains two variables X and Y. As long as we determine one of the variables first, we can find the other variable by analytical formula, that is, we can get a set of solutions. And a set of solutions is the coordinates of a point, in fact, the image of quadratic function is a graph composed of countless such points.
Familiar with the images and properties of several special quadratic functions.
1. Observe the shapes and positions of y=ax2, Y = AX2+K, Y = A (X+H) 2 images by tracing points, and get familiar with the basic features of their respective images. On the contrary, according to the characteristics of parabola, we can quickly determine which analytical formula it is.
2. Understand the translation formula of image "addition and subtraction, left plus right subtraction".
Y = AX2 → Y = A (X+H) 2+K "addition and subtraction" is K, and "adding left and subtracting right" is H. 。
In a word, if the coefficients of quadratic terms of two quadratic functions are the same, their parabolas have the same shape, but the translation of parabolas is essentially the translation of vertices because of their different coordinates and positions. If parabolas are in general form, they should be converted into vertices and then translated.
3. Through drawing and image translation, we understand and make it clear that the characteristics of analytical expressions are completely corresponding to the characteristics of images. When solving problems, we should have a picture in mind and see the function to reflect the basic characteristics of its image in our hearts.
4. On the basis of being familiar with the function image, through observing and analyzing the characteristics of parabola, we can understand the properties of quadratic function, such as increase and decrease, extreme value and so on. Distinguish the coefficients A, B, C, △ of quadratic function and the symbols of algebraic expressions composed of coefficients by images.
Third, we should make full use of the function of parabola "vertex".
1. We should be able to find the "vertex" accurately and flexibly. The form is y = a (x+h) 2+k → vertex (-h, k). For other forms of quadratic function, we can turn it into a vertex to find the vertex.
2. Understand the relationship between vertex, symmetry axis and maximum value of function. If the vertex is (-h, k) and the symmetry axis is x =-h, the maximum (minimum) value of y = k；; On the other hand, if the symmetry axis is x=m and the maximum value of y is n, then the vertex is (m, n); ; Understanding the relationship between them can achieve the effect of analyzing and solving problems.
3. Draw a sketch with vertices. In most cases, we only need to draw a sketch to help us analyze and solve problems. At this time, we can draw the approximate image of parabola according to the vertex and opening direction of parabola.
Understand and master the solution of the intersection of parabola and coordinate axis.
Generally speaking, the coordinates of a point consist of abscissa and ordinate. When we find the intersection of parabola and coordinate axis, we can give priority to one of the coordinates, and then find the other coordinate by analytical formula. If the equation has no real root, it means that the parabola and the X axis do not intersect.
From the process of finding the intersection point above, we can see that the essence of finding the intersection point is to solve the equation, which is related to the discriminant of the root of the equation, and the number of times the parabola intersects the X axis is determined by the discriminant of the root.
Fifth, flexibly use the undetermined coefficient method to find the analytical formula of quadratic function.
It is the most conventional and effective method to find the analytical formula of quadratic function by using undetermined coefficient method. There are often many ways to find analytical expressions. If we can comprehensively use the images and properties of quadratic function and flexibly use the idea of combining numbers and shapes, it will not only simplify the calculation, but also be of great benefit to further understand the essence of quadratic function and the relationship between numbers and shapes.
Quadratic function y=ax2
Learning requirements:
1. Know the meaning of quadratic function.
2. I will draw the image of function Y = AX2 by tracing points, and know the related concepts of parabola.
Analysis of key and difficult points
1. This section focuses on the concept of quadratic function and the image and properties of quadratic function y = ax2; The difficulty is to generalize the property of quadratic function Y = AX2 according to the image.
2. Functions in the form of = AX2+BX+C (where A, B and C are constants and a≠0) are all quadratic functions. There can only be two in the analytical formula.
There are variables x and y, and the coefficient of the quadratic term of x cannot be 0. The range of the independent variable x is usually a real number, but the actual quantity should be meaningful in practical problems. For example, in the relationship between circle area s and circle radius r, radius r can only be non-negative.
3. The shape of parabola y = ax2 is determined by A, and the symbol of A determines the opening direction of parabola. When a > 0, the opening is upward, the parabola is above the Y axis (the vertex is on the X axis), and it extends infinitely upward. When a < 0, the opening is downward, the parabola is below the X axis (the vertex is on the X axis), and it extends infinitely downward. The bigger the | a |, the smaller the opening; The smaller the | a |, the larger the opening.
4. When drawing a parabola y = ax2, list first, then trace points, and finally connect lines. When choosing the value of independent variable x in the list, it is often centered on 0, so choose an integer value that is convenient for calculation and tracking. When tracking points, be sure to connect them with smooth curves and pay attention to the changing trend.
The proposition in this section mainly investigates the concept of quadratic function, the application of images and the properties of quadratic function Y = AX2.
Core knowledge
Rule 1
The concept of quadratic function:
Generally speaking, if it is a constant, then y is called the quadratic function of X.
Rule 2
Related concepts of parabola:
Figure 13- 14
As shown in figure 13- 14, the image of function y=x2 is a curve that is symmetrical about y axis and is called parabola. In fact, the images of quadratic functions are all parabolas. Parabola y=x2 has an upward opening, the y axis is the symmetry axis of this parabola, and the intersection of the symmetry axis and the parabola is the vertex of the parabola.
Rule 3
Properties of parabola y=ax2;
Generally, the symmetry axis of parabola y=ax2 is the Y axis, and the vertex is the origin. When a > 0, the opening of parabola y=ax2 is upward, and when a < 0, the opening of parabola y=ax2 is downward.
Rule 4
The concept of 1. quadratic function
(1) definition: generally speaking, if y = AX2+BX+C (A, b, c are constants, and a≠0), then y is called the quadratic function of x (2). The structural characteristics of the quadratic function y = AX2+BX+C are: the function y is on the left side of the equal sign, and the independent variable is on the right side of the equal sign.
2. Images with quadratic function y = ax2
Figure 13- 1
Draw the image of quadratic function y = x2 by tracing points, as shown in figure 13- 1. This is a curve about the axis symmetry of y, and such a curve is called parabola.
Because the parabola Y = x2 is symmetrical about y axis, the y axis is the symmetry axis of this parabola, and the intersection of the symmetry axis and parabola is the vertex of the parabola. Seen from the figure, the vertex of parabola Y = x2 is the lowest point of the image. Because the parabola Y = x2 has the lowest point, the function Y = x2 has the minimum value, and its minimum value is the ordinate of the lowest point.
3. Properties of quadratic function y = ax2
function
draw
Opening direction
Vertex coordinates
axis of symmetry
Functional change
Maximum (minimum) value
y=ax2
a>0
up
(0,0)
Y axis
When x > 0, y increases with the increase of x;
When x < 0, y decreases with the increase of x.
When x = 0, y is minimum = 0.
y=ax2
When a 0, y decreases with the increase of x;
When x < 0, y increases with X.
When x = 0, y max = 0.
4. Draw an image with quadratic function y = ax2.
When using quadratic function y = ax2 to plot points, we should select the value of independent variable X symmetrically around the vertex, and then calculate the corresponding Y value. The denser the corresponding values are selected, the more accurate the drawn image will be.
Quadratic function y=ax2+bx+c
Learning requirements:
The image of 1. quadratic function will be drawn by drawing.
2. The opening direction, symmetry axis, vertex and position of parabola can be determined by images or formulas.
* 3. The analytical formula of quadratic function will be obtained from the coordinates of three points on the known image.
Important and difficult
1. This section focuses on the understanding and flexible application of the image and properties of the quadratic function Y = AX2+BX+C, but the difficulty lies in the properties of the quadratic function Y = AX2+BX+C and the transformation of the analytical formula into the form of Y = A (X-H) 2+K through the formula.
2. Learning this section requires careful observation and induction of the characteristics of images and the relationship between different images. Connect different images and find out their uniqueness.
Generally speaking, if the quadratic coefficient a of several different quadratic functions is the same, the opening direction and opening size (i.e. shape) of parabola are exactly the same, but the positions are different.
Any parabola y = a (x-h) 2+k can be obtained by appropriately translating the parabola y = ax2. The specific translation method is shown in the following figure:
Note: the law of the above translation is: "H value is positive and negative, shifting left and right; K value is positive, negative, up and down "is actually related to the translation problem of parabola, so we can't memorize the translation law by rote." It is very simple to determine the translation direction and distance according to the position relationship of their vertices.
Figure 13- 1 1
For example, to study the positional relationship between parabola L 1: y = x2-2x+3 and parabola L2: y = x2, we can change y = x2-2x+3 into vertex Y = (x- 1) 2+2 by formula, and find its vertex m 1 (/kloc-).
The image of quadratic function Y = AX2+BX+C has exactly the same shape as the image of Y = AX2, and their properties are similar. When a > 0, the openings of the two parabolas are upward and extend indefinitely, and the parabolas have the lowest point and the y has the lowest value; When a < 0, the opening extends downward and infinitely, the parabola has the highest point and y has the maximum value.
3. When drawing a parabola, we must first determine the opening direction, symmetry axis and vertex position, and then use the function symmetry list, so that we can get a complete and accurate image after drawing points and connecting lines. Otherwise, the drawn image is often only a part. For example, draw an image, y =-(x+ 1) 2- 1.
List:
x
-3
-2
- 1
1
2
three
y
-3
- 1.5
- 1
- 1.5
-3
-5.5
-9
Trace points and connect them into an image as shown in figure 13- 1 1, which cannot reflect the whole picture.
Positive solution: According to the analytical formula, the opening of the image is downward, the symmetry axis is X =-1, and the vertex coordinates are (-1,-1).
List:
x
-4
-3
-2
- 1
1
2
y
-5.5
-3
- 1.5
- 1
- 1.5
- 1.5
-5.5
Tracing line: as shown in figure 13- 12.
Figure 13- 12
4. To transform the quadratic function Y = AX2+BX+C into Y = A (X-H) 2+K by collocation method, the quadratic term coefficient A should be put forward first. Common mistakes are only mentioned in the first item, and then omitted. For example, if y =-x2+6x-2 1 is written as y =-(x2+6x-2 1) or y =-(x2+6x-21+02x-42), the symbol is wrong, mainly because the brackets are not mastered.
The proposition in this section mainly investigates the image and properties of quadratic function Y = AX2+BX+C and its application in real life. There are not only fill-in-the-blank questions, but also multiple-choice and analytical questions, as well as comprehensive questions of equations, geometry and linear functions, which are often used as the finale of the senior high school entrance examination.
Core knowledge
Rule 1
Properties of parabola y=a(x-h)2+k:
Generally, parabolas y=a(x-h)2+k and y=ax2 have the same shape but different positions. The parabola y=a(x-h)2+k has the following characteristics:
(l) When a > 0, the opening is upward; When a < 0, the opening is downward;
(2) The symmetry axis is a straight line x = h；;
(3) The vertex coordinate is (h, k).
Rule 2
Properties of quadratic function y=ax2+bx+c;
Y=ax2+bx+c (a, B, C are constants, a≠0) is a quadratic function, and the image is a parabola. The quadratic function can be expressed as y=a(x-h)2+k by the formula, from which it can be determined that the symmetry axis of this parabola is a straight line, and when the vertex coordinate is A > 0, the opening is upward. When a < 0, the opening is downward.
Rule 3
Several forms of 1. quadratic resolution function
(1) general formula: Y = AX2+BX+C (A, b, c are constants, a≠0).
(2) Vertex: y = a (x-h) 2+k (a, h, k are constants, a≠0).
(3) two expressions: y = a (X-X 1) (X-X2), where X 1, X2 is the abscissa of the intersection of parabola and x axis, that is, the two roots of quadratic equation AX2+BX+C = 0, a≠0.
Description: (1) Any quadratic function can be transformed into vertex Y = A (X-H) 2+K by formula, and the vertex coordinate of parabola is (h, k). When H = 0, the vertex of parabola Y = AX2+K is on the Y axis; When k = 0, the vertex of parabola a(x-h)2 is on the X axis; When H = 0 and K = 0, the vertex of parabola Y = AX2 is at the origin.
(2) When the parabola y = ax2+bx+c intersects the X axis, that is, the quadratic equation ax2+bx+c = 0 has the sum of the real root x 1.
When x2 exists, the quadratic function y = ax2+bx+c can be transformed into two formulas y = a (x-x 1) (x-x2) according to the decomposition formula of quadratic trinomial.
2. Determination of the second resolution function
To determine the quadratic resolution function, the undetermined coefficient method is still generally used. Because the quadratic resolution function has three undetermined coefficients a, b, c (or a, h, k or a, x 1, x2), it is necessary to know three independent conditions to determine the quadratic resolution function. When the coordinates of any three points on the parabola are known, it is very convenient to choose the general formula. When the coordinates of the vertices of parabola are known, it is more convenient to choose the vertices. When the coordinates of parabola and X axis (or abscissa x 1, x2) are known, it is convenient to choose two formulas.
Note: When finding the quadratic resolution function with vertex type or quadratic type, the general formula will be used at last.
3. Images with quadratic function y = AX2+BX+C
The image of quadratic function y = AX2+BX+C is a parabola, and its symmetry axis is parallel to (including coincident with) the Y axis.
4. Properties of quadratic function
According to the image of quadratic function y = ax2+bx+c, its properties can be summarized as follows:
function
Quadratic function y = AX2+BX+C (A, B, C are constants, a≠0)
draw
be just like
a>0
A 0, y has a minimum, when x = h, y has a minimum = k, if a < 0, y has a maximum, when x = h, y has a maximum.
② Formula method: directly use the vertex coordinate formula (-,) to find its vertex; The symmetry axis is a straight line x =-, and if a > 0, y has a minimum value; When x =-, y has a minimum value =; If a < 0, y has the maximum value; When x =-, y has the maximum value =.
6. Draw an image with quadratic function Y = AX2+BX+C.
Because the image of quadratic function is parabolic and axisymmetrical, simplified point tracing method and five-point method are often used in drawing, and the steps are as follows:
(1) First, find the vertex coordinates and draw the symmetry axis;
(2) Find four points on the parabola about the axis of symmetry (such as the intersection with the coordinate axis, etc.). );
(3) Connect these five points from left to right with a smooth curve.
7. The image position of quadratic function y = AX2+BX+C is closely related to the symbols of A, B, C and δ (see the table below):
project
eye
word
mother
alphabetic symbol
The position of the image
a
a>0
a0 ab 0 c < 0
The zero crossing point intersects with the positive semi-axis of Y axis and intersects with the negative semi-axis of Y axis.
8. The relationship between quadratic function and unary quadratic equation
The abscissas x 1 and x2 of the two intersections of the image (parabola) of the quadratic function y = ax2+bx+c and the x axis are the two real roots of the corresponding quadratic equation ax2+bx+c = 0. The intersection of parabola and X axis can be judged by the discriminant of the root of the corresponding quadratic equation:
The parabola with δ > 0 has two intersections with the X axis;
δ = 0 parabola and X axis have 1 intersection points;
The object line with δ < 0 intersects (does not intersect) the X axis zero.