How to draw the image of quadratic function

How to draw the image of quadratic function

Graph of quadratic function
Y = ax ^ 2 + BX + C (a, B, C are constants, a ≠ 0)
1、 Properties of parabola
1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line
x = -b/2a.
The only point of intersection between the axis of symmetry and the parabola is the vertex P of the parabola
In particular, when B = 0, the symmetry axis of the parabola is Y-axis (that is, the line x = 0)
2. The parabola has a vertex P whose coordinates are
P [ -b/2a ,(4ac-b^2;)/4a ].
When - B / 2A = 0, P is on the y-axis; when Δ = B ^ 2-4ac = 0, P is on the x-axis
3. The quadratic coefficient a determines the opening direction and size of the parabola
When a > 0, the parabola opens up; when a < 0, the parabola opens down
|The larger a | is, the smaller the opening of the parabola is
4. The position of the axis of symmetry is determined by the coefficient b of the first term and the coefficient a of the second term
When a and B have the same sign (AB > 0), the symmetry axis is on the left of Y axis;
When a is different from B (AB < 0), the axis of symmetry is on the right of Y axis
5. The constant term C determines the intersection of the parabola and the y-axis
The intersection of parabola and y-axis at (0, c)
6. The number of intersections of parabola and x-axis
When Δ = B ^ 2-4ac ＞ 0, there are two intersections between the parabola and X axis
When Δ = B ^ 2-4ac = 0, there is one intersection point between the parabola and X axis
When Δ = B ^ 2-4ac ＜ 0, there is no intersection between parabola and X axis
5. Quadratic function and quadratic equation of one variable
In particular, the quadratic function y = ax ^ 2; + BX + C,
When y = 0, the quadratic function is a quadratic equation of one variable with respect to X,
That is ax ^ 2; + BX + C = 0
In this case, the intersection of function image and x-axis is whether the equation has real roots
The abscissa of the intersection of function and X axis is the root of the equation
Given u = {1,2,3,4,5,6,7,8} a ∩ B = {2}) B ∩ (CUA) = {1,9} (CUA) ∩ (cub) = {4,6,8}, find the set A.B
It can be determined by the meaning of the title
Element a: 2,
Element B: 2,
The complement elements of a: 4,6,8,1,9
The complement elements of B: 4,6,8
And the complement element of a can deduce that there are 3, 5 and 7 elements of A
B has 1, 9
be
Element a is 2,3,5,7
B element is 2,1,9
The relationship between quadratic function image and ABC
For example, given two parabolas, the height of one is half of the other, what is the relationship between these two relations and ABC
For example, one is y = ax & sup2; + BX & sup2; + C. what is the other?
The height is the distance between the maximum or minimum value and the x-axis, and the maximum or minimum value is the height
(4ac-b * b) divided by 4A
y=ax²+bx²+2c
C determines the height
If one is y = ax & sup2; + BX & sup2; + C, the other is
y=ax²+bx²+2c
The height is the distance between the maximum or minimum value and the x-axis, and the maximum or minimum value is the height
(4ac-b * b) divided by 4A
Let u = {1, 2, 3, 4, 5, 6, 7, 8}, a = {5, 6, 7, 8}, B = {2, 4, 6, 8}, find a ∩ B, CUA and cub
∩ complete set u = {1,2,3,4,5,6,7,8}, set a = {5,6,7,8}, B = {2,4,6,8}, ∩ a ∩ B = {6,8}, CUA = {1,2,3,4}, Cub = {1,3,5,7}
How does ABC affect the position of the function image? How to find the maximum and minimum of a quadratic function?
The general formula of quadratic function is y = ax & sup2; + BX + C (a ≠ 0), and the image is a parabola; (1) a affects the opening direction of the parabola; (2) the axis of symmetry x = - B / 2a, so B affects the position of its axis of symmetry; (3) the maximum value (4ac-b ^ 2) / 4A; so C affects the maximum (minimum) value. There are three methods to find the maximum value: formula method, matching method and image method
Let u = {1,2,3,4,5}, a = {1,3,5}, B = {2,4,5}, then (CUA) intersection (cub)=
CUA is 2,4, cub is 1,3, the intersection of the two is completely empty In other words, is this question for relieving boredom
The relationship between the quadratic function ABC and x.y is not an image
I mean, give you three points, you can't use the undetermined coefficient method, you can only use the relationship between a and x1.x2.x3.y1.y2.y3 to find out a, B and C. I want to ask, the relationship between A.B.C and three points
According to the three points (Xi, Yi), we can directly write this function formula:
y=y1(x-x2)(x-x3)/(x1-x2)(x1-x3)+y2(x-x1)(x-x3)/(x2-x1)(x2-x3)+y3(x-x1)(x-x2)/(x3-x1)(x3-x2)
Therefore, there are:
a=y1/(x1-x2)(x1-x3)+y2/(x2-x1)(x2-x3)+y3/(x3-x1)(x3-x2)
b=-y1(x2+x3)/(x1-x2)(x1-x3)-y2(x1+x3)/(x2-x1)(x2-x3)-y3(x1+x2)/(x3-x1)(x3-x2)
c=y1x2x3/(x1-x2)(x1-x3)+y2x1x3/(x2-x1)(x2-x3)+y3x1x2/(x3-x1)(x3-x2)
They are all symmetric formulas
Given the complete set u = [123456]. CUA ∩ B = [1,6], a ∩ cub = [2,3], a ∩ B = [4], then a =? B =?
A｛2,3,4｝B｛1,6,4｝
If the intersection of a's complement and B has elements 1 and 6, then B has elements 1 and 6
If the intersection of B and a has elements 2 and 3, then a has elements 2 and 3
If the intersection of a and B contains element 4, then a {2,3,4} B {1,6,4}
If there is element 5, the intersection does not conform
The relation of analytic expression ABC of quadratic function
Internal species of different situations
f(x)=ax^2+bx+c
(1) When a > 0, the image opening of F (x) is upward;
① If B ^ 2 = 4ac (c > 0), the image of F (x) has an intersection with X axis, that is, f (x) = 0 has a unique solution;
② If B ^ 2 ＞ 4ac, the image of F (x) has two intersections with X axis, that is, f (x) = 0 has two different solutions;
③ If B ^ 2 < 4ac (c > 0), the image of F (x) has no intersection with X axis, that is, f (x) = 0 has no solution;
(2) When a < 0, the image opening of F (x) is downward;
① If B ^ 2 = 4ac (C < 0), the image of F (x) has an intersection with X axis, that is, f (x) = 0 has a unique solution;
② If B ^ 2 ＞ 4ac, the image of F (x) has two intersections with X axis, that is, f (x) = 0 has two different solutions;
③ If B ^ 2 < 4ac (in this case C < 0), the image of F (x) has no intersection with X axis, that is, f (x) = 0 has no solution;
When a > 0, the opening of parabola is upward;
When a < 0, the opening of the parabola is downward
|The larger a | is, the smaller the opening of parabola is;
|The smaller the A, the larger the opening of the parabola
|A | the same parabola must coincide by translation (or rotation, axisymmetry)
a. When B is of the same sign, the symmetry axis of the parabola is on the left side of the y-axis;
a. The symmetry axis of the parabola is on the right side of the y-axis
The intersection coordinates of B = 0 parabola and Y axis are (0, c)
When a > 0, the opening of parabola is upward;
When a < 0, the opening of the parabola is downward
|The larger a | is, the smaller the opening of parabola is;
|The smaller the A, the larger the opening of the parabola
|A | the same parabola must coincide by translation (or rotation, axisymmetry)
a. When B is of the same sign, the symmetry axis of the parabola is on the left side of the y-axis;
a. The symmetry axis of the parabola is on the right side of the y-axis
The intersection coordinates of B = 0 parabola and Y axis are (0, c)
Given the complete set u = {1,2,3,4,5,6,7}, a = {2,4,5}, B = {1,3,5,7}, find Au (cub), (CUA) ∩ (cub)
U={1,2,3,4,5,6,7},A={2,4,5},B={1,3,5,7}
CuB={2,4,6},CuA={1,3,6,7}
AU（CuB）={2,4,5,6}
(CuA）∩(CuB）={6}