What are the main contents of quadratic functions? What is the general form of a quadratic function?

Quadratic functionsThe general form is: y=ax +bx+c.

The general form of the quadratic function is usually y=ax +bx+c, also known as the analytic expression of the mega quadratic function. Now that you know the 3 points, you can get 3 equations, 3 equations, and 3 unknowns by substituting them into this analytic formula, and you can find a, b, and c. If 2 of the 3 intersections are the intersection of the quadratic function and the x-axis.

Image relationships. The relationship between a, b, and c values and images.

a>0, parabola.

The opening is upward; a<0, the parabolic limb excavation is downward.

When the parabolic axis of symmetry.

A and b have the same sign when the y-axis is on the left side of the y-axis, and a,b when the parabolic symmetry axis is on the right-hand side of the y-axis.

c>0, the intersection of the parabola and the y-axis is above the x-axis; At c<0, the intersection of the parabola and the y-axis is below the x-axis.

When a=0, this image is a one-time function.

When b=0, the parabolic vertex is on the y-axis.

When c=0, the parabola is on the x-axis.

When the axis of symmetry of the parabolic nucleus is on the left side of the y-axis, a,b have the same sign, and when the parabolic axis of symmetry is on the right side of the y-axis, a,b has different signs.

Three forms of quadratic functions:1. General formula: y=ax +bx+c (a≠0, a, b, c are constants), then y is called a quadratic function of x.

2. Vertex type.

y=a(x-h) +k(a≠0, a, h, k are constants) 3, intersection formula (with x-axis): y=a(x-x1)(x-x2) (a≠0, x1, x2 are constants).

The primary term coefficient b and the quadratic term coefficient.

a. Jointly determine the axis of symmetry.

location. 1. When A and B have the same sign (i.e., AB>0), the axis of symmetry is on the left side of the Y axis.

2. When A and B have different signs (i.e., AB<0), the axis of symmetry is on the right side of the Y axis.

The number of points where the parabola intersects with the x-axis1. When δ=b -4ac>0, there are 2 intersections between the parabola and the x-axis.

2. When δ=b -4ac=0, the parabola has 1 intersection point with the x-axis.

3. When δ=b -4ac<0, there is no intersection between the parabola and the x-axis.

With the method of pending coefficients.

Find the analytic formula of the quadratic function.

1. When the question is given to the condition that the known image passes through three known points or the three pairs of corresponding values of x and y, the analytic formula can be set to the general form

y=ax²+bx+c(a≠0)。

2. The condition is the vertex coordinates of the known image.

or symmetry axis, the analytical formula can be set as vertex: y=a(x-h)+k(a≠0).

The quadratic function is a parabola, and the general formula is y=ax +bx+c, where a determines the direction of the parabola opening, b and a together determine the position of the parabola relative to the y-axis, and c determines the intercept of the parabola on the y-axis.

y ax +bx+c,a,b,c are all arbitrary constants, and a is not equal to 0.

The vertex formula of the quadratic function is.

y a(x+b) +c,a,b,c is any number, and a is not equal to 0.

The general formula for a quadratic function is.

y=ax^2+bx+c(a≠0)。

The general form of a quadratic function is y=ax 2+bx+c. (a≠0)。

Three forms of quadratic functions:

1. The general formula ruler J: y=ax +bx+c (a≠0, a, b, c are constants), then y is called the quadratic function of x.

2. Vertex formula: y=a(x-h) +k(a≠0, a, h, k are constants).

3. Intersection (with x-axis): y=a(x-x1)(x-x2) (a≠0, x1, x2 are constants).

As shown in the figure, the cross-section of a highway tunnel is parabolic, its maximum height is 6 meters, and the bottom width OM is 12 meters. Now the Cartesian coordinate system is established with the O point as the origin and the straight line where the Om is located as the X-axis.

Directly write the coordinates of the point m and the parabolic vertex p;

Find the analytic formula for this parabola;

If you want to build a rectangular "support frame" ad-dc-cb so that points c and d are on the parabola and points a and b are on the ground om, what is the maximum total length of this "support frame"?

Solution] m(12,0),p(6,6).

Let the parabolic analytic formula be: y=a(x-6)2+6

The parabola y=a(x-6)2+6 passes through the point (0,0), 0=a(0-6)2+6, ie.

The parabolic analytic formula is:

Let a(m,0), then b(12-m,0), the total length of the "support frame" ad+dc+cb

The image opening of this quadratic function is downward.

When m = 3 m, there is a maximum value of 15 m.

Basically, the analysis of the amount of laughter is carried out through the analytical formula that is obtained, and the most value is found, and the range is eliminated, similar to the rough surface of the bridge.

Quadratic functions

i.Definitions and Definition Expressions.

In general, there is a relationship between the independent variable x and the dependent variable y:

y=ax 2+bx+c(a,b,c is constant, a≠0, and a determines the opening direction of the function, when a>0, the opening direction is up, when a<0, the opening direction is down, iai can also determine the size of the opening, the larger iai, the smaller the opening, the smaller iai, the larger the opening. ）

then y is called a quadratic function of x.

To the right of a quadratic function expression is usually a quadratic trinomial.

ii.Three expressions for quadratic functions.

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

Vertex formula: y=a(x-h) 2; +k [the vertex of the parabola p(h,k)].

Intersection: y=a(x-x1)(x-x2) [only for parabolas with intersection points a(x1,0) and b(x2,0) with the x-axis].

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

h=-b/2a k=(4ac-b^2;)/4a x1,x2=(-b±√b^2;-4ac)/2a

iii.An image of a quadratic function.

If we make an image of the quadratic function y=x in a planar Cartesian coordinate system, we can see that the image of the quadratic function is a parabola.

iv.The nature of the parabola.

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

x = -b/2a。

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

In particular, when b = 0, the axis of symmetry of the parabola is the y axis (i.e., the straight line x = 0).

2.The parabola has a vertex p with coordinates.

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 term coefficient a determines the direction and magnitude of the opening of the parabola.

When a 0, the parabola opens upwards; When a 0, the parabola opens downwards.

a|The larger it is, the smaller the opening of the parabola.

4.The primary coefficient b and the quadratic coefficient a together determine the position of the axis of symmetry.

When a and b have the same sign (i.e., ab 0), the axis of symmetry is left on the y-axis;

When A and B are different (i.e., AB 0), the axis of symmetry is to the right of the Y axis.

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

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

6.The number of points where the parabola intersects with the x-axis

b 2-4ac 0, the parabola has 2 intersections with the x-axis.

b 2-4ac=0, the parabola has 1 intersection point with the x-axis.

b 2-4ac 0, the parabola has no intersection with the x-axis.

v.Quadratic functions and unary quadratic equations.

In particular, the quadratic function (hereafter referred to as the function) y=ax 2; +bx+c, when y=0, the quadratic function is a one-dimensional quadratic equation about x (hereinafter referred to as the equation), i.e., ax 2; +bx+c=0

In this case, whether the function image intersects with the x-axis or not, that is, whether the equation has a real number root or not.

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

In mathematics, a quadratic function must be quadratic at its highest order, and a quadratic function is a polynomial function of the form y=ax +bx+c(a≠0). The image of a quadratic function is a parabola with the axis of symmetry parallel to the y-axis.

The definition of the quadratic function expression y=ax +bx+c is a quadratic polynomial, since the highest degree of x is 2.

If the value of the quadratic function is equal to zero, a quadratic equation is obtained. The solution of this equation is called the root of the equation or the zero point of the function.

Basic introduction. In general, we call a function of the form y=ax +bx+c (where a, b, c are constants, and a≠0) a quadratic function, where a is called the quadratic coefficient, b is the primary coefficient, and c is the constant term. x is the independent variable and y is the dependent variable. The highest number of independent variables to the right of the equal sign is 2.

Key takeaways: "variables" is different from "unknowns", and it cannot be said that "a quadratic function is a polynomial function with the highest degree of unknowns being quadratic". "Unknown" is just a number (the specific value is unknown, but only one value is taken), and "variable" can be arbitrarily taken within a certain range. The concept of "unknowns" is applied to equations (in functional equations and differential equations, it is an unknown function, but whether it is an unknown number or an unknown function, it generally represents a number or function - special cases are also encountered), but the letters in the function represent variables, and the meaning is different.

The difference between the two can also be seen in the definition of the function. Just like a function is not equal to a function relationship.

The case of the intersection of the two-function image with the x-axis.

When =b -4ac>0, the function image has two intersections with the x-axis.

When =b -4ac=0, the function image has only one intersection point with the x-axis.

When =b -4ac<0, the function image has no intersection with the x-axis.

Combination of numbers and shapes, general, intersecting, vertex, vertex coordinates The intersection of the image and the x-axis, the opening direction.