As shown in the figure, point P is the inverse scale function y = K / X (k)

Given that the image of the first-order function y = KX + B and the image of the inverse scale function y = - 2 / X intersect at the point (- 1, m) and pass through the point (0, - 3), the analytic expression of the first-order function is obtained

Analysis: M = 2, so the straight line passes (0, - 3), (- 1,2), so B = - 3, k = - 5, so y = - 5x-3

It is known that the vertex coordinates of the image of quadratic function are (1, - 4) and pass through points (2, - 2)
Find the relation of the quadratic function

Solution
Known vertex coordinates
Let y = a (X-H) ² + K
∵ vertex coordinates are: (1, - 4)
∴y=a(x-1)²-4
∵ over point (2, - 2)
∴a(2-1)²-4=-2
∴a=2
∴y=2(x-1)²-4
That is y = 2x & # 178; - 4x-2

It is known that the vertex coordinates of a quadratic function are (4, - 2) and its image passes through points (5,1)

Let the analytic expression of the quadratic function be y = a (x-4) 2-2; ∵ the image of the quadratic function passes through the point (5,1), ∵ a (5-4) 2-2 = 1, ∵ a = 3, ∵ y = 3 (x-4) 2-2 = 3x2-24x + 46

If the image of quadratic function y = (A-1) x ^ 2-2x + 1 does not intersect the X axis, then the value range of a is?

If the square of minus 2 minus 4 times (A-1) multiplied by 1 is less than 0 and A-1 cannot be equal to 1, then a > 2 is obtained

If the vertex of quadratic function y = x ^ 2-4x + m is on the X axis, then the value of M is zero
As the title
I also want to ask whether the ordinate of the lower vertex is the maximum or the minimum of the function?

y=x²-4x+4-4+m
=(x-2)²-4+m
So if the vertex (2, - 4 + m) is on the X axis, the ordinate is 0
-4+m=0
m=4
It depends on the direction of the opening
If the opening is upward, the vertex ordinate is the minimum
If the opening is down, the vertex ordinate is the maximum

Given that the vertex coordinates of the image of quadratic function are (4, - 1) and pass through the point (0,3), the first question is to find the expression of quadratic function, the second question is to find the projection

The vertex coordinates are (4, - 1) let the function expression be y = a (x-4) & sup2; - 1, substitute the point (0,3), 16a-1 = 3, a = 1 / 4, and the function expression be y = 1 / 4 (x-4) & sup2; - 1. I don't know what you want to ask next. The axis of symmetry x = 4, and the coordinates of the intersection of X axis, substitute y = 0,1 / 4 (x-4) & sup2; - 1 = 0x1 = 2, X2 = 6

What is the expression of vertex coordinates of quadratic function, and what is the complete square expression, expressed in letters

Come on~~
CHEER YOU UP~~
1、 Understand the connotation and essence of quadratic function
Quadratic function y = AX2 + BX + C (a ≠ 0, a, B, C are constants) contains two variables X, y, we only need to determine one of the variables, we can use the analytic formula to find another variable, that is to get a group of solutions; and a group of Solutions is the coordinates of a point, in fact, the image of quadratic function is composed of countless such points
2、 Be familiar with the image and properties of some special quadratic functions
1. By tracing points, we can observe the shape and position of y = AX2, y = AX2 + K, y = a (x + H) 2 images, and get familiar with the basic characteristics of each image. On the contrary, we can quickly determine which kind of analytical formula it is according to the characteristics of parabola
2. Understand the image translation formula "plus minus, plus left minus right."
Y = AX2 → y = a (x + H) 2 + k "plus minus below" is for K, "plus left minus right" is for H
In a word, if the quadratic coefficients of two quadratic functions are the same, then their parabola shape is the same, because the vertex coordinates are different, so the positions are different, and the translation of the parabola is essentially the translation of the vertex. If the parabola is a general form, it should be transformed into the vertex form first and then the translation
3. By drawing a picture and translating the image, we can understand and make sure that the characteristics of the analytic formula are completely corresponding to the characteristics of the image;
4. On the basis of being familiar with the image of function, by observing and analyzing the characteristics of parabola, we can understand the properties of quadratic function, such as increasing or decreasing, extremum, etc
3、 We should make full use of the "vertex" of parabola
1. In order to find the "vertex" accurately and flexibly, such as y = a (x + H) 2 + K → vertex (- H, K), for other forms of quadratic function, we can change it into vertex form and find the vertex
2. If the vertex is (- H, K), then the symmetry axis is x = - H, y max (small) = k; on the contrary, if the symmetry axis is x = m, y max = n, then the vertex is (m, n); understanding the relationship between them can achieve the effect of drawing inferences from one instance when analyzing and solving problems
3. In most cases, we only need to draw a sketch to help us analyze and solve the problem. At this time, we can draw the general image of the parabola according to the parabola vertex and the opening direction
4、 Understand the intersection of parabola and coordinate axis
Generally, the coordinates of a point are composed of abscissa and ordinate. When we find the intersection of the parabola and the coordinate axis, we can first determine one of the coordinates, and then use the analytic formula to find the other coordinate. If the equation has no real root, then there is no intersection between the parabola and the X axis
From the above process, we can see that the essence of finding the intersection point is to solve the equation, and it is connected with the discriminant of the root of the equation. Using the discriminant of the root, we can determine the number of the intersection points of the parabola and the X axis
5、 Flexible application of undetermined coefficient method for quadratic function analysis
The method of undetermined coefficient is the most common and effective way to find the analytic formula of quadratic function. Many methods can be used to find the analytic formula. If the image and properties of quadratic function can be comprehensively used and the idea of combination of number and shape can be flexibly applied, it can not only simplify the calculation, but also help us to further understand the essence of quadratic function and the relationship between number and shape
Quadratic function y = AX2
Learning requirements:
1. Know the meaning of quadratic function
2. Be able to draw the image of function y = AX2 by point tracing method, and know the concept 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 summarize the properties 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
For example, in the formula s = π R2, the radius r can only be a non negative number
3. The shape of the parabola y = AX2 is determined by A. the sign of a determines the opening direction of the parabola. When a > 0, the opening is upward, the parabola is above the y-axis (the vertex is on the x-axis), and extends upward infinitely; when a < 0, the opening is downward, the parabola is below the x-axis (the vertex is on the x-axis), and extends downward infinitely
4. When drawing a parabola y = AX2, you should first list, then trace the points, and finally connect the lines. When selecting the independent variable x value in the list, always take 0 as the center, and select the integer value that is convenient for calculation and tracing the points. When tracing the lines of the points, you must connect them with smooth curves, and pay attention to the change trend
The propositions in this section mainly examine the concept of quadratic function, the application of image and property of quadratic function y = AX2
Core knowledge
Rule 1
The concept of quadratic function:
Generally, if it is a constant, then y is called a quadratic function of X
Rule 2
The concepts of parabola are as follows
Figure 13-14
As shown in Figure 13-14, the image of the function y = X2 is a curve symmetrical about the y-axis, which is called a parabola. In fact, the image of a quadratic function is a parabola. The parabola y = X2 is open upward, the y-axis is the symmetry axis of the 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 the parabola y = AX2 is y-axis, and the vertex is the origin. When a > 0, the opening of the parabola y = AX2 is upward, and when a < 0, the opening of the parabola y = AX2 is downward
Rule 4
1. The concept of quadratic function
(1) Definition: in general, if y = AX2 + BX + C (a, B, C are constants, a ≠ 0), then y is called the quadratic function of X. (2) the structural characteristics of quadratic function y = AX2 + BX + C are: the left side of the equal sign is the function y, the right side is the quadratic expression of the independent variable x, and the highest degree of X is 2. The coefficient b of the first term and the constant term C can be arbitrary real numbers, and the coefficient a of the second term must be non-zero real numbers, that is, a ≠ 0
2. The image of quadratic function y = AX2
Figure 13-1
Draw the image of quadratic function y = x 2 with the point tracing method, as shown in Figure 13-1. It is a curve symmetrical about y axis. Such a curve is called a parabola
Because the parabola y = X2 is symmetric about the y-axis, the y-axis is the symmetry axis of the parabola, and the intersection of the symmetry axis and the parabola is the vertex of the parabola. From the graph, the vertex of the 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
image
Opening direction
Vertex coordinates
Axis of symmetry
Function 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 the smallest = 0
y=ax2
a＜0
down
(0,0)
Y axis
When x > 0, y decreases with the increase of X;
When x < 0, y increases with the increase of X
When x = 0, y max = 0
4. How to draw the image of quadratic function y = AX2
When drawing the image of quadratic function y = AX2 with point tracing method, we should select the value of independent variable x symmetrically on the left and right sides of the vertex, and then calculate the corresponding value of Y. the denser the corresponding value is selected, the more accurate the image is depicted
Quadratic function y = AX2 + BX + C
Learning requirements:
1. Be able to draw the image of quadratic function with point drawing method
2. Be able to determine the opening direction of parabola and the position of symmetry axis, vertex and center by image or formula
*3. The analytic expression of quadratic function can be obtained from the coordinates of three points on the known image
Key and difficult points
1. This section focuses on the understanding and flexible application of the image and properties of the quadratic function y = AX2 + BX + C, and the difficulties are 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 to find out their commonness
In general, for several different quadratic functions, if the quadratic coefficient a is the same, the opening direction and size (i.e. shape) of the parabola are exactly the same, but the position is different
Any parabola y = a (X-H) 2 + K can be obtained by proper translation of parabola y = AX2. The specific translation method is shown in the following figure:
Note: the above translation rule is: "H value is positive and negative, right and left; K value is positive and negative, up and down". In fact, it is related to the translation of parabola, so we can't memorize the translation rule by rote. It's very simple to change its analytical formula into vertex formula first, and then determine the translation direction and distance according to the position relationship of their vertices
Figure 13-11
For example, to study the positional relationship between the parabola L1 ∶ y = x2-2x + 3 and the parabola L2 ∶ y = X2, y = x2-2x + 3 can be changed into the vertex formula y = (x-1) 2 + 2 through the formula, and the vertex M1 (1,2) can be obtained, because the vertex of L2 is M2 (0,0). According to the position of their vertex, it is easy to see that L1 is obtained by translating L2 one unit to the right and then two units to the top; conversely, L1 is translated one unit to the left, Then translate down 2 units to get L2
The image of the quadratic function y = AX2 + BX + C has the same shape as the image of y = AX2, and their properties are similar

As shown in the figure, point P is the inverse scale function y = K / X (k)